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MTH1W Geometry & Measurement
Grade 9 Unit 5 — practice questions with full solutions

Pythagorean theorem, angle relationships, area and perimeter of composite shapes, surface area and volume of prisms, cylinders and cones.

Free MTH1W geometry & measurement practice for Grade 9 students in Ontario. These 15 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MTH1W unit test, catching up on a lesson, or reviewing before the final exam.
MTH1W · Grade 9 Strand E — Geometry and Measurement 14 core 1 challenge
1Core
A right triangle has legs of 9 cm and 12 cm. Find the length of the hypotenuse.
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  1. Use a2 + b2 = c2.
  2. Substitute: 92 + 122 = 81 + 144 = 225.
  3. Take the square root: c = √225 = 15.
Final answer15 cm
2Core
A right triangle has a hypotenuse of 26 cm and one leg of 10 cm. Find the other leg.
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  1. Rearrange: b2 = c2 − a2.
  2. Substitute: 262 − 102 = 676 − 100 = 576.
  3. Take the square root: b = √576 = 24.
  4. Note the hypotenuse must be the largest side — always subtract, don't add, when it is given.
Final answer24 cm
3Core
15 cm8 cmc = ?
Find the length of the hypotenuse c in the right triangle shown.
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  1. The small square at the corner marks the right angle, so the Pythagorean theorem applies.
  2. Use a2 + b2 = c2 with the two legs 8 and 15.
  3. Substitute: 82 + 152 = 64 + 225 = 289.
  4. Take the square root: c = √289 = 17.
Final answerc = 17 cm
4Core
The angles of a triangle are in the ratio 3 : 4 : 5. Find the three angles.
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  1. Let the angles be 3x, 4x and 5x.
  2. Angles in a triangle sum to 180°: 3x + 4x + 5x = 180.
  3. Simplify: 12x = 180, so x = 15.
  4. Multiply back: 45°, 60°, 75°. Check: they add to 180°. ✔
Final answer45°, 60° and 75°
5Core
Two parallel lines are cut by a transversal. One interior angle measures 65°. Find the co-interior (same-side interior) angle, and explain your reasoning.
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  1. Co-interior angles lie on the same side of the transversal, between the parallel lines.
  2. When the lines are parallel, co-interior angles are supplementary — they add to 180°.
  3. So the missing angle is 180° − 65°.
Final answer115° (co-interior angles are supplementary)
6Core
12 cm5 cm5 cm3 cm7 cm8 cm
For the L-shaped figure above, find (a) its perimeter and (b) its area.
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  1. (a) Perimeter means adding every outside edge: 12 + 5 + 5 + 3 + 7 + 8.
  2. That gives 40 cm.
  3. (b) For the area, think of the full 12 × 8 rectangle with a piece cut out of the top right.
  4. Full rectangle: 12 × 8 = 96 cm2. The missing corner piece is 5 × 3 = 15 cm2.
  5. Subtract: 96 − 15 = 81 cm2.
  6. Alternative method: split into a 12 × 5 rectangle (60) plus a 7 × 3 rectangle (21), giving 81 cm2 as well. ✔
Final answer(a) 40 cm   (b) 81 cm2
7Core
A shape is made of a rectangle 10 cm by 6 cm with a semicircle of radius 3 cm attached to the top. Find the total area, rounded to two decimal places.
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  1. Area of the rectangle: 10 × 6 = 60 cm2.
  2. Area of a full circle of radius 3: π(3)2 = 9π.
  3. A semicircle is half of that: 4.5π ≈ 14.14 cm2.
  4. Add the two parts together: 60 + 14.14.
Final answer≈ 74.14 cm2
8Core
Find the volume of a cylinder with radius 5 cm and height 12 cm. Leave your answer in terms of π, then give a decimal to one decimal place.
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  1. Use V = πr2h.
  2. Substitute: π(5)2(12) = π(25)(12).
  3. Simplify: 300π.
  4. As a decimal: 300 × 3.14159… ≈ 942.5.
Final answer300π ≈ 942.5 cm3
9Core
72°abcL₁L₂L₁ ∥ L₂
