Free Grade 3 Working with Cosine Rule Worksheets
Free Grade 3 worksheets for Working with Cosine Rule: 12 real questions with a full answer key and worked solutions. In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). Trigonometry links the sides to the angles.
Free
Stretch
15 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- UnderstandExplain working with cosine rule
Explain working with cosine rule accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate working with cosine rule
Calculate working with cosine rule accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare working with cosine rule
Compare working with cosine rule accurately in a familiar context.
Assessed by: Exit ticket of four short items
- EvaluateJustify working with cosine rule
Justify working with cosine rule accurately in a familiar context.
Assessed by: Exit ticket of four short items
Prerequisites
Close these gaps first — each has its own free lesson, worksheet and quiz.
How Working with Cosine Rule works
In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). Trigonometry links the sides to the angles.
12 practice questions
Easy to hard, 24 marks in total. Try them first, then open the answers.
- 1.A right-angled triangle has shorter sides 3 cm and 4 cm. Find the hypotenuse.[1]
- 2.A right-angled triangle has shorter sides 5 cm and 12 cm. Find the hypotenuse.[1]
- 3.A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.[1]
- 4.A right-angled triangle has shorter sides 8 cm and 15 cm. Find the hypotenuse.[1]
- 5.A right-angled triangle has hypotenuse 15 cm and one side 9 cm. Find the other side.[2]
- 6.A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side.[2]
- 7.A right-angled triangle has hypotenuse 25 cm and one side 7 cm. Find the other side.[2]
- 8.A right-angled triangle has hypotenuse 17 cm and one side 8 cm. Find the other side.[2]
- 9.In a right-angled triangle, the side opposite angle θ is 9 cm and the hypotenuse is 15 cm. Find θ to 1 d.p.[3]
- 10.In a right-angled triangle, the side opposite angle θ is 6 cm and the hypotenuse is 10 cm. Find θ to 1 d.p.[3]
- 11.In a right-angled triangle, the side opposite angle θ is 8 cm and the hypotenuse is 17 cm. Find θ to 1 d.p.[3]
- 12.In a right-angled triangle, the side opposite angle θ is 3 cm and the hypotenuse is 5 cm. Find θ to 1 d.p.[3]
Show answers and working
1. 5 cm
c² = a² + b² = 9 + 16 = 25. c = √25 = 5 cm.
2. 13 cm
c² = a² + b² = 25 + 144 = 169. c = √169 = 13 cm.
3. 10 cm
c² = a² + b² = 36 + 64 = 100. c = √100 = 10 cm.
4. 17 cm
c² = a² + b² = 64 + 225 = 289. c = √289 = 17 cm.
5. 12 cm
b² = c² − a² = 225 − 81 = 144. b = √144 = 12 cm.
6. 12 cm
b² = c² − a² = 169 − 25 = 144. b = √144 = 12 cm.
7. 24 cm
b² = c² − a² = 625 − 49 = 576. b = √576 = 24 cm.
8. 15 cm
b² = c² − a² = 289 − 64 = 225. b = √225 = 15 cm.
9. 36.9°
sin θ = opposite / hypotenuse = 9/15. θ = sin⁻¹(0.6) = 36.9°.
10. 36.9°
sin θ = opposite / hypotenuse = 6/10. θ = sin⁻¹(0.6) = 36.9°.
11. 28.1°
sin θ = opposite / hypotenuse = 8/17. θ = sin⁻¹(0.4706) = 28.1°.
12. 36.9°
sin θ = opposite / hypotenuse = 3/5. θ = sin⁻¹(0.6) = 36.9°.
Worked examples
A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.
- c² = a² + b² = 81 + 144 = 225.
- c = √225 = 15 cm.
- Answer: 15 cm
A right-angled triangle has hypotenuse 10 cm and one side 6 cm. Find the other side.
- b² = c² − a² = 100 − 36 = 64.
- b = √64 = 8 cm.
- Answer: 8 cm
In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 25 cm. Find θ to 1 d.p.
- sin θ = opposite / hypotenuse = 7/25.
- θ = sin⁻¹(0.28) = 16.3°.
- Answer: 16.3°
Interactive practice
Free, no account required — answer on screen or print it.
Common mistakes
What learners typically get wrong, and how to fix it.
Adds the squares when finding a shorter side.
Correction: Shorter side: subtract from the hypotenuse squared.
Forgets to square root at the end.
Correction: c² is not c — finish with √.
Teacher tips
- · Label the hypotenuse first every time.
Parent tips
- · Measure a phone screen's sides and check the diagonal.
Real-life applications
- · Ladder length against a wall, TV screen sizes.
Assessment objectives
How mastery is evidenced, and which standards it maps to.
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Frequently asked
- Is this Grade 3 Working with Cosine Rule worksheet really free?
- Yes. Every worksheets on FreeMathWorksheet is free to use, print and share — no account, no paywall and no watermark.
- What should a learner already know before this?
- Start with Introducing Cosine Rule, Sine Rule. Each one has its own free lesson, worksheet and quiz.
- How is the worksheet sequenced?
- Recognition first, then understanding, application, reasoning and finally mastery — the same ladder used across every free resource on the platform.
- What comes next after Working with Cosine Rule?
- Move on to Cosine Rule in Context, Sine Rule, Area of a Triangle, which build directly on this idea.
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