Question Video: Finding the Area of a Shaded Part Between a Semicircle and a Rectangle Mathematics • 7th Grade

Video Transcript

Using 3.14 as an approximation for 𝜋, find the area of the shaded shape.

If we look at the diagram, we can see from the shading that this shape in orange is excluded from the area. This orange shape is a semicircle. As this shaded figure or shape is composed of two geometric figures, then we can call this a composite figure. We can calculate the area of this shaded region by finding the area of the large rectangle and then subtracting the area of the semicircle. Let’s begin by calculating the area of this rectangle. And as we find the area of a rectangle by multiplying the length by the width, we’d have 52 times 23. We can evaluate this without a calculator to give us the value 1196. And the units here as it’s an area will be squared centimeters.

Now let’s focus on the area of the semicircle. We can use the fact that the area of a circle is equal to 𝜋𝑟 squared to establish that the area of a semicircle would be half of that. In other words, it’s 𝜋𝑟 squared divided by two. Before we can use this formula, however, we need to work out the value of the radius of this semicircle. The length of our rectangle is 52 centimeters. And therefore, we can see that the diameter of this circle can be found by subtracting the other two lengths of 15 from 52, which gives us the diameter of 22 centimeters. And so the length of our radius will be half of that. Half of 22 will give us 11 centimeters. Plugging this value into our formula for the area of a semicircle gives us 𝜋 times 11 squared over two, which is 𝜋 times 121 over two.

In order to evaluate this, we were told to use 3.14 as an approximation for 𝜋. We could put this directly into our calculator if we wished. But if we weren’t using a calculator, then we could simplify the calculation to 1.57 multiplied by 121 and then use any method of multiplication to work this out. Here, I’ve used the grid or area method to find the value of 189.97 square centimeters for the area of the semicircle. Now that we’ve found the two important pieces of information, the area of the rectangle and the area of the semicircle, we can work out the area of the shaded shape.

Remembering that as this shaded shape does not include the area of the semicircle, this means we must subtract it from the area of the rectangle. Therefore, we’ll have the calculation 1196 subtract 189.97, which gives us 1006.03 square centimeters. And that is our final answer for the area of the shaded shape.