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Prealgebra / LEVEL 12 · DIFFICULTY 4/5

Perimeter, Area & Circles

Choose appropriate measurements and decompose composite regions.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST 12.1Segments & PerimeterRead the lesson
MINI QUEST 12.2Area & Composite RegionsRead the lesson
MINI QUEST 12.3Circles, Arcs & SectorsRead the lesson
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STAGE 12.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Segments & Perimeter

Useful preparation: Signed Numbers & Order of Operations · Expressions & Substitution

Goal: Add lengths along a path and distinguish boundary from area.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Segment addition depends on order

If B lies between A and C on a straight line, $\displaystyle AB + BC = AC$ . On a number line, distance is the absolute difference of coordinates.

distance(a,b) =

\displaystyle \left|b - a\right|

Perimeter follows the boundary

Add each exterior edge exactly once. Shared edges between joined shapes lie inside the union and are not part of its perimeter.

rectangle perimeter = 2(length + width)

Equal area need not mean equal perimeter

Perimeter measures boundary length; area measures covered surface. Changing shape can preserve one while changing the other.

units for perimeter are length units

WORKED EXAMPLE 1

Points have coordinates −4 and 9. Find their distance.

Compute the absolute difference.
$\displaystyle \left|9 - \left(-4\right)\right| = 13$ .

WORKED EXAMPLE 2

Two 4-by-6 rectangles join along a full 6-unit side. Find the union perimeter.

The union is 8 by 6.
Its perimeter is $\displaystyle 2\, \left(8 + 6\right)$ .

Common pitfalls

Possible mix-up: Subtract signed coordinates without taking a magnitude.

Distance is nonnegative.

Possible mix-up: Add perimeters of joined shapes directly.

Remove the shared edge twice from that sum.

Explain it to yourself

Why is a shared edge removed twice when adding the perimeters of two joined shapes?

Preview the eight practice prompts

Problem 1
Find the distance between −7 and 5 on a number line.
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Problem 2
B lies between A and C. $\displaystyle AB = 8.5$ and $\displaystyle AC = 21$ . Find BC.
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Problem 3
A rectangle has length 13 and width 7. Find its perimeter.
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Problem 4
A rectangle has perimeter 54 and length 16. Find its width.
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Problem 5
An equilateral triangle has perimeter 31.5. What is one side length?
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Problem 6
Two squares of side 5 share one complete edge. Find the perimeter of their union.
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Problem 7New context
A rectangular garden is 18 m by 11 m. A 4 m gate replaces part of the fence. How many meters of fence are needed?
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Problem 8New context
A square frame uses 48 cm of wire. It is reshaped into a rectangle with length 17 cm, preserving all wire. Find the rectangle width.
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Open stage 12.1 in the student workspace →

STAGE 12.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Area & Composite Regions

Useful preparation: Segments & Perimeter · Combining & Scaling Fractions

Goal: Use perpendicular heights, decomposition and subtraction to measure regions.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

NOTICE → MARK → REASON

Base and height: a matching pair

The height is the perpendicular distance to the chosen base. This is a separate learning diagram; it does not supply the values for your practice question.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

A rectangle counts rows of unit squares

Rectangle area is length times width. A parallelogram can be cut and rearranged into a rectangle with the same base and perpendicular height.

rectangle or parallelogram: A = bh

A triangle is half a related parallelogram

Two copies of a triangle form a parallelogram. The height is the perpendicular distance to the line of the chosen base, even if it lies outside an obtuse triangle.

triangle: A = bh/2

Build or subtract nonoverlapping regions

Composite area can be found by summing parts or subtracting missing regions from a larger shape. Count overlap only once. A trapezoid uses the average of its parallel bases.

trapezoid:

\displaystyle A = \frac{\left(b_{1} + b_{2}\right)\, h}{2}

WORKED EXAMPLE 1

A triangle has base 15 and perpendicular height 8. Find its area.

Multiply base by height.
Take half of 120.

WORKED EXAMPLE 2

A 10-by-9 rectangle has a 3-by-4 rectangular corner removed. Find the remaining area.

The outer area is 90.
The removed area is 12.

Common pitfalls

Possible mix-up: A slanted side can always replace the height.

The height must be perpendicular to the chosen base.

Possible mix-up: Doubling both lengths doubles area.

Both factors double, so area is multiplied by four.

Explain it to yourself

Draw two decompositions of the same L-shaped region and explain why their areas agree.

