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CBSE Class 9 Mathematics Question 6 of 6

Heron's Formula — Question 6

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Question 6

An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Answer

Length of equal sides (a and b) = 12 cm

Let third side of triangle (c) = x cm

Given,

Perimeter of triangle = 30

∴ 12 + 12 + x = 30

⇒ 24 + x = 30

⇒ x = 30 - 24

⇒ x = 6 cm.

∴ c = 6 cm.

By formula,

Semi perimeter (s) = Perimeter of triangle2=302\dfrac{\text{Perimeter of triangle}}{2} = \dfrac{30}{2} = 15 cm.

By Heron's formula,

Area of triangle (A) = s(s−a)(s−b)(s−c)\sqrt{s(s - a)(s - b)(s - c)} sq.units

Substituting values we get :

A=15(15−12)(15−12)(15−6)=15×3×3×9=1215=915 cm2.A = \sqrt{15(15 - 12)(15 - 12)(15 - 6)} \\[1em] = \sqrt{15 \times 3 \times 3 \times 9} \\[1em] = \sqrt{1215} \\[1em] = 9\sqrt{15} \text{ cm}^2.

Hence, area of triangle = 9159\sqrt{15} cm2.

Heron's Formula - Interactive Study Notes | Bright Tutorials
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Mathematics | Heron's FormulaWeb Content • Interactive Notes

Heron’s Formula — Interactive Study Guide

The Formula

s = (a+b+c)/2
Area = √[s(s−a)(s−b)(s−c)]

Works for ANY triangle. No need to know the height!

Step-by-Step Calculation

Sides: 5, 12, 13

s = (5+12+13)/2 = 15

s−5 = 10, s−12 = 3, s−13 = 2

Area = √(15×10×3×2) = √900 = 30 sq units

Verify: This is a right triangle (5²+12²=13²), so area = ½×5×12 = 30. Matches!

Quick Self-Check

  1. Find area of equilateral Δ with side 6. (s=9, A=√(9×3×3×3)=9√3)
  2. Find area of triangle with sides 3, 4, 5. (6 sq units)

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