Video: Finding the Area of a Composite Figure Involving Two Circular Sectors and a Rectangle

Using 3.14 to approximate 𝜋, find the area of the shaded region.

02:42

Video Transcript

Using 3.14 to approximate 𝜋, find the area of the shaded region.

Let’s look at the diagram to determine what the shaded region is composed of. There is a rectangle 𝐴𝐵𝐶𝐷 and then two quarter circles cut out from it. The shaded region is the remaining area. So it’s the area of the rectangle minus the area of the two quarter circles.

Let’s calculate the area of the rectangle first of all. The area of a rectangle is found by multiplying its length by its width. The length of this rectangle is 68. And the width is the sum of 34 and 11.1. So our calculation for the area of the rectangle is 68 multiplied by 34 plus 11.1.

Next, let’s consider the area of the quarter circles. These quarter circles are identical, as indicated by the blue lines, which show that the radii of the two circles are the same. The area of a circle is calculated using the formula 𝜋𝑟 squared, where 𝑟 represents the radius of the circle.

As these are quarter circles, we need to divide by four. However, there are also two of them. So we need to multiply by two to account for both areas. The two in the numerator and the four in the denominator can both be canceled by a factor of two. This gives an overall formula for the area of the two quarter circles as 𝜋𝑟 squared over two.

The radius of the circles is 34 centimeters. And so the calculation is 𝜋 multiplied by 34 squared over two. We’re told in this question that we need to use 3.14 as an approximation for 𝜋. So the calculation is 3.14 multiplied by 34 squared over two.

Now I’ve included brackets around the two halves of this calculation, not because they’re mathematically necessary, but just to keep the area of the rectangle and the area of the quarter circles separate.

Now we just need to evaluate the two areas and subtract. The area of the rectangle is 3066.8 centimeters squared. And the area of the two quarter circles is 1814.92 centimeters squared. The area of the shaded region is 1251.88 centimeters square.

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