How do you use Heron's formula to find the area of a triangle with sides of lengths #14 #, #8 #, and #11 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #14  #, #8    #, and #11   #?

Savannah Aguilar · Accepted Answer

Area of triangle is #87.83# According to Heron's formula, area of a triangle whose three sides are #a#, #b# and #c# is given by #sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#. As in given case three sides are #14#, #8# and #11#, #s=1/2(14+8+11)=33/2# and hence area of the triangle is #sqrt(33/2xx(33/2-14)xx(33/2-8)xx(33/2-11))# = #sqrt(33/2xx5/2xx17/2xx11/2)# = #1/2sqrt(11xx3xx5xx17xx11)=11/2sqrt(15xx17)# = #11/2sqrt255=11/2xx15.969=87.83#

James Atkins · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths 14, 8, and 11, follow these steps:

1. Calculate the semi-perimeter (s) of the triangle 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 the semi-perimeter, use Heron's formula to find the area (A) of the triangle:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

Substitute the values of $s$, $a$, $b$, and $c$ into the formula and solve for $A$.

Given the side lengths $a = 14$, $b = 8$, and $c = 11$, plug these values into the formula for the semi-perimeter:
$$ s = \frac{14 + 8 + 11}{2} = \frac{33}{2} = 16.5 $$

Now, substitute $s$ and the side lengths into Heron's formula:
$$ A = \sqrt{16.5(16.5 - 14)(16.5 - 8)(16.5 - 11)} $$

Calculate the expression inside the square root:
$$ A = \sqrt{16.5 \times 2.5 \times 8.5 \times 5.5} $$

$$ A = \sqrt{16.5 \times 2.5 \times 8.5 \times 5.5} $$

$$ A \approx \sqrt{16.5 \times 2.5 \times 8.5 \times 5.5} $$

$$ A \approx \sqrt{2497.8125} $$

$$ A \approx 49.978 $$

Therefore, the area of the triangle with side lengths 14, 8, and 11 is approximately $49.978$ square units.