A composite plane figure is formed from a rectangle ABCD measuring 20\text{ cm} by 14\text… — JAMB UTME

A composite plane figure is formed from a rectangle $ABCD$ measuring $20\text{ cm}$ by $14\text{ cm}$ . A semicircle with diameter $BC = 14\text{ cm}$ is attached externally along side $BC$ . At the opposite end, an isosceles triangle $AED$ with base $AD = 14\text{ cm}$ and perpendicular height $6\text{ cm}$ is cut out from the interior of the rectangle. Taking $\pi = \frac{22}{7}$ , what is the total area of the figure in $\text{cm}^2$ ?

Answer: 315 cm²

Answer

The total area of the composite figure is 315 cm².

The area of the complete shape is found by adding the area of the external semicircle (

77\text{ cm}^2

) to the area of the rectangle (

280\text{ cm}^2

) and subtracting the area of the removed triangle (

42\text{ cm}^2

), yielding a net area of

315\text{ cm}^2

Step-by-Step Solution

Calculate the area of the rectangular portion

ABCD

Area of rectangle

= 20\text{ cm} \times 14\text{ cm} = 280\text{ cm}^2

The main section of the figure before modifications is a rectangle with length

20\text{ cm}

and width

14\text{ cm}

Calculate the area of the externally attached semicircle

Area of semicircle

= \frac{1}{2} \times \frac{22}{7} \times 7^2 = 77\text{ cm}^2

The diameter of the semicircle is

BC = 14\text{ cm}

, giving a radius

r = 7\text{ cm}

. Area is given by

\frac{1}{2}\pi r^2

Calculate the area of the removed internal triangle

AED

Area of triangle

= \frac{1}{2} \times 14\text{ cm} \times 6\text{ cm} = 42\text{ cm}^2

The triangle has base

AD = 14\text{ cm}

and height

6\text{ cm}

. Area is calculated as

\frac{1}{2} \times \text{base} \times \text{height}

Combine the areas to find the total remaining area of the figure

Total Area

= 280\text{ cm}^2 + 77\text{ cm}^2 - 42\text{ cm}^2 = 315\text{ cm}^2

The external semicircle increases the total area while the internal cut-out triangle reduces it.

Key Concept

Area of Composite Plane Figures

Estimated Time:2m 0s

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