Perimeter and Area UPCAT Reviewer: Composite Figures and Shaded Regions

TEACHER ABI UPCAT MATHEMATICS

Perimeter, Area, and Composite Figures

Decide whether a problem needs boundary length or covered space, break composite figures into familiar parts, and avoid counting hidden or shared edges.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Perimeter and Area

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Perimeter follows the boundary; area covers the region

Perimeter measures distance around a figure. Area measures the amount of surface inside it.

For composite figures, either add nonoverlapping pieces or subtract a removed region from a larger familiar shape.

Composite area = large area − removed area

Rectangle

A = lw; P = 2l + 2w

Triangle

A = (12)bh, using the perpendicular height

Trapezoid

A = (12)(b₁ + b₂)h

Circle

A = πr²; C = 2πr = πd

Scaling

If lengths scale by k, perimeter scales by k and area by k².

DO IT FAST

Trace for perimeter; shade for area

1. Mark what is being measured. Fence, trim, frame, and distance around signal perimeter. Tile, paint, cover, and region signal area.

2. Complete missing lengths. In step-shaped figures, total horizontal and vertical movement must balance.

3. Split or subtract. Use rectangles, triangles, and circle parts whose formulas you know.

4. Remove shared internal edges from perimeter.

Why it works

This prevents formula-first errors, especially counting an internal divider as part of the outside boundary or adding areas that overlap.

WORKED EXAMPLES

Five forms you should recognize

1. Composite floor

Area = 14(8)−5(2) = 102 m²

For perimeter, first recover the missing 9-m and 2-m sides.

2. Path around a garden

Path area = 20(10)−18(8) = 56 m²

3. Rectangle and semicircle

The diameter is shared internally. Include it for neither the outside perimeter nor as an extra area.

4. Inscribed circle

Shaded area = 12²−π(6²) = 144−36π

COMMON TRAPS

Check before you commit

• Using perimeter when the question asks for coverage
• Using diameter as radius
• Counting a shared internal edge in the outside perimeter
• Forgetting to subtract a cutout or unpainted region
• Using slanted side length as a triangle’s height
• Scaling area by k instead of k²

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

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A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Perimeter, Area, and Composite Figures FAQ

Should I use 3.14, 22/7, or π?

Follow the instruction. If none is given and choices use π, leave the answer in terms of π.

Why is the shared diameter excluded from the window perimeter?

It lies inside the combined figure rather than on its outside boundary.

Can two shapes have the same perimeter but different areas?

Yes. Perimeter does not uniquely determine area.

RELATED COMPETENCIES

Continue your mathematics review.

Quadrilaterals and PolygonsCirclesSimilar TrianglesSurface Area and Volume

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