Use 3.14 to approximate 𝜋 and calculate the perimeter of the shaded part.
In this question, we’re given this diagram which has a shaded section. It could be very tempting here to think we’re calculating the area, but we’re asked for the perimeter. That’s the distance around the outside edge. We’ll, therefore, need to add up the lengths of the sections of this rectangle, excluding this section in the middle, and we’ll need to find the length of this curved section and add it on.
This curved section is in fact a semicircle. So, how would we find the perimeter of a semicircle? We’ll need to recall that the perimeter or circumference of a whole circle is equal to 𝜋 times the diameter. Therefore, if we want to calculate half of this to find the perimeter of a semicircle, we’d calculate 𝜋 times the diameter over two or 𝜋𝑑 over two. The diameter of a circle is the distance from one side to the other side going through the center. To calculate this for our semicircle, we have a value of 16 centimeters along the base of this rectangle, and we then subtract three centimeters and five centimeters, which gives us the distance, the diameter, as eight centimeters. We can then plug this value into our perimeter.
We’re told to use 3.14 for 𝜋. So, we’ll be calculating 3.14 multiplied by eight all over two. Simplifying this calculation, we’ll have 3.14 times four which we can do without a calculator. This will give us the perimeter of the semicircle to be 12.56. And as we’re dealing with a perimeter, we’ll still have the length units of centimeters. To find the perimeter of the whole shaded area then, we need to add on all the other parts. The length of the side opposite, 13 centimeters, will be of the same length.
To work out the perimeter of the shaded part then, we have five plus 13 plus 16 plus 13 plus three plus 12.56 centimeters. Remember that we don’t include this value of eight centimeters as it’s not included in the shaded part. Adding all these values, we can give our answer for the perimeter of the shaded part as 62.56 centimeters.