Question

The sides of a triangle are 3 cm, 5 cm and 6 cm. What is its area?

Answer 1

• The area of the triangle = 2√14 cm²

Given :

The three sides of a triangle :

• The first side = 3 cm.
• The second side = 5 cm.
• The third side = 6 cm.

To Find :

• The area of a triangle.

Solution :

By heron's formula,

$\huge{ \star} \: \: \large{\boxed{ \orange{ \sf{ \sqrt{s(s - a)(s - b)(s - c)} }}}}$

Where,

• s = semi-perimeter

We know that,

$\sf{Perimeter \: of \: triangle = a + b + c}$

So,

$\sf{Semi \: perimeter = \dfrac{a + b + c}{2} }$

We have,

• a = 3 cm.
• b = 5 cm.
• c = 6 cm.

$: \implies \rm s = \dfrac{3 + 5 + 6}{2} \: cm \\ \\ : \implies \rm s = \dfrac{14}{2} \: cm\\ \\ : \implies \rm s = 7 \: cm$

Now, substitute the value of semi perimeter (s) amd the given value of a, b and c in the heron's formula.

$: \implies \rm \sqrt{7 \: cm(7 \: cm - 3 \: cm)(7 \: cm - 5 \: cm)(7 \: cm - 6 \: cm)} \\ \\ : \implies \rm \sqrt{7 \: cm(4 \: cm)(2 \: cm)(1 \: cm)} \\ \\ : \implies \rm \sqrt{7(2 \times 2)(2)(1) \: {cm}^{4} } \\ \\ : \implies \rm \sqrt{7 \times 2 \times 2 \times 2 \times 1 \: {cm}^{4} } \\ \\ : \implies \rm 2 \sqrt{7 \times 2 \times 1} \: {cm}^{2} \\ \\ : \implies \rm 2 \sqrt{14} \: {cm}^{2}$

Hence,

The area of the triangle is 2√14 cm².