Find the area of the circle when opposite sides of the square are tangent and chord

One side of a square is the chord of a circle, and the opposite side is the tangent of the same circle. Find the area of the circle if the sides of the square are 8 cm.

From figure ABCD is a square, AD is the tangent of the circle and BC is the chord of the circle. Calculate the area of the circle, if the sides of a square are 8 cm long.

Solution: Area of the circle

We can draw a bisector to chord BC, then

Here PQ is the diameter of the circle because PQ is the bisector of chord BC

Assume diameter of the circle, PQ = 2r

From figure

OP = AB = 8 cm

OQ = PQ − OP = 2r – 8

OB = OC = 8/2 = 4 cm

Apply intersecting chords theorem in figure

OP × OQ = OB × OC

⇒ 8(2r − 8) = 4 × 4

⇒ 16r − 64 = 16

⇒ 16r = 80

⇒ r = 5 cm

Now we got radius of the circle = 5 cm

so Area of the circle = πr²

⇒ Area of the circle = π × 5²

⇒ Area of the circle = 25π cm²