Examples with solutions for Area of a Circle: Subtraction or addition to a larger shape

Exercise #1

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?

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Video Solution

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:

4448881234Therefore, 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:

12πr2 \frac{1}{2}\pi r^2

12π42=8π \frac{1}{2}\pi4^2=8\pi

Let's now calculate the area of semicircle A3:

12πr2 \frac{1}{2}\pi r^2

12π22=2π \frac{1}{2}\pi2^2=2\pi

Therefore the area of the rectangle equals:

4×8=32 4\times8=32

Finally we can calculate the total area of the shape:

32+2×8π+2×2π=32+16π+4π=32+20π 32+2\times8\pi+2\times2\pi=32+16\pi+4\pi=32+20\pi

Answer

32+20π 32+20\pi cm².

Exercise #2

A trapezoid is shown in the figure below.

On its upper base there is a semicircle.

What is the area of the entire shape?

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Video Solution

Answer

24+3.125π 24+\text{3}.125\pi cm².

Exercise #3

Two circles share a center point, marked O.

What is the area of the shaded part of the orange circle?

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Video Solution

Answer

12π 12\pi cm²

Exercise #4

Look at the circle in the figure.

Its center is O.

Inside the circle there is a square.

What is the area of the white parts combined?
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Video Solution

Answer

23.085 cm²

Exercise #5

From the point O on the circle we take the radius to the point D on the circle. Given the lengths of the sides in cm:

DC=8 AE=3 OK=3 EK=6

EK is perpendicular to DC

Calculate the area between the circle and the trapezoid (the empty area).

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Video Solution

Answer

36.54

Exercise #6

The trapezoid ABCD is drawn inside a circle.

The radius can be drawn from point O to point C.

DC = 12 cm
OK = 3 cm
NB = 4 cm
NK = 5 cm

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Calculate the white area between the trapezoid and the circle's edge.

Video Solution

Answer

91.37

Exercise #7

The drawing shows an equilateral triangle

The length of each of its sides 7 cm

a semicircle is placed on each side of each side

What is the area of the entire shape? Replace π=3.14 \pi=3.14

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Video Solution

Answer

78.91 cm².