A designer made this design from circles and semicircles. He claims that two-thirds of the design is shaded. Is he correct? Show all your working.
In order to calculate the area of a circle, we use the formula 𝜋𝑟 squared, 𝜋 multiplied by the radius squared. The two pink semicircles will make a circle of radius five centimeters. And the two orange semicircles will make a circle with radius three centimeters.
To calculate the area of the whole shape, we multiply 𝜋 by 10 squared, as the radius of the whole shape is 10 centimeters. 10 squared is equal to 100. So the area of the whole shape is 100𝜋.
The area of the nonshaded sections is 𝜋 multiplied by three squared plus 𝜋 multiplied by five squared. Three squared is equal to nine and five squared is equal to 25. This gives us nine 𝜋 plus 25𝜋. As nine plus 25 is equal to 34, the nonshaded area is 34𝜋.
As the whole shape has an area of 100𝜋 and the nonshaded area is 34𝜋, subtracting these gives us the area of the shaded region. 100 minus 34 is equal to 66. So the shaded area is 66𝜋.
We should note at this stage that all three of these answers are in centimeters squared. However, for the purpose of this question, we don’t need to include that.
The designer claims that two-thirds of the design is shaded. As we know the area of the shaded section and the whole shape, we can say that the fraction that is shaded is 66𝜋 out of 100𝜋. Cancelling the 𝜋 gives us 66 out of 100, or sixty-six hundredths. This is not equal to two-thirds, as 66 out of 100 is equal to 66 percent and two-thirds is equal to 66.6 recurring percent.
As these two fractions are not equal, we can say that the designer is incorrect. His claim that two-thirds of the design is shaded is wrong.