Before you read

Answer first — see what you already know.

A hotel lobby has a rectangular floor (20 m $\times$ 10 m) with a circular ornamental fountain (radius 3 m) in the center. What area of the floor is available for furniture placement? (Use $π \approx 3.14$ )

A
171.74 m $^{2}$
B
181.74 m $^{2}$
C
191.74 m $^{2}$
D
200 m $^{2}$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A warehouse has a cross-section shaped like a rectangle (8 cm wide, 5 cm tall) topped by a semicircular roof with diameter 8 cm. What is the total cross-sectional area? (Use $π \approx 3.14$ )

A
40 cm $^{2}$
B
51.12 cm $^{2}$
C
65.12 cm $^{2}$
D
90.24 cm $^{2}$

Check your answer

Answer:

Theory

Decomposition Strategy

For irregular shapes, decompose them into basic shapes you know:
1. Identify rectangles, triangles, semicircles, trapezoids within the shape
2. Calculate each area separately
3. Add areas if the shape is a combination, subtract if parts are cut out

Composite figure: identify basic shapes within the irregular shape

Alternatively, enclose the shape in a rectangle and subtract the parts outside.

$A_{\text{total}} = A_1 + A_2 - A_{\text{overlap}}$ Add areas, subtract overlap
$A_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}}$ Shaded region strategy

Two main strategies for composite area problems

Example 1

An L-shaped room can be split into two rectangles: one 10 ft $\times$ 6 ft and one 4 ft $\times$ 8 ft. What is the total area?

$A_1 = 10 \times 6 = 60$ ft $^{2}$

$A_2 = 4 \times 8 = 32$ ft $^{2}$

Total $= 60 + 32 = 92$ ft $^{2}$

92 ft

^{2}

Strategy

SAT Pro Tip

On shaded region problems, area of shaded = area of outer shape − area of inner shape. This is the most common composite area question type on the SAT.

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