Preview the eight practice prompts

Problem 1
A rectangle measures 9 by 14 units. Find its area.
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Problem 2
A triangle has base 18 and perpendicular height 7. Find its area.
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Problem 3
A parallelogram has base 11, slanted side 8 and perpendicular height 6. Find its area.
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Problem 4
A trapezoid has parallel bases 5 and 13, with height 4. Find its area.
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Problem 5
A triangle has area 54 and base 12. Find its perpendicular height.
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Problem 6
Both side lengths of a rectangle are multiplied by 3. By what factor is its area multiplied?
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Problem 7New context
A 16 m by 10 m garden contains a nonoverlapping 4 m by 7 m pond inside it. What area remains for planting?
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Problem 8New context
A banner consists of a 12-by-5 rectangle and a triangle attached along the full 5-unit edge. The triangle extends 4 units perpendicular to that edge. Find the total area.
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Open stage 12.2 in the student workspace →

STAGE 12.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Circles, Arcs & Sectors

Useful preparation: Area & Composite Regions · Triangle & Polygon Angle Sums

Goal: Relate radius, full-circle measures and fractional turns.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

NOTICE → MARK → REASON

Radius, boundary and inside

Mark what your question gives, and which measurement you need. This is a separate learning diagram; it does not supply the values for your practice question.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

Radius and diameter are different lengths

A diameter passes through the center and consists of two radii. Circumference measures the full circular boundary.

\displaystyle d = 2\, r

;

\displaystyle C = 2\, \pi\, r

Circle area scales with radius squared

The area of a circle is $\displaystyle \pi\, r^{2}$ . Doubling the radius doubles circumference but quadruples area. Keep $\displaystyle \pi$ exact unless an approximation is specified.

\displaystyle A = \pi\, r^{2}

A central angle selects a fraction

A sector with central angle $\displaystyle \theta$ uses $\displaystyle \frac{\theta}{360}$ of the circle area; its arc uses the same fraction of the circumference. The sector perimeter also includes two radii.

sector area =

\displaystyle \left(\frac{\theta}{360}\right)\, \pi\, r^{2}

WORKED EXAMPLE 1

A circle has diameter 14. Find its area.

The radius is 7.
Use $\displaystyle \pi \times 7^{2}$ .

$\displaystyle 49\, \pi$

WORKED EXAMPLE 2

$\displaystyle A\, 90^{\circ}$ sector has radius 6. Find its arc length.

The arc is one quarter of the circumference.
$\displaystyle \left(\frac{1}{4}\right)\, \left(2\, \pi \times 6\right) = 3\, \pi$ .

$\displaystyle 3\, \pi$

Common pitfalls

Possible mix-up: Use diameter in $\displaystyle \pi\, r^{2}$ .

Halve the diameter first.

Possible mix-up: A sector perimeter is just its arc length.

Include the two radius segments as well.

Explain it to yourself

Why do circumference and area change by different factors when the radius doubles?

Preview the eight practice prompts

Problem 1
A circle has radius 9. Write its circumference as $\displaystyle k\, \pi$ . Find k.
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Problem 2
A circle has diameter 10. Write its area as $\displaystyle k\, \pi$ . Find k.
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Problem 3
A circle has radius 7. Using $\displaystyle \pi = \frac{22}{7}$ , find its circumference.
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Problem 4
A semicircle has radius 8. Write its area as $\displaystyle k\, \pi$ . Find k.
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Problem 5
$\displaystyle A\, 60^{\circ}$ sector has radius 12. Write its arc length as $\displaystyle k\, \pi$ . Find k.
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Problem 6
$\displaystyle A\, 120^{\circ}$ sector has radius 9. Write its area as $\displaystyle k\, \pi$ . Find k.
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Problem 7New context
A circular path lies between concentric circles of radii 10 m and 6 m. Write the path area as $\displaystyle k\, \pi\, m^{2}$ . Find k.
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Problem 8New context
A wheel has radius 0.5 m and makes 20 complete rotations without slipping. Write its travel distance as $\displaystyle k\, \pi$ meters. Find k.
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Open stage 12.3 in the student workspace →

Browse another level →

Perimeter, Area & Circles

Segments & Perimeter

Segment addition depends on order

Perimeter follows the boundary

Equal area need not mean equal perimeter

Points have coordinates −4 and 9. Find their distance.

Two 4-by-6 rectangles join along a full 6-unit side. Find the union perimeter.

Common pitfalls

Explain it to yourself

Area & Composite Regions

Base and height: a matching pair

A rectangle counts rows of unit squares

A triangle is half a related parallelogram

Build or subtract nonoverlapping regions

A triangle has base 15 and perpendicular height 8. Find its area.

A 10-by-9 rectangle has a 3-by-4 rectangular corner removed. Find the remaining area.

Common pitfalls

Explain it to yourself

Circles, Arcs & Sectors

Radius, boundary and inside

Radius and diameter are different lengths

Circle area scales with radius squared

A central angle selects a fraction

A circle has diameter 14. Find its area.

A 90∘\displaystyle A\, 90^{\circ}A90∘ sector has radius 6. Find its arc length.

Common pitfalls

Explain it to yourself

$\displaystyle A\, 90^{\circ}$ sector has radius 6. Find its arc length.