Area Calculation: 4×8 Rectangle with Four Semicircles

Observe the rectangle in the figure below.

A semicircle has been added to each side of the rectangle.

Determine the area of the entire shape?

Video Solution

Solution Steps

00:00 What is the total area of the shape?

00:07 The area equals the rectangle area plus the areas of 4 half-circles

00:20 Each half-circle equals its opposite one, because they are on equal sides

00:28 Therefore, this is the correct formula

00:32 Our formula is half of the circle area formula

00:37 The diameter is the rectangle side which equals 8

00:43 Let's solve using the circle area formula where the radius is 4

00:50 This is the area of half-circle 1 (and also 2 since they are equal)

00:53 Now let's use the same method to calculate the area of half-circle 3

00:57 Here the circle diameter is side 4

01:01 Let's substitute the appropriate values and solve to find the area of half-circle 3

01:05 This is the area of half-circle 3 (and also 4)

01:09 Now let's calculate the rectangle area using side(4) multiplied by side(8)

01:14 Now let's sum all the areas to find the total area of the entire shape

01:23 And this is the solution to the question

Step-by-Step Solution

The area of the entire shape equals the area of the rectangle plus the area of each of the semicircles.

Let's begin by labelling each semicircle with a number:

Therefore, we can determine that:

The area of the entire shape equals the area of the rectangle plus 2A1+2A3

Let's proceed to calculate the area of semicircle A1:

$\frac{1}{2}\pi r^2$

$\frac{1}{2}\pi4^2=8\pi$

Let's now calculate the area of semicircle A3:

$\frac{1}{2}\pi r^2$

$\frac{1}{2}\pi2^2=2\pi$

Therefore the area of the rectangle equals:

$4\times8=32$

Finally we can calculate the total area of the shape:

$32+2\times8\pi+2\times2\pi=32+16\pi+4\pi=32+20\pi$