The figures which we can be drawn on a flat surface or that lie on a plane are called Plane Figure.
Example – Circle, Square, Rectangle etc.
Solid figures
The 3D shapes which occupy some space are called Solid Figures.
Example – Cube, Cuboid, Sphere etc. (Scroll down to continue …)
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Volume
Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.
Surface Area
The area of all the faces of the solid shape is its total surface area. The unit of surface area is a square unit.
Lateral or Curved Surface Area
The surface area of the solid shape after leaving the top and bottom face of the figure is called the lateral surface of the shape. The unit of lateral surface area is a square unit.
Surface Area and Volume of a Cube
Cube is a solid shape having 6 equal square faces.
Lateral surface area of a cube
4s2
Total surface area of a cube
6s2
The volume of a cube
s3
Diagonal
√3 s, s = edge of the cube = side length of face of cube
Surface Area and Volume of a Cube
Example
What is the capacity of a cubical vessel having each side of 8 cm?
Solution
Given side = 8 cm So, Volume of the cubical vessel = l3 = (8)3 = 256 cm3.
Surface Area and volume of a Cuboid
Cuboid is a solid shape having 6 rectangular faces at a right angle.
Lateral surface area of a cuboid
2h(l + b)
Total surface area of a cuboid
2(lb + bh + lh)
Volume of a cuboid
lbh
Diagonal
l = length, b = breadth, h = height
Surface Area and volume of a Cuboid
Example
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
Solution
Given, length = 20 cm, breadth = 8 cm, Height = 30 cm
Total surface area of the cereal box = 2(lb + bh + lh)
= 2(20 × 8 + 8 × 30 + 20 × 30)
= 2(160 + 240 + 600)
= 2(1000) = 2000 cm2.
Surface Area and Volume of a Right Circular Cylinder
If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder.
Curved surface area of a Right circular cylinder
2πrh
Total surface area of a Right circular cylinder
2πr2 + 2πrh = 2πr(r + h)
The volume of a Right circular cylinder
πr2h
r = radius, h = height
Surface Area and Volume of a Right Circular Cylinder
Surface Area and Volume of a Hollow Right Circular Cylinder
If a right circular cylinder is hollow from inside then it has different curved surface and volume.
Curved surface area of a Right circular cylinder
2πh (R + r)
Total surface area of a Right circular cylinder
2πh (R + r) + 2π(R2 – r2)
R = outer radius, r = inner radius, h = height
Surface Area and Volume of a Hollow Right Circular Cylinder
Example
Find the Total surface area of a hollow cylinder whose length is 22 cm and the external radius is 7 cm with 1 cm thickness. (π = 22/7)
Solution
Given, h = 22 cm, R = 7 cm, r = 6 cm (thickness of the wall is 1 cm).
Total surface area of a hollow cylinder = 2πh(R + r) + 2π(R2 – r2)
= 2(π) (22) (7+6) + 2(π)(72 – 62)
= 572 π + 26 π = 598 π
= 1878.67 cm2
Surface Area and Volume of a Right Circular Cone
If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.
Curved surface area of a Right Circular Cone
πrl = πr[√(h2 + r2)]
Total surface area of a Right Circular Cone
πr2 + πrl = πr(r + l)
The volume of Right Circular Cone
(1/3) πr2h
r = radius, h = height, l = slant height
Surface Area and Volume of a Right Circular Cone
Surface Area and Volume of a Sphere
A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
Curved or Lateral surface area of a Sphere
4πr2
Total surface area of a Sphere
4πr2
Volume of a Sphere
(4/3) πr3
R = radius
Surface Area and Volume of a Sphere
Surface Area and Volume of a Hemisphere
If we cut the sphere in two parts then is said to be a hemisphere.
Curved or Lateral surface area of a Sphere
2πr2
Total surface area of a Sphere
3πr2
Volume of a Sphere
(2/3) πr3
r = radius
Surface Area and Volume of a Hemisphere
Example
If we have a metal piece of cone shape with volume 523.33 cm3 and we mould it in a sphere then what will be the surface area of that sphere?
