How do you use Heron's formula to find the area of a triangle with sides of lengths #7 #, #5 #, and #8 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #7   #, #5    #, and #8    #?

Autumn Armstrong · Accepted Answer

#10sqrt3# #sq.cm^2# Use Heron's formula:- #sqrt[s(s-a)(s-b)(s-c)]# Where #s#=semi-perimeter of triangle=#(a+b+c)/2# So,#a=7,b=5,c=8,s=10=(7+5+8)/2=20/2# #rarrsqrt[s(s-a)(s-b)(s-c)]# #rarrsqrt[10(10-7)(10-5)(10-8)]# #rarrsqrt[10(3)(5)(2)]# #rarrsqrt[10(30)]# #rarrsqrt300=10sqrt3#

Mason Adams · Answer

undefined To use Heron's formula to find the area of a triangle with side lengths 7, 5, and 8, you first need to calculate the semi-perimeter of the triangle. The semi-perimeter, denoted as $ s $, is calculated as the sum of all the side lengths divided by 2.

In this case, the side lengths are 7, 5, and 8. So, the semi-perimeter is:

$$ s = \frac{7 + 5 + 8}{2} = \frac{20}{2} = 10 $$

Next, you can use Heron's formula to find the area of the triangle, which is given by:

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

Where:
- $ s $ is the semi-perimeter of the triangle.
- $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle.

Substituting the values:

$$ A = \sqrt{10(10 - 7)(10 - 5)(10 - 8)} $$

$$ A = \sqrt{10(3)(5)(2)} $$

$$ A = \sqrt{300} $$

$$ A \approx 17.32 $$

So, the area of the triangle with side lengths 7, 5, and 8 is approximately 17.32 square units.