The perimeter of a regular pentagon is 85 centimeters. Find the area, giving the answer to the nearest square centimeter.
The key piece of information given in this question is that this pentagon is regular. We have a formula that we can use for calculating the area of any regular polygon.
For a regular polygon with 𝑛 sides each of length 𝑥, its area is equal to one-quarter multiplied by 𝑛 multiplied by 𝑥 squared multiplied by cot of 𝜋 over 𝑛. Remember, cot is the reciprocal of tan. Cot of 𝜋 over 𝑛 is equal to one over tan of 𝜋 over 𝑛.
This question is about the pentagon. And therefore, the number of sides 𝑛 is equal to five. We haven’t been given the side length directly in the question. But instead, we’ve been told the perimeter of the pentagon.
However, we can form a simple equation to calculate this side length. The regular pentagon has five sides, all of length 𝑥. And therefore, its perimeter will be equal to five 𝑥. So we have the equation five 𝑥 is equal to 85. Dividing both sides of this equation by five tells us that the side length of this pentagon is 17 centimeters.
Now that we know the values of both 𝑛 and 𝑥, we can substitute them into our formula for the area. This gives the area of the pentagon is equal to one-quarter multiplied by five multiplied by 17 squared multiplied by cot of 𝜋 over five.
Now we need to use a calculator to evaluate this. And the important thing is that you make sure the calculator is in the correct mode. As the angle has been specified using 𝜋, this means that we’re using radians to measure angles, not degrees.
So the first part of this simplifies to 1445 over four and then multiplied by cot 𝜋 over five. The simplest way to evaluate this on your calculator is probably to recall that cot 𝜋 over five is equal to one over tan of 𝜋 over five. So this whole thing is equal to 1445 divided by four tan 𝜋 over five. Evaluating this gives 497.21796.
Now if you haven’t got that and you’re certain that you’ve typed it into your calculator correctly, then again check that your calculator is in the correct mode. It needs to be in radians.
Now the question has asked for the answer to the nearest square centimeter. So the final step is to round this value. And we have that the area of this regular pentagon to the nearest square centimeter is 497 square centimeters.