Video Transcript
A regular pentagon is circumscribed around a circle of radius two centimeters. Find the length of one side of the pentagon. Give your answer correct to one decimal place.
Here we have a circle. First, we can add a center to this circle. And then we have five places where the pentagon is tangent to this circle. We know that the distance from the center to any point on the outside circle is the radius. And we know that the radius measures two centimeters. We notice that a vertex of the pentagon is an intersection of two tangent lines, which tells us that each of those tangent lines are of the same length. We should also remember that when a tangent intersects a radius, the intersection is 90 degrees.
At this point, it might seem like there’s nothing else we can add. But we know that this is a regular pentagon. And that means we can either remember each of these angles or know how to calculate them. If you don’t remember what the interior angles of a regular polygon are, you can calculate the interior angles of any polygon with the formula 𝑠 equals 𝑛 minus two times 180 degrees. Where 𝑠 represents the sum of the interior angles and 𝑛 is the number of sides in the polygon.
A pentagon has five sides. So we need five minus two times 180 degrees. Five minus two is three. Three times 180 is 540. But remember, that’s the sum of all five of these angles. But since this is a regular pentagon, all five of the angles are equal. And that means if we divide 540 by five, we can find the measure of each of these angles. 540 divided by five is 108. In a regular pentagon, all of the angles measure 108 degrees. But that still doesn’t totally help us.
What would happen though if we drew a line between one of the vertexes and the center of the circle? That line divides these angles in half. Instead of 108, each of them would be 54 degrees. At this point, it’s probably helpful if we zoom in on what we’re talking about.
We have our two-centimeter radius, one part of the pentagon, and the line we’ve just drawn from the vertex of the pentagon to the center of the circle. We know that the radius and the tangent meet at a right angle and that this angle is 54 degrees. In any triangle, all angles need to add to 180 degrees. That means 90 degrees plus 54 degrees plus some angle has to equal 180 degrees. When we add 90 and 54, we get 144. And if we subtract 144 from 180, we find out our missing angle measures 36 degrees.
Remember, our goal is to find the length of one side of this regular polygon. Because it’s a regular polygon, we know that all sides are the same length. And that means if we find the length of one side, we would find the length of all the sides. If we add another radius, we can see that again we have two tangents, which will be of equal length. And that means if we can find this distance, we need to multiply it by two to find the length of one side of the pentagon. And that means we need to consider this right triangle. And we need to find this missing side length.
Because we’re now dealing with right triangles, we can use our right triangle trigonometry. If we think about three trig relationships, we have sine, cosine, and tangent. Sine is the opposite over the hypotenuse. Cosine is the adjacent over the hypotenuse. And tangent equals the opposite over the adjacent. The yellow side length is the hypotenuse, and we don’t know how long it is. We know the radius, the pink side, and we want to know the other side length. We’re not interested in the hypotenuse. And that means our best option is to use the tangent relationship.
If we start at the 54-degree angle, we could say the tangent of 54 degrees equals the opposite side length, two centimeters. And the adjacent side length is our missing side, 𝑠. But that’s not our only option. We could also take the tangent of 36 degrees, where the opposite side is side length 𝑠 and the adjacent side length is the radius of two centimeters. Either one of these will work. But I’m going to choose the tangent of 36 degrees as the math will be a little bit simpler, because the side length is in the numerator of the fraction and not the denominator. So we’ll have one less step than if we use the tangent of 54 degrees.
In this case, we just need to multiply both sides of the equation by two centimeters. And we’ll have two centimeters times the tangent of 36 degrees equals the side length. When we multiply two times tangent of 36 degrees, we get 1.45308 continuing. If you don’t get this, you should check that the calculator you’re using is set to degrees and not to radians.
We need to be careful here because this 1.45308 continuing is only half of this side of the pentagon. Both of these distances measure 1.45308 continuing centimeters. And together they form the length of one side of this regular pentagon. And so the length of one side of this pentagon is equal to two times two centimeters times the tangent of 36 degrees, or two times 1.45308 continuing centimeters. When we do that multiplication, we get 2.90616.
We’re rounding to the first decimal place. That means we need to look to their right, to the hundreds place. Since there’s a zero in the hundreds place, we round down to 2.9 centimeters.
