 # Heron’s formula

## Application of Heron’s Formula

When you look around, you find various materials with different shapes. Some of the most common shapes are circle, quadrilaterals, triangle, sphere, etc. In your previous academic sessions, you have already learnt about these shapes - how to calculate their perimeter, area, etc.

Here, the discussion will be on various properties of closed figures and shapes so that you can easily understand the basic concept of Heron’s formula. This popular geometrical formula can help you to calculate the area of both triangle and quadrilaterals. However, there are some other ways as well to find out the area of these two shapes that you learn in this content.

Let us start learning CBSE 9th maths Heron’s formula.

Heron’s Theorem

Heron’s theorem or formula was named after Heron of Alexandria who found the area of triangles can be measured in terms of the length of their sides. By deriving Heron’s formula, you can calculate the area of a triangle without measuring the angles or any other distances. This formula is known for its simple calculation based on the length of three sides of a triangle.

Let say, if the length of three sides are A, B, and C, then its semi-perimeter is

S = (A + B + C) / 2

Thus, the area would be

A = √{S (S - A)(S - B)(S - C)}

Heron’s Formula Triangle Measurements

Now, as you have learnt all the terms related to Heron’s formula of triangles, you can derive all of their values as per the given information. For your convenience, you can find an example below explaining the derivation and application of this formula.

Example

1. There is a triangular park with 40 m, 24 m, and 32 m long walls surrounded by three sides respectively. What is the area of the park?

Ans. From the given information, the perimeter of the park will be (40 + 24 + 32) m = 96 m

Hence, the semi-perimeter of the park will be 96 / 2 m = 48 m

Now, by deriving Heron’s formula,

Area of this triangular park will be

√{S (S - A)(S - B)(S - C)}

= √{48 (48 – 40)(48 - 24)(48 - 32)} m2

= √(48 X 8 X 24 X 16) m2

= 384 m2

This is how you can derive Heron’s formula 9th class NCERT syllabus and find out the answer to your question. Once you understand how to use Heron’s formula to find the area of the triangle, you can also calculate the value of all the other terms by creating the equation as per data provided.

However, not only for triangle measurements but you can also use Heron’s formula for quadrilateral measurements. From your past classes, you have already learnt to divide a quadrilateral into two triangles. Thus, by calculating the area of each triangle with the Heron mathematician formula and then summing them up, you can find out the area of a quadrilateral.

Check out the diagram for reference.

From the above diagram, you can see that quadrilateral ABCD is divided into two triangles that are ABD and BCD. Now, you can use Heron’s formula NCERT syllabus-wise, and calculate the area of these two triangles. And finally, by adding these two areas, you can find out the area of quadrilateral ABCD.

Example

1. What will be the area of quadrilateral ABCD, if its AB = 4 cm, BC = 3 cm, CD = 3 cm, DA = 4 cm, and AC = 5 cm?

Solution

Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ADC

Firstly, area of triangle ABC

From Heron’s formula,

The semi-perimeter is s = (AB + BC + AC) / 2 cm

=  (4 + 3 + 5) / 2 = 12 /2 = 6 cm

Area = √ {6 X (6-4)(6 -3)(6 - 5)} cm2

= √ (6 X 2 X 3 X1)

= 6 cm2

From Heron’s formula,

The semi-perimeter is s = (CD + DA + AC) / 2

(3 + 4 + 5) / 2 cm = 6 cm

So, the area is

√{6 (6 - 3)(6 - 4)(6 - 5)} cm2

= √(6 X 3 X 2 X 1) cm2

= 6 cm2

Hence, the area of quadrilateral ABCD is (6 + 6) cm2 = 12 cm2.

From the above-mentioned explanations and examples, you must have understood NCERT class 9 Heron’s formula and its derivatives. If you need more study materials on this topic, you can visit our Vedantu website anytime. Furthermore, with our new online classes, you can clear any doubts from our experts and be confident about your answers.

1. What do you understand from Heron’s formula?

Ans. Earlier, the area of a triangle was measured with the help of its base and length. The formula was – Area = ½ X Base X Length. However, Heron of Alexandria found a simple way to calculate the area of triangles if the lengths of all sides are given. In this formula, first, you have to calculate the semi-perimeter of the triangle. Here the semi-perimeter is referred to as “s” and a, b, and c are the length of three sides. s = (a + b + c) /2. Then, area will be √{s (s - a)(s - b)(s - c)}.

2. Why do you use this Heron’s formula?

Ans. Several formulas are there to calculate the area of triangles. For example, you can use the length of the base and the height of a triangle to measure its area. Or, by calculating its angles also, you can find out the area. However, these formulas are not easy to apply. In Heron’s formula, you only need to measure the three sides and then by simple addition, subtraction and multiplication; you can find out the area of a triangle. Thus, this is a preferable formula for easy and quick mathematics.

3. How do you use Heron’s formula?

Ans. To use Heron’s formula, you need to know the three sides of a triangle. Let us explain the use of this formula with the help of an example. What will be the cost to paint a triangle board with 15 m, 10 m, 17m lengthy sides respectively, if the price for painting the surface at Rs.5 per m2? Now, by using Heron’s formula, first you need to calculate the semi-perimeter s = ( 15 + 10 + 17) /2 = 21. The area is √{21 X (21 – 15)(21 - 10)(21 - 17)} = √(21 x 6 X 10 X 4) = 12 √35 m2. So, the total cost will be 5 X 12 √35 = Rs.360.

4. What is the meaning of “s” in Heron’s formula?

Ans. In Heron’s formula, you have come across the term “s”. This means semi-perimeter. Perimeter is the summation of the three sides of a triangle, whereas semi-perimeter is the half of the perimeter. So, if the three sides of a triangle are a, b, and c; then the perimeter will be (a + b + c) and the semi-perimeter will be (a + b + c) / 2. In Heron’s formula, the measurement of semi-perimeter is necessary to find out the area.