HERON’S FORMULA | Study

Perimeter

Perimeter is defined as the outside boundary of any closed shape. 

To calculate the perimeter of a given shape we need to add all the sides of the shape.

Example: The perimeter of a rectangle is the sum of its all four sides. The unit of the perimeter is the same as its length.

Perimeter of the Given rectangle = 3 + 7 + 3 + 7 cm

Perimeter of rectangle = 20 cm. (Scroll down to continue …)

Area

Area of any closed figure is the surface enclosed by the perimeter. Unit of Area is the square of the unit of length.

Area of a triangle

The general formula to find the area of a triangle, if the height is given, is

Area of a Right Angled Triangle

To find the area of a right-angled triangle, we use the formula:

 right-angled triangle, we take the two sides having the right angle, one as the base and one as height.

Example: Calculate area of a triangle of the Figure.

Data: base = 3 cm and height = 4 cm

Formula: Area of triangle = 1/2 × 3 × 4= 6 cm ²

Result: Area of a triangle of the Figure is 6 cm ² .

Remark: If you take base as 4 cm and height as 3 cm then also the area of the triangle will remain the same.

Area of Equilateral Triangle

Equilateral Triangle: Equilateral Triangle is defined as a triangle having three equal sides.

To calculate the area of the Equilateral Triangle ABC,

We calculate the height (altitude), AD by making the median of the triangle.

In the given example, the Height (altitude), AD touches Base of the equilateral triangle at the midpoint of BC, Say point, D. 

Here the equilateral triangle ABC has three equal sides, such as:

AB = BC = AC = 10 cm.

Since, midpoint of BC divides the triangle into two right angle triangles.

The height, AD, is calculated using Pythagoras theorem. 

According to Pythagoras theorem, AB² = AD² + BD²

On substituting the values we get, 

(10)² = AD² + (5)²

AD² = (10)² – (5)²

AD² = 100 – 25 = 75

AD = 5√3

Now we can find the area of the triangle using the formula:

Area of triangle = 1/2 × base × height

On substituting the values we get, 

Area of triangle = 1/2 × 10 × 5√3

 25√3 cm²

Area of Isosceles Triangle

In the isosceles triangle also we need to find the height of the triangle then calculate the area of the triangle.

Here,

Area of a Triangle — by Heron’s Formula

The formula of the area of a triangle given by herons is called  Heron’s Formula.

Heron’s Formula:

where a, b and c are the sides of the triangle and s is the semiperimeter 

Generally, this formula is used when the height of the triangle is not possible to find or you can say if the triangle is a scalene triangle.

Here the sides of triangle are

AB = 12 cm

BC = 14 cm

AC = 6 cm

Application of Heron’s Formula in Finding Areas of Quadrilaterals

If we know the sides and one diagonal of the quadrilateral then we can find its area by using the Heron’s formula.

Find the area of the quadrilateral if its sides and the diagonal are given as follows.

Given, the sides of the quadrilateral

AB = 9 cm

BC = 40 cm

DC = 28 cm

AD = 15 cm

Diagonal is AC = 41 cm

Here, ∆ABC is a right angle triangle, so its area will be

Area of Quadrilateral ABCD = Area of ∆ABC + Area of ∆ADC

= 180 cm² + 126 cm²

= 306 cm²