Use 3.14 to approximate 𝜋 and calculate the perimeter of the following figure.
Let’s start by taking a look at the diagram and seeing what shapes we can identify. Externally, we have what looks like four semicircles. But are we certain that these are all going to have the same radius or diameter? If we take a look at this length which is marked as 15 centimeters, we can see that the center of this circle which forms the semicircle is given. The radius of the circle here has two lines on it. And we can see that this is also true of the other radii on the other three circles. So therefore, all four semicircles have the same radius and therefore will be the same size.
So let’s start thinking then of how we can calculate the perimeter. And we remember that the perimeter is the distance around the outside of a shape. When it comes to finding the perimeter of this shape, even though we can see that there’s a square within it, we don’t count it as part of the perimeter as it’s not on the outside of the shape. When we’re thinking in terms of the perimeter of a circle, we use the word circumference instead. We find the circumference of a circle by multiplying 𝜋 times the diameter or indeed two times 𝜋 times the radius.
There are two different ways in which we can think of approaching this question of finding the perimeter of four semicircles. One method would be to consider how we would find the perimeter of one semicircle and then multiply it by four. The second method would be to realize that if we joined two of these semicircles, we would get one circle. And since we have two of those, then we’d find the perimeter or circumference of a circle and then multiply it by two.
Let’s begin by considering how we would find the perimeter of one semicircle. We know that our perimeter is going to be based on the circumference, which is 𝜋 times the diameter. But because it’s only half of that, we divide by two. We can fill in the values that we’re given. And we’re told to use 3.14 as an approximation for 𝜋 and the diameter of the circle is 15. As we’re going to use a non-calculator method here, we can simplify this calculation, recognizing that 15 divides by two 7.5 times. Working out the calculation of 3.14 times 7.5 would give us a perimeter of 23.55 centimeters.
So if that’s the perimeter of one semicircle, then to find the whole perimeter, we’d need to multiply that value by four. 23.55 multiplied by four gives us a perimeter of 94.2 centimeters. Let’s see if it would’ve been quicker to realize that our two semicircles make a full circle and then multiply that by two. This time, we’re finding the perimeter of the whole circle, which is this circumference. And we use the formula 𝜋 times diameter.
Filling in the values for 𝜋 and the diameter, we’ll be calculating 3.14 multiplied by 15. Evaluating that would give us an answer of 47.1 centimeters. In this method, remember, we’ve found the value of one circle and we need to multiply it by two, which would also give us a value of 94.2 centimeters. So using either method would allow us to find the total perimeter.