A cuboid is a three-dimensional geometric shape that resembles a rectangular box or a rectangular prism. A cuboid has 3 Pairs of opposite, congruent and parallel rectangular faces, 12 edges, and 8 vertices.
Note 1: All squares are rectangles.
Note 2: Cuboid may have one, or Three equal pairs of squares. (Square is a special type of Rectangle.
Note 3: If All three pairs of faces of a cuboid are squares then it it becomes a Cube.
Note 4: A cube is a special case of cuboid.
Parts And Their Alignment Of A Cuboid
Faces
The flat surfaces of a cuboid are known as its faces.
A cuboid has six faces, and each face is a rectangle.
These faces are arranged such that three pairs of opposite faces are parallel to each other.
The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.
Note 1: All squares are rectangles.
Note 2: Rectangle may have one or two pairs of squares.
Note 3: If All three pairs of faces of a rectangle are squares then it it becomes a Cube.
Note 4: A cube is a special case of cuboid.
Edges
An edge is a line segment where the two surfaces of a cuboid meet.
There are 12 edges in a cuboid, where three edges meet at each vertex.
All edges form right angles with the adjacent edges and faces.
Vertices
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cuboid has eight vertices.
Diagonals
Diagonal of a cuboid is a line segment that joins two opposite vertices.
The cuboid has four space diagonals.
Length of the diagonal of cuboid = √(length2 + breadth2 + height2) units.
Symmetry
Cuboids exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
Features of a Cuboid
It is a three-dimensional, Rectangular figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are rectangles.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cuboid are right angles.
Mensuration of Cuboid
Surface Area of a Cuboid
The total surface area of a cuboid is defined as the area of its surface (Appearing face).
The Lateral Surface Area of a Cube.
Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)
Derivation of Total Surface Area of a Cuboid
Since the total surface area of a cuboid (TSA) is the area of its surface.
Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]
Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.
The volume of a cuboid can be calculated using the formula, V = lbh, where,
l = length, b = breadth or width, h = height
This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.
The volume is calculated by multiplying the object’s length, breadth, and height.
Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height
Cuboids in Our Daily Life
Cuboids are commonly used in everyday objects, such as boxes, books, and building blocks.
They are used in architectural and engineering designs for modeling rooms, buildings, and structures.
In mathematics and geometry, cuboids serve as fundamental examples for teaching and understanding concepts related to three-dimensional shapes.
Similar Shapes:
A cube is a special type of cuboid where all sides are equal in length, making it a regular hexahedron.
Real-world Examples:
A shoebox is an example of a cuboid.
Most refrigerators, ovens, and TV screens have cuboidal shapes.
Buildings and houses often have cuboidal rooms.
Fun Fact:
Cuboids are among the simplest and most familiar three-dimensional shapes, making them a fundamental concept in geometry.
Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.
What is a cube?
A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.
Parts And Their Alignment Of In A Cube
Faces
The flat surfaces of a cube are known as its faces.
A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.
The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.
All the edges have the same length.
A cube also has 8 vertices and 12 edges.
Edges
An edge is a line segment where the two surfaces of a cube meet.
There are twelve edges in a cube, where three edges meet at each vertex.
All edges have equal length and form right angles with the adjacent edges and faces.
Vertices
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cube has eight vertices.
Diagonals
The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.
Symmetry
Cubes exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
Features of a Cube
It is a three-dimensional, square-shaped figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are squares with equal area.
All sides have the same length.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cube are right angles.
Mensuration of Cube
Surface Area of a Cube
The total surface area of a cube is defined as the area of its outer surface.
Derivation of Total Surface Area of a Cube
Since the total surface area of a cube is the area of its outer surface.
total surface area of a cube = 6 ✕ area of one face.
We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.
Let the length of each edge is “s”.
