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)

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.

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

Shaded Area – Square Inscribed in a Circle