How do you use Heron's formula to find the area of a triangle with sides of lengths #15 #, #16 #, and #12 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #15   #, #16    #, and #12    #?

Mason Adams · Accepted Answer

#Area=85.45137# square units

Heron's formula for finding area of the triangle is given by #Area=sqrt(s(s-a)(s-b)(s-c))# Where #s# is the semi perimeter and is defined as #s=(a+b+c)/2# and #a, b, c# are the lengths of the three sides of the triangle. Here let #a=15, b=16# and #c=12# #implies s=(15+16+12)/2=43/2=21.5# #implies s=21.5# #implies s-a=21.5-15=6.5, s-b=21.5-16=5.5 and s-c=21.5-12=9.5# #implies s-a=6.5, s-b=5.5 and s-c=9.5# #implies Area=sqrt(21.5*6.5*5.5*9.5)=sqrt7301.9375=85.45137# square units #implies Area=85.45137# square units

Sadie Allen · Answer

undefined To use Heron's formula to find the area of a triangle with side lengths $a$, $b$, and $c$, where $s$ is the semiperimeter, the formula is as follows:

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

First, calculate the semiperimeter ($s$) of the triangle using the formula:

$$
s = \frac{a + b + c}{2}
$$

Substitute the given side lengths into the formula:

$$
s = \frac{15 + 16 + 12}{2} = \frac{43}{2} = 21.5
$$

Next, substitute the values of $s$, $a$, $b$, and $c$ into Heron's formula:

$$
\text{Area} = \sqrt{21.5(21.5 - 15)(21.5 - 16)(21.5 - 12)}
$$

Simplify the expression inside the square root:

$$
\text{Area} = \sqrt{21.5 \times 6.5 \times 5.5 \times 9.5}
$$

$$
\text{Area} = \sqrt{6327.75}
$$

$$
\text{Area} \approx 79.95
$$

Therefore, the area of the triangle with side lengths 15, 16, and 12 is approximately 79.95 square units.