Area of one square face = length of edge ✕ length of edge
Area of one square face == s ✕ s = s²
Therefore, the total surface area of the cube = 6s²
The total surface area of the cube will be equal to the sum of all six faces of the cube.
The Lateral Surface Area of a Cube.
Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = 4a²
The Volume of a Cube
Volume
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.
The volume of a cube can be calculated using the formula V = s3, where “s” represents the length of one side of the cube.
This formula shows that the volume of a cube is directly proportional to the cube of its side length.
The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.
Hence the volume of the cube is s ✕ s ✕ s = s³
Cubes in Our Daily Life
We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.
Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.
Also, Cubes have many applications in mathematics, engineering, architecture and art etc.
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Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.
In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.
Let’s start with the basic definition of a right circular cylinder.
A cylinder is a solid shape that has two parallel circular bases of equal size and shape.
The curved surface that connects the two bases is called the lateral surface.
The axis of the cylinder is a line passing through the center of both bases.
Types Of Cylinders
(i). Solid Cylinder
(ii). Hollow Cylinder
Area Of A Solid Cylinder
Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.
Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)
Or
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)
Therefore the formula to calculate the surface area of a cylinder is expressed as the following:
$$SA = 2\pi r h + 2\pi r^2$$
where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.
Here, Curved Surface Area, $$CSA = 2\pi r h$$ and
Cicular Base Area = Top Surface Area = $$\pi r^2$$
Volume Of A Solid Cylinder
The formula to calculate the volume of a cylinder is given by:
$$V = \pi r^2 h$$
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:
$$V = \pi (4)^2 (10) = 160\pi$$
Therefore, the volume of the cylinder is 160π cubic cm.
To calculate its surface area, we can use the formula:
$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$
Therefore, the surface area of the cylinder is 120π square cm.
In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.
Hollow Cylinder
A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.
Volume Of A Hollow Cylinder
The volume of a hollow cylinder can be calculated using the formula V = πh(R2-r2), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Surface Area Of A Hollow Cylinder
The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.
In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.
Cone
A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.
Surface Area of a Cone
The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:
$$A = \pi r (r + l)$$
Where:
( A ) is the surface area of the cone
$$\pi = 3.14159$$
( r ) is the radius of the base of the cone
( l ) is the slant height of the cone
Volume of a Cone
The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
( V ) is the volume of the cone
( \pi ) is a mathematical constant approximately equal to 3.14159
( r ) is the radius of the base of the cone
( h ) is the height of the cone
Example Equations
Here are a few example equations related to cones:
Equation for calculating the slant height of a cone: $$l = \sqrt{r^2 + h^2}$$
Equation for calculating the radius of a cone given its slant height and height: $$r = \sqrt{l^2 – h^2}$$
Equation for calculating the height of a cone given its volume and radius: $$h = \frac{3V}{\pi r^2}$$
Sphere
Solid Sphere
Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.
Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.
Volume of Solid Sphere The formula to calculate the volume (\(V_s\)) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: \(V_s\) = Volume of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere.
Surface Area of Solid Sphere The formula to calculate the surface area (\(A_s\)) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]
Where: \(A_s\) = Surface area of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere
Mass of Solid Sphere The mass (\(m_s\)) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: \(m_s\) = Mass of the solid sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_s\) = Volume of the solid sphere (calculated using the previous formula)
Hollow Sphere
Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.
Characteristics – The hollow sphere has a well-defined outer radius (\(R\)), inner radius (\(r\)), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.
Volume of Hollow Sphere The formula to calculate the volume (\(V_h\)) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – \(V_h\) = Volume of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Surface Area of Hollow Sphere The formula to calculate the surface area (\(A_h\)) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: \(A_h\) = Surface area of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Mass of Hollow Sphere The mass (\(m_h\)) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: \(m_h\) = Mass of the hollow sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_h\) = Volume of the hollow sphere (calculated using the previous formula)
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A cuboid is a three-dimensional geometric shape that resembles a rectangular box or a rectangular prism. A cuboid has 3 Pairs of opposite, congruent and parallel rectangular faces, 12 edges, and 8 vertices.
