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CBSE Class 9 Mathematics Question 4 of 6

Heron's Formula — Question 4

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Question

Question 4

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.

Answer

Let a, b and c be the sides of the triangle.

Let a = 18 cm, b = 10 cm.

Given,

Perimeter = 42 cm

∴ a + b + c = 42

⇒ 18 + 10 + c = 42

⇒ 28 + c = 42

⇒ c = 42 - 28 = 14 cm.

By formula,

Semi Perimeter (s) = Perimeter of triangle2=422\dfrac{\text{Perimeter of triangle}}{2} = \dfrac{42}{2} = 21 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=21(21−18)(21−10)(21−14)=21×3×11×7=4851=2111 cm2A = \sqrt{21(21 - 18)(21 - 10)(21 - 14)} \\[1em] = \sqrt{21 \times 3 \times 11 \times 7} \\[1em] = \sqrt{4851} \\[1em] = 21\sqrt{11} \text{ cm}^2

Hence, area of triangle = 211121\sqrt{11} 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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