Find The Area Of The Rectangle Formed By Connecting The Tangents Of The Circles

Three equal circles are touching each other. A rectangle is formed by connecting the tangents of these circles, Then find the area of the rectangle

Find the area of the rectangle formed by connecting the tangents of the circles

Three circles have a 1 cm radius, These circles are touch one another as shown in the figure. We can create a rectangle ABCD by connecting the tangents of the circles, Now find the area of the rectangle ABCD

Solution

Draw PQ and RS as shown in the figure

From figure

PO = 2 cm (Diameter of the circle)

TR = OU = UQ = 1 cm (Radius of the circle)

Connect UT then we get a right angle triangle OUT

Apply Pythagorean theorem in triangle OUT

OT² = UT² – OU²

⇒ OT² = 2² – 1² = 4 – 1 = 3

⇒ OT = √3

PQ = PO + OU + UQ

⇒ PQ = 2 + 1 + 1 = 4 cm

SR = SO + OT + TR

⇒ SR = 1 + √3 + 1 = 2 + √3 cm

Area of rectangle = PQ × SR = 4(2 + √3) = 8 + 4√3 cm²