Note 1: All squares are rectangles.
Note 2: Cuboid may have one, or Three equal pairs of squares. (Square is a special type of Rectangle.
Note 3: If All three pairs of faces of a cuboid are squares then it it becomes a Cube.
Note 4: A cube is a special case of cuboid.
Parts And Their Alignment Of A Cuboid
Faces
The flat surfaces of a cuboid are known as its faces.
A cuboid has six faces, and each face is a rectangle.
These faces are arranged such that three pairs of opposite faces are parallel to each other.
The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.
Note 1: All squares are rectangles.
Note 2: Rectangle may have one or two pairs of squares.
Note 3: If All three pairs of faces of a rectangle are squares then it it becomes a Cube.
Note 4: A cube is a special case of cuboid.
Edges
An edge is a line segment where the two surfaces of a cuboid meet.
There are 12 edges in a cuboid, where three edges meet at each vertex.
All edges form right angles with the adjacent edges and faces.
Vertices
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cuboid has eight vertices.
Diagonals
Diagonal of a cuboid is a line segment that joins two opposite vertices.
The cuboid has four space diagonals.
Length of the diagonal of cuboid = √(length2 + breadth2 + height2) units.
Symmetry
Cuboids exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
Features of a Cuboid
It is a three-dimensional, Rectangular figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are rectangles.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cuboid are right angles.
Mensuration of Cuboid
Surface Area of a Cuboid
The total surface area of a cuboid is defined as the area of its surface (Appearing face).
The Lateral Surface Area of a Cube.
Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)
Derivation of Total Surface Area of a Cuboid
Since the total surface area of a cuboid (TSA) is the area of its surface.
Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]
Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.
The volume of a cuboid can be calculated using the formula, V = lbh, where,
l = length, b = breadth or width, h = height
This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.
The volume is calculated by multiplying the object’s length, breadth, and height.
Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height
Cuboids in Our Daily Life
Cuboids are commonly used in everyday objects, such as boxes, books, and building blocks.
They are used in architectural and engineering designs for modeling rooms, buildings, and structures.
In mathematics and geometry, cuboids serve as fundamental examples for teaching and understanding concepts related to three-dimensional shapes.
Similar Shapes:
A cube is a special type of cuboid where all sides are equal in length, making it a regular hexahedron.
Real-world Examples:
A shoebox is an example of a cuboid.
Most refrigerators, ovens, and TV screens have cuboidal shapes.
Buildings and houses often have cuboidal rooms.
Fun Fact:
Cuboids are among the simplest and most familiar three-dimensional shapes, making them a fundamental concept in geometry.
Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.
What is a cube?
A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.
Parts And Their Alignment Of In A Cube
Faces
The flat surfaces of a cube are known as its faces.
A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.
The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.
All the edges have the same length.
A cube also has 8 vertices and 12 edges.
Edges
An edge is a line segment where the two surfaces of a cube meet.
There are twelve edges in a cube, where three edges meet at each vertex.
All edges have equal length and form right angles with the adjacent edges and faces.
Vertices
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cube has eight vertices.
Diagonals
The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.
Symmetry
Cubes exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
Features of a Cube
It is a three-dimensional, square-shaped figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are squares with equal area.
All sides have the same length.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cube are right angles.
Mensuration of Cube
Surface Area of a Cube
The total surface area of a cube is defined as the area of its outer surface.
Derivation of Total Surface Area of a Cube
Since the total surface area of a cube is the area of its outer surface.
total surface area of a cube = 6 ✕ area of one face.
We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.
Let the length of each edge is “s”.
Area of one square face = length of edge ✕ length of edge
Area of one square face == s ✕ s = s²
Therefore, the total surface area of the cube = 6s²
The total surface area of the cube will be equal to the sum of all six faces of the cube.
