How do you use Heron's formula to find the area of a triangle with sides of lengths #14 #, #9 #, and #13 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #14  #, #9    #, and #13   #?

Mason Adams · Accepted Answer

#color(blue)("Area"=18sqrt(10)" square units")#

The Heron's Formula is as follows:

#"Area=sqrt(s(s-a)(s-b)(s-c))#

Where #a, b and c# are the lengths of the triangles sides.

#s="semiperimeter"=(a+b+c)/2#

Let: #a=14#, #b=13#. #c=9#

Then:

#s=(14+13+9)/2=#18

#"Area"=sqrt(18(18-14)(18-13)(18-9))#

# \ \ \ \ \ \ \ \ \ =sqrt(18(4)(5)(9)#

# \ \ \ \ \ \ \ \ \ =sqrt(3240)=18sqrt(10) # square units

Carson Arnold · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths 14, 9, and 13, follow these steps:

1. Calculate the semi-perimeter of the triangle. The semi-perimeter (s) is given by the formula: $ s = \frac{a + b + c}{2} $, where $ a $, $ b $, and $ c $ are the lengths of the triangle's sides.

2. Once you have the semi-perimeter, use Heron's formula to find the area (A) of the triangle. Heron's formula states that the area of a triangle with side lengths $ a $, $ b $, and $ c $ is given by: 
   $$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

3. Substitute the values of the sides and the semi-perimeter into Heron's formula, and then calculate the area.

Using the given side lengths:
- $ a = 14 $
- $ b = 9 $
- $ c = 13 $

Calculate the semi-perimeter:
$$ s = \frac{14 + 9 + 13}{2} = 18 $$

Now, substitute into Heron's formula:
$$ A = \sqrt{18(18 - 14)(18 - 9)(18 - 13)} $$

$$ A = \sqrt{18 \times 4 \times 9 \times 5} $$

$$ A = \sqrt{3240} $$

$$ A \approx 56.78 $$

Therefore, the area of the triangle with side lengths 14, 9, and 13 is approximately $ 56.78 $ square units.