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Geometry and Measurement Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 50 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Angle pairs: complementary angles add to $90^\circ$ , supplementary angles add to $180^\circ$ . When two lines are marked parallel, corresponding and alternate angles are equal and co-interior angles add to $180^\circ$ .
Polygons with $n$ sides: the interior angles add to $(n-2)\times180^\circ$ . The exterior angles always add to $360^\circ$ , whatever $n$ is.
Pythagorean theorem: $a^2+b^2=c^2$ , where $c$ is the hypotenuse, opposite the right angle. To find a leg, subtract: $a^2=c^2-b^2$ .
Formulas: circle $C=2\pi r$ , $A=\pi r^2$ ; cylinder $SA=2\pi r^2+2\pi rh$ , $V=\pi r^2h$ ; pyramid or cone $V=\frac{1}{3}Bh$ .
A height is always perpendicular to the base. A slanted edge or slant height is not the $h$ in an area or volume formula.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Angles · Polygons · Pythagoras · Area · Volume

K1 Angle Pairs

a)
Find the complement and the supplement of a $38^\circ$ angle.
b)
Lines $\ell$ and $m$ are parallel. Determine the angles $a$ and $b$ .
c)
In $\triangle PQR$ , the exterior angle at $R$ is $131^\circ$ and $\angle P=57^\circ$ . Determine $\angle Q$ and the interior angle $\angle PRQ$ .

Stuck? Peek at a hint

Complementary pairs make a right angle and supplementary pairs make a straight line. In (c), the exterior angle equals the sum of the two interior angles that are not next to it.

K2 Polygon and Circle Angles

a)
Determine the sum of the interior angles and the sum of the exterior angles of an octagon.
b)
Each exterior angle of a regular polygon measures $24^\circ$ . Determine the number of sides and the measure of each interior angle.
c)
In a circle with centre $O$ , the central angle $\angle AOB=130^\circ$ . Point $C$ lies on the circle, on the major arc. Determine the inscribed angle $\angle ACB$ .

Stuck? Peek at a hint

From one vertex, split the polygon into triangles and count them. Walking once around any convex polygon, you turn a total of $360^\circ$ .

K3 Missing Sides

Find each unknown side length.

a)
A right triangle has legs $7$ cm and $24$ cm. Find the hypotenuse.
b)
Find $x$ in the right triangle below.
c)
A right triangle has a hypotenuse of $11$ m and one leg of $6$ m. Find the other leg, to the nearest tenth of a metre.

Stuck? Peek at a hint

Before writing any equation, mark the side opposite the right angle as $c$ .

K4 Perimeter and Area

Round answers to the nearest tenth where needed.

a)
A circle has a diameter of $14$ cm. Find its circumference and its area.
b)
Find the perimeter and the area of the parallelogram below.
c)
A window is a semicircle with a diameter of $80$ cm. Find its area and its perimeter.

Stuck? Peek at a hint

Write down the radius before you use $\pi r^2$ . In (b), the height is the side that meets the base at a right angle.

K5 Volume

Round answers to the nearest tenth where needed.

a)
Find the volume of a cylinder with radius $4$ cm and height $10$ cm.
b)
Find the volume of a cone with the same radius and height as the cylinder in (a).
c)
Find the volume of the square-based pyramid below. Its height is $12$ m and the slant height of each face is $13$ m.

Stuck? Peek at a hint

Find the base area $B$ first. For a cone or a pyramid, $h$ is the perpendicular height from the apex to the base.

Real World (A)

Pythagoras in Context · Surface Area · Best Dimensions

A1 Accessible Ramps

An accessibility rule says a wheelchair ramp may rise at most 1 m for every 12 m of horizontal run.

a)
A library entrance is $0.5$ m above the sidewalk. Find the shortest horizontal run the rule allows, then the length of the ramp surface, to the nearest hundredth of a metre.
b)
An old ramp has a surface $3.4$ m long and a horizontal run of $3.0$ m. How high does it rise?
c)
A rink entrance in Sudbury is $0.9$ m up, with only $8$ m of flat ground in front of it. A student designs a straight ramp of length $\sqrt{8^2+0.9^2}\approx8.05$ m. Can it be built under the rule? What would the rink need instead?

Stuck? Peek at a hint

The ramp surface is the hypotenuse. In (a), use the rule to find the run first.

A2 Maple Syrup Can

A cylindrical can of maple syrup from a sugar bush near Lanark has a radius of $5$ cm and a height of $12$ cm. Round answers to the nearest tenth.

a)
A paper label covers the curved side of the can exactly, with no overlap. Find the area of the label.
b)
Find the area of metal needed to make the can, including the top and the bottom.
c)
The designer has label sheets that are $30$ cm by $15$ cm, an area of $450$ cm², more than the label needs. Can one sheet, cut but not taped, make the label? Justify.

Stuck? Peek at a hint

Unroll the label in your head: it becomes a rectangle. Which length of the can is its width?

