The areas of two similar polygons are 361 square centimeters and 81 square centimeters. Given that the perimeter of the first is 38 centimeters, find the perimeter of the second.
We recall that similar shapes have the same number of sides, corresponding angles are congruent, and corresponding sides are in the same proportion. We are given here that there are two similar polygons. We don’t know what shape these polygons are. For example, they could be squares, rectangles, kites, or even hexagons. Let’s call them polygon one and polygon two. We are given some pieces of information about these two polygons. We’re told that the first one has an area of 361 square centimeters and the second one has an area of 81 square centimeters. We are also told that the perimeter of the first one is 38 centimeters, and we need to work out the perimeter of this second polygon.
Usually, if we know what shape a polygon is, it’s possible that we could work out the perimeter from its area. However, we don’t know what shape these polygons are. So we’ll need to use the ratio of their areas to help us work out the perimeter. We can use the fact that if the length ratio of two similar polygons is 𝑎 to 𝑏, then the ratio of their areas is 𝑎 squared to 𝑏 squared. So let’s take the two polygons and write their area ratio. This would be 361 to 81.
We know from the statement above that this area ratio can be written as 𝑎 squared to 𝑏 squared when the length ratio is given as 𝑎 to 𝑏. Let’s see if we could calculate the values of 𝑎 and 𝑏. Well, we might notice that 361 and 81 are both square numbers. The square root of 361 is 19, and the square root of 81 is nine. This ratio of 19 to nine is in fact the length ratio of these two polygons.
Now that we have the length ratio between the two polygons, we might wonder how this will help us to work out the perimeter of the second polygon. To do this, we’ll need to remember that perimeter is simply a length as well. And so the perimeters will be in the same ratio as the lengths. And that means what we’re really dealing with is two equivalent ratios. We need to find the value in the ratio of the perimeters 38 to what which makes it equivalent to the ratio of the lengths of 19 to nine. We notice that 19 multiplied by two gives us 38. We would then perform the equivalent calculation on the other side of the ratio, which gives a value of 18. Therefore, the answer is that the perimeter of the second polygon must be 18 centimeters.