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