If the perimeter of an equilateral triangle is 21.19 centimeters and its area is 15.6 centimeters squared, find its height to the nearest hundredth.
So this question is specifically about an equilateral triangle. What’s special about equilateral triangles is that all three of their sides at the same length. As they’re all the same, we’ll give them all the same letter and refer to them as 𝑥 centimeters. We’re given two pieces of information about this triangle. Firstly, we’re told that its perimeter, the sum of its three sides, is 21.19 centimeters.
Secondly, we’re told that the area of this triangle is 15.6 centimeters squared. Our task is to calculate the height of the triangle, which we’ll refer to as ℎ centimeters. So let’s think about how to approach this problem. The area of a triangle is calculated by multiplying the base by the perpendicular height and dividing by two.
Using the letters in this question, this would be 𝑥 multiplied by ℎ divided by two. Now we know the area of the triangle; it’s 15.6 centimeters squared. We don’t know the height, and we don’t know the value of 𝑥; however, if we can calculate the value of 𝑥, we can then work out the value of ℎ, which is our objective in this question.
Let’s return to the perimeter of the triangle. Remember the perimeter is found by adding all three sides of the triangle together. This means that 𝑥 plus 𝑥 plus 𝑥 must be equal to 21.19; or in a simplified format, three 𝑥 is equal to 21.19. We can work out the value of 𝑥 by dividing both sides of this equation by three.
This tells me that 𝑥 is equal to 7.06 and then a recurring three. So now I know the value of 𝑥. I can therefore substitute this into the equation for the area. I have that the value of 𝑥 I just calculated multiplied by ℎ divided by two is equal to 15.6.
Now we want to solve this equation to work out the value of ℎ. So the first step is I’m going to multiply both sides of the equation by two. I now have that 7.063 recurring multiplied by ℎ is equal to 31.2. Next, I’m going to divide by this value of 7.063 recurring.
ℎ is equal to 31.2 divided by 7.063 recurring. Now hopefully you’ll have kept that value on your calculator screen from earlier, which means now you can just type 31.2 divided by “answer” in order to get an exact value for ℎ.
This gives ℎ is equal to 4.417177, and it’s continuing decimal. The question asked us to give the height of the triangle to the nearest hundredth. So rounding this value as requested gives us our answer to the problem: the height of the triangle to the nearest hundredth is 4.42 centimeters.