Question

a belt is drawn tightly around three circles of radius 10 cm each, as shown. the length of the belt, in cm, can be written in the form a + b\pi for rational numbers a and b. what is the value of a + b?

Answer 1

80 cm

Explanation:

You want a function of the length of the perimeter belt around three packed circles of radius 10 cm.

Perimeter

Examination of the attached figure shows the length of the belt is the sum of the lengths of the sides of an equilateral triangle with sides 20 cm, and the circumference of a circle with diameter 20 cm.

triangle perimeter = 3s = 3(20 cm) = 60 cm

circle circumference = πd = π(20 cm) = 20π cm

The length of the belt is the sum of these, or

60 cm +20π cm = (60 +20π) cm

Function

Compared to (a +bπ) cm, we find a=60, and b=20. The desired sum is ...

a +b = 60 +20 = 80

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Additional comment

You will notice that equilateral triangle PQR in the attached figure has side lengths that are twice the radius of the circle. The straight segments AB, A'B', A"B" are parallel to the triangle sides, and are the same length. (ABPR is a rectangle.)

Each circular arc, BA', for example, is 1/3 of the circumference of the circle, so the total length of all the arcs is the circumference of one 20 cm circle.