Year 8 Measurement and Pythagoras' theorem: free practice test with answers

Sample measurement and pythagoras' theorem questions with answers

Try each one before you open the answer.

1. Length and perimeter

   An L-shaped garden bed has the following side lengths, going clockwise from the top-left corner: 5 m, 2 m, 3 m, 4 m, 2 m, and 6 m. Find the perimeter of the garden bed.

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   Key idea: The perimeter is the total distance around the outside of the shape. Add every side length together.

   1. List all six side lengths: 5, 2, 3, 4, 2, 6 (in metres).
   2. Add them:\

   $$5 + 2 + 3 + 4 + 2 + 6 = 22$$

   Answer: The perimeter is $22$ m.

2. Circumference of circles

   (a) A circle has a radius of $7 \text{ cm}$. What is its diameter?

   (b) A circle has a diameter of $30 \text{ m}$. What is its radius?

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   Answer: (a) $14 \text{ cm}$ (b) $15 \text{ m}$

3. Area of special quadrilaterals

   Find the area of a parallelogram with a base of $12 \text{ cm}$ and a perpendicular height of $7 \text{ cm}$.

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   Answer: $84 \text{ cm}^{2}$ ($12 \times 7$)

4. Area of circles

   A circle has a radius of $9$ cm. Find its circumference and area. Give both answers correct to one decimal place.

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   Key idea: Circumference $C = 2\pi r$ and area $A = \pi r^2$, where $r$ is the radius.

   1. Write the radius: $r = 9$ cm.
   2. Find the circumference:\

   $$C = 2\pi r = 2 \times \pi \times 9 = 18\pi \approx 56.5 \text{ cm}$$

   3. Find the area:\

   $$A = \pi r^2 = \pi \times 9^2 = 81\pi \approx 254.5 \text{ cm}^2$$

   Answer: Circumference $\approx 56.5$ cm, Area $\approx 254.5$ cm$^2$

5. Surface area of prisms

   Find the surface area of a rectangular prism that is $8$ cm long, $5$ cm wide and $3$ cm high.

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   Key idea: A rectangular prism has three pairs of identical rectangular faces: $SA = 2(lw + lh + wh)$.

   1. $lw = 8 \times 5 = 40$, $lh = 8 \times 3 = 24$, $wh = 5 \times 3 = 15$
   2. $SA = 2(40 + 24 + 15) = 2 \times 79 = 158 \text{ cm}^2$

   Answer: $158 \text{ cm}^2$

6. Volume of prisms and cylinders

   A triangular prism has a right-angled triangular cross-section with base $6$ cm and perpendicular height $4$ cm. The prism is $10$ cm long. Find its volume.

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   Key idea: The volume of any right prism is $V = A_{\text{cross-section}} \times \text{length}$.

   1. Find the area of the triangular cross-section:\

   $$A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$$

   2. Multiply by the length of the prism:\

   $$V = 12 \times 10 = 120 \text{ cm}^3$$

   Answer: $120 \text{ cm}^3$

7. Units of time and time zones

   Convert these times: (a) 7:45 am to 24-hour time, (b) 20:30 to 12-hour time.

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   Key idea: In 24-hour time, hours run from 00 to 23. For pm times after 12 noon, add 12 to the hour. For 24-hour times from 13:00 onwards, subtract 12 and write pm.

   1. (a) 7:45 am is before noon, so the hour stays the same. Write the hour with two digits: $07{:}45$.
   2. (b) $20 - 12 = 8$, so 20:30 is 8:30 pm.

   Answer: (a) $07{:}45$ and (b) 8:30 pm

8. Using Pythagoras' theorem

   A right-angled triangle has shorter sides of length $9$ m and $12$ m. Find the length of the hypotenuse.

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   Key idea: In a right-angled triangle, $c^2 = a^2 + b^2$, where $c$ is the hypotenuse.

   1. Substitute the known sides: $c^2 = 9^2 + 12^2$
   2. Evaluate: $c^2 = 81 + 144 = 225$
   3. Take the square root: $c = \sqrt{225} = 15$

   Answer: The hypotenuse is $15$ m.

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 8 area and perimeter worksheet
• Year 8 circles worksheet
• Year 8 volume worksheet
• Year 8 time and time zones worksheet
• Year 8 pythagoras' theorem worksheet
• Year 8 properties of quadrilaterals worksheet