A3 Riverside Garden

A farm near Peterborough has $60$ m of fencing for a rectangular garden along a river. The river side needs no fence. The width $w$ is perpendicular to the river and the length $L$ runs along it.

$w\text{ (m)}$	$5$	$10$	$15$	$20$	$25$
$L\text{ (m)}$	$50$
$\text{Area (m}^2\text{)}$	$250$

a)
Write $L$ in terms of $w$ . Then complete the table.
b)
A student says the best garden is a square. Find the area of the square garden that uses all $60$ m of fencing, and compare it with the best garden in your table.
c)
The straight part of the riverbank is only $24$ m long, so $L\le24$ . State the dimensions that give the largest area now, and that area.

Stuck? Peek at a hint

Only three sides need fencing: two widths and one length.

Think It Through (T)

Error Analysis · Explain Why

T1 Spot the Flaw

Lines $\ell$ and $m$ are cut by the transversal $t$ . Nothing in the diagram says $\ell$ and $m$ are parallel. A student writes:
"Alternate interior angles are equal, so $2x+20=3x-10$ . Then $x=30$ , and both angles measure $80^\circ$ ."

a)
Use the $112^\circ$ angle to find the value of $x$ .
b)
Find the angle at $A$ . Are $\ell$ and $m$ parallel? Explain.
c)
Identify the student's error. How does the $112^\circ$ angle show that $x=30$ is impossible?

Stuck? Peek at a hint

Start at $B$ : the $112^\circ$ angle and $(3x-10)^\circ$ sit on a straight line.

T2 Glued Cubes

A student claims: "When you glue two solids together, the surface area of the new solid is the sum of their surface areas." Each cube below has edges of $3$ cm.

a)
Two cubes are glued face to face. Find the sum of their surface areas, and the surface area of the new solid.
b)
Explain why the student's rule overcounts, and by how much in (a).
c)
Four of these cubes are glued face to face into one solid. Find the smallest possible surface area and describe the arrangement that gives it.

Stuck? Peek at a hint

Count only the faces you could still touch from outside the new solid.

Boss Level

Optional challenge

Half-Full Cup

A cone-shaped paper cup has a top radius of $4$ cm and a height of $12$ cm. It is held point down. When the water is $d$ cm deep, the water surface is a circle of radius $\frac{d}{3}$ cm.

a)
Find the volume of the cup, in terms of $\pi$ .
b)
The cup is filled to half its height. What fraction of the cup is full?
c)
Find the depth of water that makes the cup exactly half full, to the nearest tenth of a centimetre.

Stuck? Peek at a hint

Write the volume of the water as a formula in $d$ alone.

Answer key

Final answers only. Try first, then check.

Core Skills

a) Complement $52^\circ$ ; supplement $142^\circ$
b) $a=64^\circ$ , $b=116^\circ$
c) $\angle Q=74^\circ$ , $\angle PRQ=49^\circ$

a) Interior: $1080^\circ$ ; exterior: $360^\circ$
b) $15$ sides; each interior angle $156^\circ$
c) $\angle ACB=65^\circ$

a) $25$ cm
b) $x=15$ cm
c) $\sqrt{85}\approx9.2$ m

a) $C\approx44.0$ cm; $A\approx153.9$ cm²
b) $P=38$ cm; $A=72$ cm²
c) $A\approx2513.3$ cm²; $P\approx205.7$ cm

a) $160\pi\approx502.7$ cm³
b) $\frac{160\pi}{3}\approx167.6$ cm³
c) $400$ m³

Real World

a) Run $6$ m; ramp surface $\approx6.02$ m
b) $1.6$ m
c) No: a $0.9$ m rise needs at least $12\times0.9=10.8$ m of run, and only $8$ m is available. A straight ramp can't fit; the rink needs a ramp with a turn (switchback) or a lift.

a) $120\pi\approx377.0$ cm²
b) $170\pi\approx534.1$ cm²
c) No: unrolled, the label is a rectangle about $31.4$ cm long (the circumference $2\pi(5)$ ) and $12$ cm high. Both sides of the sheet are shorter than $31.4$ cm, so it can't reach all the way round, even though its area is larger.

a) $L=60-2w$ ; $L$ : $50, 40, 30, 20, 10$ ; area: $250, 400, 450, 400, 250$ m²
b) Square: $20$ m by $20$ m, $400$ m². The $15$ m by $30$ m garden gives $450$ m², which is more.
c) $L=24$ m, $w=18$ m; area $432$ m²

Think It Through

a) $3x-10=180-112=68$ , so $x=26$
b) $72^\circ$ . Not parallel: the alternate interior angles are $72^\circ$ and $68^\circ$ , which are not equal.
c) The student assumed $\ell\parallel m$ from the Z shape; alternate angles are equal only when the lines are given as parallel. With $x=30$ , the angles at $B$ would be $80^\circ+112^\circ=192^\circ$ , but a straight line is $180^\circ$ .

a) Sum: $108$ cm²; new solid: $90$ cm²
b) Where the cubes touch, one face of each cube is hidden inside the solid. The rule counts those two $9$ cm² faces, so it is $18$ cm² too high.
c) $144$ cm², a $2\times2$ square slab (four touching pairs hide $8$ faces). A straight row gives $162$ cm².

