Find a formular for the area of the haded region, which represents one-fourth of an annulus (ring).

Question

Sally Cresswell · Accepted Answer

Formula:
Area of circle:
If r is the radius of circle, then area of circle is given by the formula:
$A_{circle} = π r^{2}$ .
Calculation:
To find the area of shaded region, let's find the area of annulus (ring) between the two circles.
The area of ring between the two circles can be determined by subtracting the area of inner circle from the area of outer circle.
$A_{ring} = A_{outer circle} = A_{inner circle}$
The radius of outer circle is R.
So, the area of outer circle $= π R^{2}$ .
The radius of inner circle is r.
So, the area of inner circle $= π r^{2}$
$A_{ring} = A_{outer circle} - A_{inner circle}$
$A_{ring} = π R^{2} - π r^{2}$
Simplify: $A_{ring} = π (R^{2} - r^{2})$
The shaded region represents one-fourth of the area of ring.
Thus, $A_{shaded region = \frac{A_{ring}}{4}}$
$A_{shaded region} = \frac{π (R^{2} - r^{2})}{4}$
Answer: The formula for area of shaded region $== \frac{π (R^{2} - r^{2})}{4}$

Find a formular for the area of the haded region,

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