Video: Calculating the Perimeter of a Shaded Part in a Square

Using 3.14 to approximate πœ‹ and the fact that 𝐴𝐡𝐢𝐷 is a square, calculate the perimeter of the shaded part.

02:11

Video Transcript

Using 3.14 to approximate πœ‹ and the fact that 𝐴𝐡𝐢𝐷 is a square, calculate the perimeter of the shaded part.

So looking at the diagram, we can see that we have the square 𝐴𝐡𝐢𝐷 and within it we have four quarter circles. The region of the square outside this four quarter circles is the part that has been shaded and the part whose perimeter we’re looking to calculate. The perimeter is made up of four arcs, each of which is a quarter of a circumference of a full circle. The sum of these four arcs is just the same as the circumference of a full circle.

So in order to calculate the perimeter of this shaded part, we are just looking to calculate the circumference of a circle. We have two possible formulae that we can use in order to calculate the circumference of a circle: either πœ‹π‘‘, where 𝑑 represents the diameter of a circle, or two πœ‹π‘Ÿ, where π‘Ÿ represents the radius. Let’s determine either the diameter or the radius of our circle.

We know that 𝐴𝐡𝐢𝐷 is a square, so each of its sides is of length 68 centimetres. The radius of our circle is half the length of each of the sides of the square and therefore it’s equal to 34 centimetres. So now we have the necessary information to be able to calculate the perimeter of the shaded part. So the perimeter is equal to two multiplied by πœ‹ multiplied by π‘Ÿ.

Now we’re told to use 3.14 to approximate πœ‹ and we now know that the radius π‘Ÿ is 34. So we have our calculation for the perimeter: two multiplied by 3.14 multiplied by 34; this gives us a value of 213.52. Finally, the perimeter is a length and the units in this problem are centimetres. So we have our answer to the question; the perimeter of the shaded part is 213.52 centimetres.

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