SAT — mathHard3 min read

Composite Figures and Shaded Regions

Find areas and volumes of complex shapes by adding or subtracting simpler shapes for the Digital SAT.

Tutor AI Team2026-02-25T01:33:50.839Z

Composite figures are shapes formed by combining or removing simpler shapes. Shaded region problems — a favourite of the SAT — ask you to find the area of a region by subtracting one shape from another. These problems combine your knowledge of area formulas with spatial reasoning.

Core Concepts

Strategy for Composite Areas

Identify the simpler shapes that make up the figure.
Calculate each area separately.
Add or subtract as appropriate.

Common Composite Patterns

Rectangle with semicircle on one end: rectangle area + semicircle area.
Square with circle removed: square area − circle area.
Circle with inscribed polygon: circle area − polygon area.
Overlapping shapes: add both areas − overlap area.

Shaded Region Formula

$A_{\text{shaded}} = A_{\text{outer shape}} - A_{\text{inner shape(s)}}$

Worked Example: Example 1

Problem

A rectangle 10 × 6 has a semicircle of diameter 6 attached to one short side. Total area?

$A = 10(6) + \frac{1}{2}\pi(3)^2 = 60 + 4.5\pi \approx 74.1$

Solution

Worked Example: Example 2

Problem

A circle of radius 8 has a square inscribed (corners touching the circle). Find the shaded area outside the square inside the circle.

Square diagonal = circle diameter = 16. Side of square = $\frac{16}{\sqrt{2}} = 8\sqrt{2}$ .

Square area = $(8\sqrt{2})^2 = 128$ .

Circle area = $\pi(8)^2 = 64\pi$ .

Shaded = $64\pi - 128 \approx 73.1$ .

Solution

Worked Example: SAT-Style

Problem

A square of side 12 has four quarter-circles of radius 3 removed from each corner. Find the remaining area.

Four quarter-circles = one full circle of radius 3.

$A = 12^2 - \pi(3)^2 = 144 - 9\pi \approx 115.7$

Solution

Worked Example: Example 4

Problem

A running track is a rectangle 100 m × 60 m with semicircles on each short side. Find the total area enclosed.

$A = 100(60) + \pi(30)^2 = 6000 + 900\pi \approx 8827.4$ m²

(Two semicircles = one full circle with $r = 30$ .)

Solution

Practice Problems

Problem 1

A square of side 20 has a circle of radius 10 inscribed. Find the area between the square and circle.

Problem 2

A rectangle 8 × 5 with a triangle of base 8 and height 3 on top. Total area?

Problem 3

A circle of radius 6 with a regular hexagon inscribed. Side of hexagon = 6. Find shaded area between circle and hexagon.

Problem 4

Two overlapping circles of radius 5, with centres 6 apart. (This is advanced — set up the calculation.)

Want to check your answers and get step-by-step solutions?

Common Mistakes

Subtracting when you should add (or vice versa). Carefully determine if shapes are combined or removed.
Using the wrong radius. For a semicircle on a side of length $d$ , the radius is $d/2$ .
Forgetting to account for all removed shapes. If four corners are cut, that's four pieces.
Not recognising that two semicircles = one full circle.

Key Takeaways

✓
Decompose complex shapes into simpler ones.
✓
Shaded area = outer − inner.
✓
Two semicircles of same radius = one full circle.
✓
Four quarter-circles of same radius = one full circle.
✓
Always draw and label the component shapes.

Tags:

#sat#math#geometry#composite#shaded-regions

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Core Concepts

Strategy for Composite Areas

Common Composite Patterns

Shaded Region Formula

Worked Example: Example 1

Worked Example: Example 2

Worked Example: SAT-Style

Worked Example: Example 4

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Common Mistakes

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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