How do you use Heron's formula to find the area of a triangle with sides of lengths #12 #, #15 #, and #18 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #12   #, #15    #, and #18    #?

Sadie Allen · Accepted Answer

#"Area"_triangle= 135/4sqrt(7)~~89.3# sq.units

Heron's formula says that a triangle with sides of length, #a, b, c# has an area: #color(white)("XXX")"Area"_triangle=sqrt(s(s-a)(s-b)(s-c))# where #s# is the semi-perimeter #=(a+b+c)/2# Given sides #12, 15, 18# #color(white)("XXX")s=45/2# and #color(white)("XXX")"Area"_triangle=sqrt(45/2 * (45/2-12) * (45/2-15) * (45/2-18)# #color(white)("XXXXXXX")=sqrt(45/2) * (21/2) * (15/2) * (9/2))# #color(white)("XXXXXXX")=sqrt(((3 * 3 * 5) * (3 * 7) * (3 * 5) * * (3 * 3))/(2 * 2 * 2 * 2))# #color(white)("XXXXXXX")=sqrt(((3^3)^2 * 5^2 * 7)/((2^2)^2))# #color(white)("XXXXXXX")=(3^3 * 5)/2 sqrt(7)# #color(white)("XXXXXXX")=135/4 sqrt(7)# #color(white)("XXXXXXX")~~89.3#

Wyatt Anderson · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths 12, 15, and 18:

1. Calculate the semi-perimeter, $ s $, of the triangle:
   $ s = \frac{a + b + c}{2} = \frac{12 + 15 + 18}{2} = 22.5 $.

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

3. Substitute the values into the formula:
   $ A = \sqrt{22.5(22.5 - 12)(22.5 - 15)(22.5 - 18)} $.

4. Perform the calculations:
   $ A = \sqrt{22.5 \times 10.5 \times 7.5 \times 4.5} $.

5. Simplify:
   $ A = \sqrt{33075} $.

6. Calculate the square root:
   $ A \approx 181.019 $.

So, the area of the triangle is approximately $ 181.019 $ square units.