The Lateral Surface Area of a Cube.
Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = 4a²
The Volume of a Cube
Volume
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.
The volume of a cube can be calculated using the formula V = s3, where “s” represents the length of one side of the cube.
This formula shows that the volume of a cube is directly proportional to the cube of its side length.
The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.
Hence the volume of the cube is s ✕ s ✕ s = s³
Cubes in Our Daily Life
We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.
Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.
Also, Cubes have many applications in mathematics, engineering, architecture and art etc.
Study Tools
Audio, Visual & Digital Content
Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.
In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.
Let’s start with the basic definition of a right circular cylinder.
A cylinder is a solid shape that has two parallel circular bases of equal size and shape.
The curved surface that connects the two bases is called the lateral surface.
The axis of the cylinder is a line passing through the center of both bases.
Types Of Cylinders
(i). Solid Cylinder
(ii). Hollow Cylinder
Area Of A Solid Cylinder
Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.
Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)
Or
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)
Therefore the formula to calculate the surface area of a cylinder is expressed as the following:
$$SA = 2\pi r h + 2\pi r^2$$
where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.
Here, Curved Surface Area, $$CSA = 2\pi r h$$ and
Cicular Base Area = Top Surface Area = $$\pi r^2$$
Volume Of A Solid Cylinder
The formula to calculate the volume of a cylinder is given by:
$$V = \pi r^2 h$$
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:
$$V = \pi (4)^2 (10) = 160\pi$$
Therefore, the volume of the cylinder is 160π cubic cm.
To calculate its surface area, we can use the formula:
$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$
Therefore, the surface area of the cylinder is 120π square cm.
In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.
Hollow Cylinder
A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.
Volume Of A Hollow Cylinder
The volume of a hollow cylinder can be calculated using the formula V = πh(R2-r2), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Surface Area Of A Hollow Cylinder
The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.
In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.
Cone
A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.
Surface Area of a Cone
The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:
$$A = \pi r (r + l)$$
Where:
( A ) is the surface area of the cone
$$\pi = 3.14159$$
( r ) is the radius of the base of the cone
( l ) is the slant height of the cone
Volume of a Cone
The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
( V ) is the volume of the cone
( \pi ) is a mathematical constant approximately equal to 3.14159
( r ) is the radius of the base of the cone
( h ) is the height of the cone
Example Equations
Here are a few example equations related to cones:
Equation for calculating the slant height of a cone: $$l = \sqrt{r^2 + h^2}$$
Equation for calculating the radius of a cone given its slant height and height: $$r = \sqrt{l^2 – h^2}$$
Equation for calculating the height of a cone given its volume and radius: $$h = \frac{3V}{\pi r^2}$$
Sphere
Solid Sphere
Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.
Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.
Volume of Solid Sphere The formula to calculate the volume (\(V_s\)) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: \(V_s\) = Volume of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere.
Surface Area of Solid Sphere The formula to calculate the surface area (\(A_s\)) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]
Where: \(A_s\) = Surface area of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere
Mass of Solid Sphere The mass (\(m_s\)) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: \(m_s\) = Mass of the solid sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_s\) = Volume of the solid sphere (calculated using the previous formula)
Hollow Sphere
Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.
Characteristics – The hollow sphere has a well-defined outer radius (\(R\)), inner radius (\(r\)), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.
Volume of Hollow Sphere The formula to calculate the volume (\(V_h\)) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – \(V_h\) = Volume of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Surface Area of Hollow Sphere The formula to calculate the surface area (\(A_h\)) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: \(A_h\) = Surface area of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Mass of Hollow Sphere The mass (\(m_h\)) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: \(m_h\) = Mass of the hollow sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_h\) = Volume of the hollow sphere (calculated using the previous formula)
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The figures which we can be drawn on a flat surface or that lie on a plane are called Plane Figure.
Example – Circle, Square, Rectangle etc.
