How do you use Heron's formula to find the area of a triangle with sides of lengths #19 #, #14 #, and #13 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #19   #, #14    #, and #13    #?

Claire Akin · Accepted Answer

The area of the triangle would be #55.3 units^2#

First we would find S which is the sum of the 3 sides divided by 2.

#S = (19 + 14 + 13)/2 # = #46/2# = #23#

Then use Heron's Equation to calculate the area.

#Area = sqrt(S(S-A)(S-B)(S-C)) #

#Area = sqrt(23(23-19)(23-14)(23-13)) #

#Area = sqrt(8.5(4)(9)(10)) #

#Area = sqrt(3060) #

#Area = 55.3 units^2 #

James Atkins · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths 19, 14, and 13, follow these steps:

1. Calculate the semi-perimeter of the triangle, denoted by $ s $, using the formula:
$$ s = \frac{a + b + c}{2} $$
where $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle.

2. Once you have $ s $, use Heron's formula to find the area ($ A $) of the triangle:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

3. Substitute the values of $ a $, $ b $, and $ c $ into the formula and calculate the area.

For the given triangle with side lengths 19, 14, and 13:
$$ s = \frac{19 + 14 + 13}{2} = 23 $$

$$ A = \sqrt{23(23 - 19)(23 - 14)(23 - 13)} $$

$$ A = \sqrt{23(4)(9)(10)} $$

$$ A = \sqrt{8280} $$

$$ A \approx 90.95 $$

Therefore, the area of the triangle is approximately 90.95 square units.