𝐴𝐵𝐶𝐷 is a square of side 140. Calculate the perimeter of the shaded region, taking 22 over seven as an approximation for 𝜋.
We’re asked to find the perimeter of the shaded region. We can remember that the perimeter is the distance around the outside. Therefore, we’ll need to calculate the distance of these two curved sections. So, let’s have a think about one of these curved sections. This will in fact be made of a circle that has the same radius as the length of the side in the square. Therefore, we know that the radius of this circle would be 140 units. As 𝐴𝐵𝐶𝐷 is a square, we know that the angle at 𝐷𝐶𝐵 will be 90 degrees, which means that the shape is a quarter circle. The arc created at 𝐷𝐵 is also part of a quarter circle.
In order to answer the question, to find the perimeter of the shaded region, we’ll then need to calculate the perimeter of two of these quarter circles. In order to find the perimeter or circumference of a whole circle, we use the formula that this is equal to 𝜋 times the diameter. To find the perimeter of a quarter circle, we take the circumference and divide it by four. So, we’ll be calculating 𝜋 times the diameter over four. In the question, we’re given that the radius is 140. So, to find the diameter, we multiply 140 by two.
We can then plug this into our calculation to give us the perimeter is equal to 𝜋 times 280 over four. We can simplify this calculation so that we’re calculating 70 times 𝜋. We’re told to take 22 over seven as an approximation for 𝜋. So, we’ll have the calculation of 70 times 22 over seven. This will simplify to give us 220 units. And now we’ve found the perimeter of a quarter circle, we can find the perimeter of two quarter circles. We take our value of 220, and we multiply it by two, which gives us a value of 440 units. And so, our answer for the perimeter of the shaded region would be 440 units.
There is, of course, an alternative method that we could’ve used. And that is by understanding that two quarter circles added together would give us half of a circle. In this method, we take the circumference 𝜋 times the diameter and divide it by two. We would’ve calculated 22 over seven multiplied by 280 all over two. The numerator of the fraction would cancel to give us 22 times 40 over two. This can be further simplified to 11 times 40, which would also give us the answer of 440 units. So, we could use either method of finding two quarter circles or finding one half circle to give us the perimeter of this shaded region.