In the diagram, L₁ is parallel to L₂ and one angle measures 72°. Find angles a, b and c, giving the angle relationship you used each time.
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  1. Angle a and the 72° angle sit on a straight line, so they are supplementary: a = 180° − 72° = 108°.
  2. Angle b is between the parallel lines on the opposite side of the transversal from 72° — it is the alternate interior angle, so b = 72°.
  3. Angle c is on a straight line with b, so c = 180° − 72° = 108°.
  4. Sanity check: c is also the co-interior angle with 72°, and co-interior angles are supplementary. ✔
Final answera = 108° (supplementary), b = 72° (alternate interior), c = 108° (co-interior)
10Core
Find the surface area of a rectangular prism measuring 8 cm by 5 cm by 3 cm.
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  1. A rectangular prism has three pairs of identical faces.
  2. Face areas: 8×5 = 40, 8×3 = 24, 5×3 = 15.
  3. Add one of each: 40 + 24 + 15 = 79.
  4. Double it for the matching pairs: 2(79).
Final answer158 cm2
11Core
Find the volume of a cone with radius 6 cm and height 10 cm, to two decimal places.
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  1. Use V = 1⁄3πr2h.
  2. Substitute: 1⁄3π(6)2(10) = 1⁄3π(36)(10).
  3. Simplify: 360⁄3π = 120π.
  4. As a decimal: 120 × 3.14159… ≈ 376.99.
Final answer120π ≈ 376.99 cm3
12Core
r = 4 cmh = 9 cm
For the cylinder shown, calculate (a) its volume and (b) its total surface area. Give both in terms of π and as decimals to one decimal place.
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  1. (a) Volume of a cylinder: V = πr2h = π(4)2(9) = π(16)(9) = 144π.
  2. As a decimal: 144 × 3.14159… ≈ 452.4 cm3.
  3. (b) Surface area = two circular ends + the curved side: SA = 2πr2 + 2πrh.
  4. Ends: 2π(4)2 = 32π. Curved side: 2π(4)(9) = 72π.
  5. Add: 32π + 72π = 104π ≈ 326.7 cm2.
Final answer(a) 144π ≈ 452.4 cm3   (b) 104π ≈ 326.7 cm2
13Core
A square-based prism has a volume of 500 cm3 and a height of 5 cm. Find the side length of its square base.
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  1. Volume of a prism = base area × height.
  2. So base area = 500 ÷ 5 = 100 cm2.
  3. The base is a square, so side2 = 100.
  4. Take the square root: side = 10 cm.
Final answer10 cm
14Core
A ladder 13 m long leans against a wall with its base 5 m from the wall. How far up the wall does the ladder reach?
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  1. The wall, ground and ladder form a right triangle, with the ladder as the hypotenuse.
  2. Apply the Pythagorean theorem: height2 = 132 − 52.
  3. Calculate: 169 − 25 = 144.
  4. Take the square root: 12.
Final answer12 m
15Challenge
A grain silo is a cylinder of radius 3 m and height 10 m, topped by a cone of radius 3 m and height 4 m. Find the total volume in terms of π, then to one decimal place.
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  1. Cylinder volume: π(3)2(10) = 90π.
  2. Cone volume: 1⁄3π(3)2(4) = 1⁄3π(36) = 12π.
  3. Add the two parts: 90π + 12π = 102π.
  4. As a decimal: 102 × 3.14159… ≈ 320.4.
Final answer102π ≈ 320.4 m3

MTH1W Geometry & Measurement — common questions

Short answers to the things students ask most about this unit.

When do you add and when do you subtract in the Pythagorean theorem?

Add the two legs squared to find the hypotenuse. If the hypotenuse is already given, subtract the known leg squared from it.

How do you find the area of a composite shape?

Split it into rectangles, triangles and parts of circles, or take a full rectangle and subtract the missing piece. Both methods should give the same answer.

More MTH1W practice — grade 9 math

MTH1W Number SenseMTH1W Algebraic Expressions & Exponent LawsMTH1W Solving Linear EquationsMTH1W Linear Relations & GraphingAll MTH1W units
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