Boss Level

Boss

a) $64\pi$ cm³
b) $\frac{8\pi}{64\pi}=\frac{1}{8}$
c) $\frac{\pi d^3}{27}=32\pi$ , so $d^3=864$ and $d\approx9.5$ cm

Free cheat sheet

Geometry and Measurement Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

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Build your own

The same page with blanks: fill it in from memory, then check.

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Phone card

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Most testedA $6.5$ m ladder leans against a wall with its foot $2.5$ m from the wall. How high up the wall does it reach? (card 4)

1Angle pairs

F, Z, C: only for lines marked parallel.

complementary	add to $90^\circ$
supplementary (straight line)	add to $180^\circ$
vertically opposite (X)	equal
corresponding (F)	equal
alternate (Z)	equal
co-interior (C)	add to $180^\circ$

TrapCo-interior angle $70^\circ$ : the other is $110^\circ$ , not $70^\circ$ .

2Triangle angles

$\angle A+\angle B+\angle C=180^\circ$

$e=\angle A+\angle B$

Use $\angle A$ , $\angle B$ : the interior angles not next to $e$ .

Trap $e=125^\circ$ and $\angle A=48^\circ$ : $\angle B=125^\circ-48^\circ=77^\circ$ , not $173^\circ$ .

3Polygon and circle angles

$\text{interior sum}=(n-2)\times180^\circ$

$\text{exterior sum}=360^\circ$

$\text{inscribed}=\dfrac{1}{2}\times\text{central}$

UseRegular polygon: divide by $n$ for one angle. Circle: both angles on the same arc.

TrapDecagon: exterior angles add to $360^\circ$ , not $8\times180^\circ$ .

TrapCentral angle $110^\circ$ : inscribed angle $55^\circ$ , not $110^\circ$ .

4Pythagorean theorem

$a^2+b^2=c^2$

$a^2=c^2-b^2$

Use $c$ is the hypotenuse: opposite the right angle, the longest side.

ExampleLadder $6.5$ m, foot $2.5$ m out: $h^2=6.5^2-2.5^2=36$ , so $h=6$ m up the wall.

TrapHypotenuse $13$ , leg $5$ : the other leg is $\sqrt{169-25}=12$ , not $\sqrt{194}$ , longer than $13$ .

5Area of polygons

Use $h$ meets $b$ at $90^\circ$ . Composite: split, then add or subtract.

rectangle	$A=lw$
parallelogram	$A=bh$
triangle	$A=\dfrac{bh}{2}$
trapezoid	$A=\dfrac{(a+b)h}{2}$

TipFixed fencing: a square gives the most area. Along a river (3 sides): length $=2\times$ width.

TrapParallelogram, base $10$ , slanted side $6$ , height $5$ : $A=10\times5=50$ , not $10\times6=60$ .

6Circles

$C=2\pi r=\pi d$

$A=\pi r^2$

$\text{sector: }A=\dfrac{\theta}{360^\circ}\times\pi r^2$

UseFind $r=d\div2$ first. A semicircle's perimeter is half the circumference plus the diameter.

TrapDiameter $10$ cm: $A=\pi(5)^2\approx78.5$ cm², not $\pi(10)^2\approx314.2$ cm².

7Surface area

$\text{cylinder: }SA=2\pi r^2+2\pi rh$

$\text{cone: }SA=\pi r^2+\pi rs$

UsePrism, pyramid: add the areas of the net. $s$ : slant height, along the side.

TrapTwo $2$ cm cubes glued: $48-2\times4=40$ cm², not $48$ cm². Glued faces are hidden.

8Volume

$\text{prism, cylinder: }V=Bh$

$\text{pyramid, cone: }V=\dfrac{1}{3}Bh$

Use $B$ : base area ( $\pi r^2$ for a circle). $h$ : perpendicular height. $1$ cm³ $=1$ mL.

ExampleSilo: cylinder $r=2$ , $h=10$ , cone $h=3$ on top: $40\pi+4\pi=44\pi\approx138.2$ m³.

TrapCone, $r=3$ , $h=4$ , slant $5$ : $V=1/3\times\pi(9)(4)=12\pi$ , not $15\pi$ (slant) or $36\pi$ (no $1/3$ ).

9Changing dimensions

Multiply every length by $k$ :

perimeter	$\times k$
area, surface area	$\times k^2$
volume	$\times k^3$

TipCylinder, cone: $r\times2$ alone gives $V\times4$ ; $h\times2$ alone gives $V\times2$ .

TrapCone filled to half its height: $(1/2)^3=1/8$ full, not $1/2$ .

Which measure?

The distance around a shape Perimeter, in m

The space covering a surface Area, in m²

The space inside a solid Volume, in m³

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