Question

lesson practice
7.3.04
name: _____________________________ date: _____ period: _______
problems 1–2: here is a quarter-circle. it was created
using a circle with a 12-inch diameter. the circle was folded
in half and then folded in half again.

1. label the quarter-circle with any important measurements.
2. what is the perimeter of this quarter-circle?

Explanation:

Step1: Find the radius

The diameter of the original circle is 12 inches, so the radius \( r=\frac{12}{2} = 6 \) inches. The two straight sides of the quarter - circle are both radii, so they are 6 inches long.

Step2: Calculate the length of the arc

The formula for the circumference of a circle is \( C = 2\pi r\) or \( C=\pi d\). For a quarter - circle, the length of the arc is \( \frac{1}{4}\) of the circumference of the whole circle. Using \( C=\pi d\) (where \( d = 12\) inches), the length of the arc \( l=\frac{1}{4}\times\pi\times12= 3\pi\approx3\times3.14 = 9.42\) inches.

Step3: Calculate the perimeter of the quarter - circle

The perimeter of the quarter - circle is the sum of the length of the arc and the lengths of the two straight sides. The two straight sides each have length 6 inches, so their total length is \( 6 + 6=12\) inches. Then the perimeter \( P=12 + 9.42=21.42\) inches (or using \( \pi\) exactly, \( P = 12+3\pi\) inches).

Answer:

The perimeter of the quarter - circle is \( 12 + 3\pi\approx21.42\) inches. If we use \( \pi\approx3.14\), the perimeter is approximately 21.42 inches.