Math: Geometry
Ontario Mathematics Proficiency Test — study tool.
Scroll down past the quiz for the complete study guide on this topic.
Scroll down past the quiz for the complete study guide on this topic.
1
A CFL football field is 60 m wide and 100 m long. What is the shortest distance from one corner to the opposite corner?
2
A parallelogram with base 12 m, height 11 m, and side 15 m is cut into two congruent triangles. What is the area of each triangle?
3
A cylinder has diameter 8 cm and height 25 cm. What is the total surface area?
4
A shape is made of a trapezoid with a semicircle removed. The top = 14 cm, sides = 5 cm, semicircle diameter = 6 cm. What is closest to the area of the shaded parts?
5
A triangular prism has a triangle base of 10 cm, triangle height of 8 cm, and length of 12 cm. What is the volume?
6
A composite shape has a parallelogram (base 23 cm) and a triangle (base 41 cm, height 36 cm). What is the total area?
7
Which set of measurements could be the side lengths of a right triangle?
8
An L-shaped object has an upper prism (36 × 48 cm) and lower prism (60 × 48 cm), total height 96 cm, lower height 36 cm. What is the volume?
9
Two fence sections of 6.2 m and two of 3.1 m. Fence costs $4.17/m. What is the total cost?
10
A right triangle has a hypotenuse of 73 cm and a base of 55 cm. What is the height?
11
Two cylinders have the same height (8 cm). The larger radius is 3× the smaller (r = 3 cm). What is the approximate difference in surface area?
12
A composite shape has a semicircle and a triangle. The left side is 6 cm, the bottom is 12 cm. What is the perimeter?
13
A rectangular sandbox is 9 m × 6 m × 2 m. It currently has 40 m³ of sand. How much more is needed to fill it?
14
Which rectangle has the smallest perimeter? All have the same area (1,200 m²).
0
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Quiz Complete!
Study Guide: Geometry
PYTHAGOREAN THEOREM
For any RIGHT triangle (one angle = 90°):
a² + b² = c²
where c is the HYPOTENUSE (the longest side, opposite the right angle)
and a, b are the other two sides.
Finding a missing side:
Know hypotenuse and one side → other side = √(c² - a²)
Know both short sides → hypotenuse = √(a² + b²)
Example: Hypotenuse = 73 cm, base = 55 cm. Find height.
h² = 73² - 55² = 5,329 - 3,025 = 2,304
h = √2,304 = 48 cm
Example: Field is 60 m × 100 m. Diagonal = ?
d = √(60² + 100²) = √(3,600 + 10,000) = √13,600 ≈ 116.6 m
IDENTIFYING RIGHT TRIANGLES
To check if three sides form a right triangle, test if a² + b² = c²
(where c is the largest number).
Example: 9, 12, 15
9² + 12² = 81 + 144 = 225 = 15² ✓ → right triangle
Common Pythagorean triples (worth memorizing):
3, 4, 5 (and multiples: 6,8,10 and 9,12,15 and 12,16,20)
5, 12, 13
8, 15, 17
AREA FORMULAS
Rectangle: A = length × width Parallelogram: A = base × height (NOT base × side) Triangle: A = 1/2 × base × height Circle: A = π × r² Semicircle: A = 1/2 × π × r² Trapezoid: A = 1/2 × (top + bottom) × height IMPORTANT: Height is always PERPENDICULAR to the base (straight up), not along the slanted side.
AREA OF A TRIANGLE FROM A PARALLELOGRAM
A diagonal cuts a parallelogram into two congruent triangles.
Parallelogram area = base × height
Each triangle = half the parallelogram
Example: Parallelogram with base 12 m, height 11 m.
Parallelogram area = 12 × 11 = 132 m²
Each triangle = 132 / 2 = 66 m²
AREA OF COMPOSITE SHAPES
Break the shape into simpler shapes, find each area, then add or subtract.
Shape with a hole: total area = outer shape - hole
Shape made of parts: total area = shape 1 + shape 2
Example: Trapezoid with a semicircle removed.
Area = trapezoid area - semicircle area
Example: Parallelogram + triangle together.
Area = parallelogram area + triangle area
PERIMETER
Perimeter = total distance around the outside of a shape.
Rectangle: P = 2(length + width)
Example: Which rectangle has the smallest perimeter?
10 × 120: P = 2(10 + 120) = 260
30 × 40: P = 2(30 + 40) = 140 ← smallest
50 × 24: P = 2(50 + 24) = 148
60 × 40: P = 2(60 + 40) = 200
TIP: For a given area, the most "square-like" rectangle has the
smallest perimeter.
PERIMETER OF COMPOSITE SHAPES
Add up all the outer edges. For curved parts:
Full circle circumference: C = 2πr = πd Semicircle arc: C = πr (half the circumference, no diameter) Trace the outside of the shape with your finger — every edge you touch gets added to the perimeter.
VOLUME FORMULAS
Rectangular prism: V = length × width × height
Triangular prism: V = (1/2 × base × height of triangle) × length
Cylinder: V = π × r² × height
Example: Sandbox is 9 m × 6 m × 2 m.
V = 9 × 6 × 2 = 108 m³
Currently has 40 m³. Needs 108 - 40 = 68 m³ more.
Example: Triangular prism, triangle base 10 cm, triangle height 8 cm,
prism length 12 cm.
Triangle area = 1/2 × 10 × 8 = 40 cm²
Volume = 40 × 12 = 480 cm³
VOLUME OF COMPOSITE SHAPES
Break into simpler prisms, find each volume, then add.
Example: L-shaped object = two rectangular prisms stacked.
Upper prism volume + lower prism volume = total volume
SURFACE AREA OF CYLINDERS
SA = 2πr² + 2πrh
Where:
2πr² = area of the two circular ends
2πrh = area of the curved side (like unrolling a label)
IMPORTANT: They might give you the DIAMETER, not the radius.
Radius = diameter / 2
Example: Diameter 8 cm, height 25 cm.
Radius = 4 cm
SA = 2π(4²) + 2π(4)(25) = 2π(16) + 2π(100) = 32π + 200π = 232π ≈ 728.8 cm²
Comparing two cylinders: calculate each SA separately, then subtract.
FENCING / COST PROBLEMS
These combine perimeter with unit cost.
Step 1: Find the perimeter (total length)
Step 2: Multiply by cost per unit length
Example: Two sides of 6.2 m and two of 3.1 m, fence costs $4.17/m.
Perimeter = 2(6.2) + 2(3.1) = 18.6 m
Cost = 18.6 × $4.17 = $77.56
KEY FORMULAS SUMMARY
Pythagorean theorem: a² + b² = c² Triangle area: A = 1/2 × base × height Parallelogram area: A = base × height Rectangle perimeter: P = 2(l + w) Rectangular prism volume: V = l × w × h Triangular prism volume: V = (1/2 × b × h) × length Cylinder surface area: SA = 2πr² + 2πrh Semicircle area: A = 1/2 × πr² Always check: did they give diameter or radius?
