Calculate the area of a ring where the inner circle's radius is half the outer circle's radius.

Question

ScanSolve AI Tutor · Accepted Answer

Let's break down this explanation of finding the area of a ring.

The core idea is that a ring is like a flat donut shape – it's a larger circle with a smaller circle removed from its center. So, to find the area of just the ring, you calculate the area of the big outer circle and then subtract the area of the small inner hole.

Big Area = πR²
- This is the standard formula for the area of a circle: A = πr², where 'r' is the radius.
- Here, the Big Circle has a radius of R. So, its area is simply πR².
Small Hole Area = π(R/2)² = πR²/4
- This is the crucial part. It implies that the radius of the Small Hole is half the radius of the Big Circle, or R/2.
- Using the area formula A = πr² for the small hole, we substitute r = (R/2): Area = π × (R/2)²
- When you square (R/2), you square both the R and the 2: (R/2)² = (R × R) / (2 × 2) = R²/4
- So, the Small Hole Area becomes πR²/4.
Target Area (A) = πR² - πR²/4 = (3/4)πR²
- Now we subtract the area of the small hole from the area of the big circle: A = (Big Area) - (Small Hole Area) A = πR² - πR²/4
- To subtract these, think of πR² as (4/4)πR². A = (4/4)πR² - (1/4)πR²
- Subtracting the fractions gives: A = (4 - 1)/4 × πR² A = (3/4)πR²
So, the area of the ring is three-quarters of the area of the big circle, because the hole takes up one-quarter of that area.

Calculate the area of a ring where the inner circle's radius is half the outer circle's radius.

Step-by-step solution

Big Area = πR²

Small Hole Area = π(R/2)² = πR²/4

Target Area (A) = πR² - πR²/4 = (3/4)πR²

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