The figure shows the beginning of a spiral created by starting with a semicircle

of radius 1 unit and endlessly attaching semicircles that are each
half the radius of the previous semicircle. What is the length of
the whole spiral? Express your answer in terms of π.

1 answer

The length of each semicircle can be calculated using the formula for the circumference of a circle:

C = πd

Since the radius of the first semicircle is 1 unit, its diameter is 2 units. Therefore, the length of the first semicircle is:

C1 = π(2) = 2π

The radius of the second semicircle is half of the radius of the first semicircle, so it is 1/2 units. Therefore, the length of the second semicircle is:

C2 = π(1) = π

Similarly, the radius of the third semicircle is half of the radius of the second semicircle, so it is 1/4 units. Therefore, the length of the third semicircle is:

C3 = π·(1/2) = π/2

This pattern continues, with each semicircle having a radius that is half of the previous semicircle:

C4 = π·(1/4) = π/4
C5 = π·(1/8) = π/8
C6 = π·(1/16) = π/16
...

To find the length of the entire spiral, we need to add up all these lengths. Since the lengths form a geometric series with a common ratio of 1/2, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio. In this case, a = 2π and r = 1/2, so we have:

S = (2π) / (1 - 1/2)
S = (2π) / (1/2)
S = 4π

Therefore, the length of the whole spiral is 4π units.