Question

How do you use Heron’s formula to find the area of triangle with sides of lengths $ 6,6 $ and $ 7? $

Answer 1

Hint: As we know that we have to find the area of the triangle using the Heron’s formula. The heron’s formula is $ \sqrt {s(s - a)(s - b)(s - c)} $ , where $ a,b $ and $ c $ are the lengths of the triangle. The term ” $ s $ ” here means the semi-perimeter. The value of s or the semi-perimeter is $ s = \dfrac{{a + b + c}}{2} $ . We will substitute the values according to the question and then we will find the area.

Complete step by step solution:
Here in this question we need to find the area of the triangle using the heron’s formula of which sides are $ 6,6 $ and $ 7 $ .
We have $ a = 6,b = 6 $ and $ c = 7 $ . First we will find the value of the semi-perimeter i.e.
 $ s = \dfrac{{6 + 6 + 7}}{2} $ .
It gives the value of $ s = \dfrac{{19}}{2} $ i.e. $ 9.5 $ .
Now the area of the triangle is $ \sqrt {s(s - a)(s - b)(s - c)} $ , by putting all the values in the formula we have:
$\Rightarrow \sqrt {9.5(9.5 - 6)(9.5 - 6)(9.5 - 7)} $ .
On further solving,
$\Rightarrow \sqrt {9.5 \times 3.5 \times 3.5 \times 2.5} = \sqrt {290.9375} $ .
Hence the required area of the triangle is $ 17.06 $ square units.
So, the correct answer is “ $ 17.06 $ square units”.

Note: We should know that Heron’s formula is valid If only all the three sides are given. Also if the question does not mention anything then we should not use Heron's formula in case of right angled triangles. We can directly use the formula i.e. $ Area = \dfrac{1}{2} \times b \times h $ , where $ b $ is the base of the triangle and $ h $ is the height.