Shaded Area – Square Inscribed in a Circle

The diagonal of the square inscribed in the circle below is 8cm. Find the shaded area. (Use pi = 3.14)

square inscribed in a circle

 

 

 

 

 

 

 

Solution

From the diagram above, we can get the shaded area by subtracting the area of the square from the area of the circle.

We let the diagonal of the square be the base of two the triangles. Next, we draw the height of one of the triangles.

triangle

Since the height of the triangle is the radius of the circle, it is therefore ½(8) = 4cm.

(1) Area of the Circle (A_c)

The formula for finding the area of a circle is

A_c = pi \times r \times r

where r is the radius.

Substituting the 4 cm to radius, we have

A_c = 3.14 \times 4 \times 4

A_c = 50.24 sq. cm.

(2) Area of the Square (A_s)

To find the area of the square, we can first find the area of the triangle. To find the area of triangle ABC, we have

A_s = \frac{1}{2} \times b \times h

where b is the base and h is the height.

Substituting we have

A_s = \frac{1}{2} \times 8 \times 4

A_s = 16

Since there are two triangles, each with area 16 sq. cm, the area of the whole square is

16 \times 2 = 32 sq cm.

(3) Shaded area

shaded area = area the circle – area of the square

shaded area = 50.24 – 32
shaded area = 18.24

Answer: 18.24 sq.cm

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