EQN-04-10

Composite Figures & Shaded Regions

Break complex figures into simple shapes and add or subtract their areas — the key to shaded-region questions.

~6 min read•Difficulty 2/5

Decompose, then combine

Split an irregular figure into rectangles, triangles, and circle parts whose areas you know. Add the pieces for the total.

Shaded $=$ whole $-$ unshaded

Most shaded-region questions are "big shape minus a hole".

square (side 10) with a quarter-circle (radius 10) removed :

shaded = 1 0^{2} - (1/4) \cdot π \cdot 1 0^{2} = 100 - 25 π

Watch for shared dimensions

The radius of an inscribed circle equals half the square's side; the circle's diameter equals the square's side. Read which length the figure actually gives you.

Keep $π$ symbolic

Leave answers like " $100 - 25 π$ " rather than approximating — the GAT-E choices are usually written that way, and it avoids rounding error.

Diagram

Shaded area $=$ rectangle $-$ circle.

Worked examples

Example 1 (difficulty 2/5)

A circle of radius 3 is cut out of a rectangle 10 by 6. What is the shaded (remaining) area? (Leave in terms of

π .)

◎ Predict the answer first:

Example 2 (difficulty 2/5)

An L-shape is made of an

8 \times 8

square with

a 3 \times 3

square removed from a corner. What is its area?

◎ Predict the answer first:

Tips

Identify simple shapes first, then decide what to add and what to subtract.
Shaded region is usually (outer area $-$ inner area).
Leave $π$ in the answer to match the choices and avoid rounding.

Common mistakes

Using the diameter where the radius is needed (or vice versa)

Why: Figures give whichever is convenient to draw.

Fix: Area uses the radius; halve a diameter before squaring.

Forgetting to subtract the removed piece

Why: The whole-shape area feels like the answer.

Fix: A "shaded" or "remaining" region means subtract the cut-out.

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