How do you use Heron's formula to find the area of a triangle with sides of lengths #23 #, #21 #, and #7 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #23  #, #21    #, and #7   #?

Claire Akin · Accepted Answer

Area of triangle #color(green)(A_t = 72.85# sq units

Given : #a = 23, b = 21, c = 7# Heron’s formula for area of triangle #A_t = sqrt(s (s-a) (s-b) (s-c))# where Semi perimeter #s = (a + b + c) / 2# #s = (23 + 21 + 7)/2 = 25.5# #A_t = sqrt(25.5 * (25.5-23) * (25.5-21) * (25.5-7))# Area of triangle #color(green)(A_t = 72.85# sq units

Caroline Aucoin · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths $ a $, $ b $, and $ c $, where $ s $ is the semi-perimeter, the formula is:

$$ \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} $$

First, calculate the semi-perimeter $ s $ using the formula:

$$ s = \frac{a + b + c}{2} $$

Then, substitute the values of $ a $, $ b $, and $ c $ into the formula to find the area.

For the given triangle with side lengths $ 23 $, $ 21 $, and $ 7 $:

$$ s = \frac{23 + 21 + 7}{2} = \frac{51}{2} = 25.5 $$

$$ \text{Area} = \sqrt{25.5(25.5 - 23)(25.5 - 21)(25.5 - 7)} $$

$$ \text{Area} = \sqrt{25.5(2.5)(4.5)(18.5)} $$

$$ \text{Area} = \sqrt{25.5 \times 2.5 \times 4.5 \times 18.5} $$

$$ \text{Area} \approx \sqrt{5206.25} $$

$$ \text{Area} \approx 72.12 \text{ square units} $$