Solid figures
The 3D shapes which occupy some space are called Solid Figures.
Example – Cube, Cuboid, Sphere etc. (Scroll down to continue …)
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Volume
Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.
Surface Area
The area of all the faces of the solid shape is its total surface area. The unit of surface area is a square unit.
Lateral or Curved Surface Area
The surface area of the solid shape after leaving the top and bottom face of the figure is called the lateral surface of the shape. The unit of lateral surface area is a square unit.
Surface Area and Volume of a Cube
Cube is a solid shape having 6 equal square faces.
Lateral surface area of a cube
4s2
Total surface area of a cube
6s2
The volume of a cube
s3
Diagonal
√3 s, s = edge of the cube = side length of face of cube
Surface Area and Volume of a Cube
Example
What is the capacity of a cubical vessel having each side of 8 cm?
Solution
Given side = 8 cm So, Volume of the cubical vessel = l3 = (8)3 = 256 cm3.
Surface Area and volume of a Cuboid
Cuboid is a solid shape having 6 rectangular faces at a right angle.
Lateral surface area of a cuboid
2h(l + b)
Total surface area of a cuboid
2(lb + bh + lh)
Volume of a cuboid
lbh
Diagonal
l = length, b = breadth, h = height
Surface Area and volume of a Cuboid
Example
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
Solution
Given, length = 20 cm, breadth = 8 cm, Height = 30 cm
Total surface area of the cereal box = 2(lb + bh + lh)
= 2(20 × 8 + 8 × 30 + 20 × 30)
= 2(160 + 240 + 600)
= 2(1000) = 2000 cm2.
Surface Area and Volume of a Right Circular Cylinder
If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder.
Curved surface area of a Right circular cylinder
2πrh
Total surface area of a Right circular cylinder
2πr2 + 2πrh = 2πr(r + h)
The volume of a Right circular cylinder
πr2h
r = radius, h = height
Surface Area and Volume of a Right Circular Cylinder
Surface Area and Volume of a Hollow Right Circular Cylinder
If a right circular cylinder is hollow from inside then it has different curved surface and volume.
Curved surface area of a Right circular cylinder
2πh (R + r)
Total surface area of a Right circular cylinder
2πh (R + r) + 2π(R2 – r2)
R = outer radius, r = inner radius, h = height
Surface Area and Volume of a Hollow Right Circular Cylinder
Example
Find the Total surface area of a hollow cylinder whose length is 22 cm and the external radius is 7 cm with 1 cm thickness. (π = 22/7)
Solution
Given, h = 22 cm, R = 7 cm, r = 6 cm (thickness of the wall is 1 cm).
Total surface area of a hollow cylinder = 2πh(R + r) + 2π(R2 – r2)
= 2(π) (22) (7+6) + 2(π)(72 – 62)
= 572 π + 26 π = 598 π
= 1878.67 cm2
Surface Area and Volume of a Right Circular Cone
If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.
Curved surface area of a Right Circular Cone
πrl = πr[√(h2 + r2)]
Total surface area of a Right Circular Cone
πr2 + πrl = πr(r + l)
The volume of Right Circular Cone
(1/3) πr2h
r = radius, h = height, l = slant height
Surface Area and Volume of a Right Circular Cone
Surface Area and Volume of a Sphere
A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
Curved or Lateral surface area of a Sphere
4πr2
Total surface area of a Sphere
4πr2
Volume of a Sphere
(4/3) πr3
R = radius
Surface Area and Volume of a Sphere
Surface Area and Volume of a Hemisphere
If we cut the sphere in two parts then is said to be a hemisphere.
Curved or Lateral surface area of a Sphere
2πr2
Total surface area of a Sphere
3πr2
Volume of a Sphere
(2/3) πr3
r = radius
Surface Area and Volume of a Hemisphere
Example
If we have a metal piece of cone shape with volume 523.33 cm3 and we mould it in a sphere then what will be the surface area of that sphere?
