Question

This figure shows three identical spherical billiard balls of diameter 6 cm inside an equilateral triangular frame.

Work out the perimeter of the frame, leaving your answer
in surd form.

Answer 1

Perimeter = 18(1 + √3 ) cm

Explanation:

The radius of each ball = 1/2 * 6 = 3 cm.

Lines drawn from the 2 points of contact for one billiard ball to the center of the ball are at right angles to the sides of the triangle ( Tangent/radius theorem).

If we now draw a line from the vertex of the big triangle to the center of the ball we get 2 right triangles, and they are 30-60-90 triangles.

If the adjacent side of a triangle ( which is part of the side of the big triangle) = x:

tan 30 = 3 / x

x = 3 / tan 30

= 3 / 1/√3

= 3√3 cm.

There are 6 of these sides in the big triangle so their total length =

18√3 cm.

The three 'middle' sides joining 2 billiard balls each have a length of 2 radii = 6 cms ( as they form a rectangle with the radii of 2 billiard balls).

So the perimeter of the triangle = 18√3 + 3(6)

= 18(1 + √3 ) cm

I would have liked to transfer a diagram but I can't get to copy it to this site.