# 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**