Area of a Circle: Subtraction or addition to a larger shape

To solve this problem, we start by finding the area of the trapezoid:

• The formula for the area of a trapezoid is $A = \frac{1}{2} \times (b_1 + b_2) \times h$, where $b_1$ and $b_2$ are the lengths of the parallel sides, and $h$ is the height.
• Let's substitute the given values: $b_1 = 5$ cm, $b_2 = 11$ cm, and $h = 3$ cm.
• Calculate the area: $A_{\text{trapezoid}} = \frac{1}{2} \times (5 + 11) \times 3 = \frac{1}{2} \times 16 \times 3 = 24$ cm².

Next, we calculate the area of the semicircle:

• The formula for the area of a semicircle is $A = \frac{1}{2} \times \pi \times r^2$.
• The radius $r$ is half of the upper base, so $r = \frac{5}{2} = 2.5$ cm.
• Calculate the area: $A_{\text{semicircle}} = \frac{1}{2} \times \pi \times (2.5)^2 = \frac{1}{2} \times \pi \times 6.25 = 3.125\pi$ cm².

Combine the areas to find the total area of the shape:

Total Area = $A_{\text{trapezoid}} + A_{\text{semicircle}} = 24 + 3.125\pi$ cm².

Thus, the area of the entire shape is $24 + 3.125\pi$ cm².

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:

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