A triangle has sides with lengths: 2, 8, and 3. How do you find the area of the triangle using Heron's formula?

Question

A triangle has sides with lengths: 2, 8, and 3.  How do you find the area of the triangle using Heron's formula?

Zoe Austin · Accepted Answer

There is no such triangle since #2+3 < 8#

Such side lengths are not possible to form into a triangle.

If the lengths of the sides of a triangle are #a#, #b# and #c# then all of the following must hold:

#a+b > c#

#b+c > a#

#c+a > b#

Caroline Adams · Answer

#s=1/2 (8+2+3) = 13/2#
#sqrt(13/2 (6.5-8)(6.5-2)(6.4-3)#
Oops the this is not going to work, why do you think it is?

heron's formula is #sqrt(s(s-a)(s-b)(s-c) # Where s = the semi-perimeter And a,b,c are the sides. This is obviously not going to work; however, the issue lies not in Heron's formula but in your arbitrary apparent triangle. However, according to the triangle inequality, a+b>c, meaning that any two sides of the triangle added together must always be greater than any one side; thus, the sum of sides 2 and 3 must add up to more than 8 in order for the semi perimeter to be greater than 8.

Violet Anderson · Answer

undefined Using Heron's formula, you first need to calculate the semi-perimeter of the triangle, which is the sum of the lengths of all three sides divided by 2. Then, you can use Heron's formula, which states that the area of the triangle is the square root of the semi-perimeter multiplied by the semi-perimeter minus the length of each side:

$$ s = \frac{2 + 8 + 3}{2} = 6.5 $$

$$ \text{Area} = \sqrt{s(s - 2)(s - 8)(s - 3)} $$

$$ \text{Area} = \sqrt{6.5(6.5 - 2)(6.5 - 8)(6.5 - 3)} $$

$$ \text{Area} = \sqrt{6.5 \times 4.5 \times -1.5 \times 3.5} $$

$$ \text{Area} = \sqrt{6.5 \times 4.5 \times 1.5 \times 3.5} $$

$$ \text{Area} \approx \sqrt{43.3125} $$

$$ \text{Area} \approx 6.58 $$

Therefore, the area of the triangle is approximately 6.58 square units.