Unit 6 · GeometryG4 · Area & Volume · Aerodynamics & Navigation

Composite Area & Perimeter

Break unfamiliar regions into familiar pieces and track boundaries carefully.

30 min
Unlocks: Payload Bay

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

composite figure: a figure built from familiar shapes

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Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

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The big idea

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The decision that matters

For area, add or subtract regions; for perimeter, trace only the outside boundary. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

For area, add or subtract regions; for perimeter, trace only the outside boundary. Keep exact values through the setup; round only when the stem asks.

Core relationship

A_{shaded}=A_{whole}-A_{removed}

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Sketch and label each familiar piece before calculating. Desmos can evaluate a long area expression, but it cannot decide which regions are removed or which edges belong to the outside perimeter.

Explore in Desmos

Subtract two specific cutout areas

Use Desmos for arithmetic only after writing whole minus removed region.

The calculator is live: change anything and see what happens.

Subtract two specific cutout areas

Degree mode

Loading Desmos…

Trap

The main trap

A cutout removes area but may add exposed perimeter edges.

For perimeter, trace the boundary with a finger and count only exposed edges. In the 14-by-10 corner-cut example below, the removed outer lengths $4+3$ are replaced by new inner lengths $4+3$ , so the perimeter remains $2(14+10)=48$ even though the area decreases.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

A 14-by-10 rectangle has a 4-by-3 corner removed. What area remains?

Whole area is

140

Which expression gives the area of the removed corner?

The remaining area is

140-12=128

Why is the removed area subtracted?

If the question instead asked for perimeter, trace the L-shaped boundary: the

4+3

units removed from the old outside are replaced by

4+3

units along the cut, so the perimeter is still 48.

Worked example

Example 2: ACT twist

Problem

A square of side 12 contains a circle of radius 6. Find area outside the circle.

Square area is 144.

Which expression gives the circle area that must be removed?

The area outside the circle is

144-36\pi

Why is the circle area $36\pi$ rather than $144\pi$ ?

Faded example · you finish it

Example 3: You finish it

Problem

A 20 by 15 rectangle has two nonoverlapping 3 by 5 holes. Find remaining area.

300-2(15)

Complete the calculation.

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.