🔍 Key Concepts

• Use the circle circumference formula C = 2πR to find the radius from 157.6 inches. • Recognize that the shaded region is part of a sector of 90° (a quarter circle) with a right triangle cut out. • Area of shaded part = area of sector − area of right triangle formed by the two radii.

💭 Think About

• What is the radius of the circle when you solve 157.6 = 2πR using π ≈ 3.14? • Once you know R, what is the area of a quarter circle with that radius? • The triangle inside has both legs equal to the radius and a right angle between them; what is its area, and what happens when you subtract that from the sector area?

✅ Before You Answer

• Be sure you actually convert circumference to radius R first, not directly to area. • Confirm the central angle of the sector is 90°, so its area is ( \frac{1}{4}πR^2 ). • Confirm the triangle is a right isosceles triangle with legs = R, so its area is ( \frac{1}{2}R^2 ).