A sector of a circle of radius 6\text{ cm} has a total perimeter whose numerical value is … — JAMB UTME

A sector of a circle of radius $6\text{ cm}$ has a total perimeter whose numerical value is equal to the numerical value of its area. What is the area of the sector, in $\text{cm}^2$ ?

Answer: 18 cm^2

Answer

The area of the sector is

18\text{ cm}^2

For a sector of radius

r = 6\text{ cm}

, its total perimeter is

P = 2(6) + s = 12 + s

, where

s

is the arc length. Its area is

A = \frac{1}{2}(6)s = 3s

. Setting

P = A

gives

12 + s = 3s \implies 2s = 12 \implies s = 6\text{ cm}

. Substituting

s = 6

into the area expression gives

A = 3(6) = 18\text{ cm}^2

Step-by-Step Solution

Express the perimeter and area of the sector in terms of the arc length

s

P = 2r + s = 12 + s

and

A = \frac{1}{2}rs = 3s

The total perimeter of a sector includes two straight radii plus the arc length, while its area is given by

\frac{1}{2}rs

Equate the numerical values of perimeter and area

12 + s = 3s

The question specifies that the numerical value of the total perimeter equals the numerical value of its area.

Solve for the arc length

s

2s = 12 \implies s = 6\text{ cm}

Subtracting

s

from both sides yields

2s = 12

, so

s = 6\text{ cm}

Calculate the sector area

A = 3(6) = 18\text{ cm}^2

Substituting

s = 6

into

A = 3s

yields an area of

18\text{ cm}^2

Key Concept

Perimeter and Area of a Circular Sector

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