A composite plane figure is formed by joining a major sector of a circle of radius 14\text… — JAMB UTME

A composite plane figure is formed by joining a major sector of a circle of radius $14\text{ cm}$ (having a central angle of $270^\circ$ and center $O$ ) to a square $OACB$ . The vertices $A$ and $B$ of the square lie on the circle such that $OA$ and $OB$ serve as the bounding radii of the major sector. What is the total area of the combined figure in $\text{cm}^2$ ? (Take $\pi = \frac{22}{7}$ )

A
$350\text{ cm}^2$
B
$504\text{ cm}^2$
C
$560\text{ cm}^2$
$658\text{ cm}^2$ Answer

Answer

658\text{ cm}^2

The total area of the composite figure is the sum of the non-overlapping regions: the major sector of central angle

270^\circ

(

462\text{ cm}^2

) and the square of side length

14\text{ cm}

(

196\text{ cm}^2

), giving

462 + 196 = 658\text{ cm}^2

Step-by-Step Solution

Calculate the area of the major sector of central angle

270^\circ

and radius

14\text{ cm}

\text{Area}_{\text{sector}} = \frac{270^\circ}{360^\circ} \times \pi r^2 = \frac{3}{4} \times \frac{22}{7} \times 14^2 = \frac{3}{4} \times 616 = 462\text{ cm}^2

The major sector covers

\frac{270}{360} = \frac{3}{4}

of the full circle.

Calculate the area of the square

OACB

with side length

s = OA = 14\text{ cm}

\text{Area}_{\text{square}} = s^2 = 14^2 = 196\text{ cm}^2

The radii

OA

and

OB

form two adjacent sides of the square

OACB

of length

14\text{ cm}

Sum the areas of the major sector and the square to find the total composite area.

\text{Total Area} = 462\text{ cm}^2 + 196\text{ cm}^2 = 658\text{ cm}^2

The major sector and square share boundaries

OA

and

OB

without interior overlap.

Key Concept

Perimeter and Area of Plane Shapes

Estimated Time:2m 0s

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