How do you use Heron's formula to find the area of a triangle with sides of lengths #9 #, #4 #, and #8 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #9   #, #4    #, and #8    #?

Sadie Allen · Accepted Answer

#Area=15.998# 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=9, b=4# and #c=8# #implies s=(9+4+8)/2=21/2=10.5# #implies s=10.5# #implies s-a=10.5-9=1.5, s-b=10.5-4=6.5 and s-c=10.5-8=2.5# #implies s-a=1.5, s-b=6.5 and s-c=2.5# #implies Area=sqrt(10.5*1.5*6.5*2.5)=sqrt255.9375=15.998# square units #implies Area=15.998# square units

Parker Allen · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths 9, 4, and 8, you first need to calculate the semi-perimeter of the triangle. The semi-perimeter, denoted as $ s $, is found by adding the lengths of the three sides and dividing by 2.

$$
s = \frac{{9 + 4 + 8}}{2} = \frac{21}{2} = 10.5
$$

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

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

Substituting the given side lengths and the calculated semi-perimeter into the formula:

$$
A = \sqrt{10.5(10.5 - 9)(10.5 - 4)(10.5 - 8)}
$$

$$
A = \sqrt{10.5 \times 1.5 \times 6.5 \times 2.5}
$$

$$
A = \sqrt{10.5 \times 1.5 \times 6.5 \times 2.5} = \sqrt{264.375}
$$

$$
A \approx 16.26
$$

So, the area of the triangle with sides of lengths 9, 4, and 8 is approximately 16.26 square units.

Maverick Smith · Answer

undefined To use Heron's formula to find the area of a triangle with side lengths of 9, 4, and 8, first calculate the semi-perimeter of the triangle using the formula:

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

where $ a $, $ b $, and $ c $ are the lengths of the sides. For this triangle:

$ a = 9 $,
$ b = 4 $,
$ c = 8 $.

Substitute these values into the formula:

$ s = \frac{9 + 4 + 8}{2} = \frac{21}{2} = 10.5 $.

Next, calculate the area using Heron's formula:

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

Substitute the values of $ s $, $ a $, $ b $, and $ c $ into the formula:

$ A = \sqrt{10.5(10.5 - 9)(10.5 - 4)(10.5 - 8)} $

$ A = \sqrt{10.5 \times 1.5 \times 6.5 \times 2.5} $

$ A = \sqrt{198.75} $

$ A \approx 14.09 $ square units.

So, the area of the triangle with side lengths of 9, 4, and 8 is approximately 14.09 square units.