Question Video: Calculating the Perimeter of a Shaded Part in a Square Mathematics • Second Year of Preparatory School

Video Transcript

Using 3.14 to approximate 𝜋 and the fact that 𝐴𝐵𝐶𝐷 is a square, calculate the perimeter of the shaded part.

At first, it may seem that the perimeter of this shaded region will be tricky to work out as it’s an unusual shape. If we look carefully though, we see that each of these unshaded portions are quarter circles. The shaded region is enclosed by the curved portions of these four quarter circles. Each of these arc length is one-quarter of the circumference of a circle, but as there are four of them, together they form the full circumference of a circle.

We know that the formula for calculating the circumference of a circle is 𝐶 equals 𝜋𝑑, where 𝑑 represents the circle’s diameter. So the question is, what is the diameter of this circle?

Considering the quarter circle in the bottom left of the figure, we can see that the radius of this circle will be half of the side length of the square. That’s 68 over two. So the radius is 34 centimeters. The diameter of a circle is twice the radius, so the diameter is two times 34; it’s 68 centimeters. In fact, we could have deduced this from the figure without halving and then doubling again. Two of the radii of the quarter circles lie along the side length of the square, so twice the radius is 68 centimeters. And as the diameter is twice the radius, we find again that the diameter is 68 centimeters. We’ve already said that the perimeter of the shaded region is equal to the circumference of the full circle made up of these four quarter circles.

So using the formula 𝐶 equals 𝜋𝑑, we find that the perimeter of the shaded region is 𝜋 multiplied by 68. We’re told in the question to use 3.14 as an approximation for 𝜋. So multiplying gives 213.52. The units for this perimeter are the same as the length units used in the question. So we find that the perimeter of the shaded part using 3.14 as an approximation for 𝜋 is 213.52 centimeters.