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Calculate a hypotenuse from two perpendicular lengths · Check your progress
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Calculate a shorter side from a hypotenuse and another side · Check your progress
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Test the Pythagorean equality using the longest side · Check your progress
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Choose a right-triangle ratio and calculate a side · Check your progress
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Choose a right-triangle ratio and calculate an acute angle · Check your progress
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Calculate a trigonometric height difference and the height from its reference · Check your progress
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Calculate a side from opposite angle-side pairs using the sine rule · Check your progress
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Find every possible triangle angle from sine-rule givens · Check your progress
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Calculate the opposite side from two sides and their included angle · Check your progress
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Calculate an acute or obtuse angle from three triangle sides · Check your progress
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Calculate triangle area from two sides and their included angle · Check your progress
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Calculate the requested distance and bearing in a two-leg journey · Check your progress
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Calculate the angle between a line and a plane in a cuboid or right pyramid · Check your progress
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Calculate a hypotenuse from two perpendicular lengths · Check again
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Calculate a shorter side from a hypotenuse and another side · Check again
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Test the Pythagorean equality using the longest side · Check again
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Choose a right-triangle ratio and calculate a side · Check again
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Choose a right-triangle ratio and calculate an acute angle · Check again
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Calculate a trigonometric height difference and the height from its reference · Check again
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Calculate a side from opposite angle-side pairs using the sine rule · Check again
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Find every possible triangle angle from sine-rule givens · Check again
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Calculate the opposite side from two sides and their included angle · Check again
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Calculate an acute or obtuse angle from three triangle sides · Check again
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Calculate triangle area from two sides and their included angle · Check again
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Calculate the requested distance and bearing in a two-leg journey · Check again
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Calculate the angle between a line and a plane in a cuboid or right pyramid · Check again
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Retention check
Come back later to see whether your learning has lasted.
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Calculate a hypotenuse from two perpendicular lengths · Retention check
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Calculate a shorter side from a hypotenuse and another side · Retention check
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Test the Pythagorean equality using the longest side · Retention check
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Choose a right-triangle ratio and calculate a side · Retention check
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Choose a right-triangle ratio and calculate an acute angle · Retention check
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Calculate a trigonometric height difference and the height from its reference · Retention check
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Calculate a side from opposite angle-side pairs using the sine rule · Retention check
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Find every possible triangle angle from sine-rule givens · Retention check
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Calculate the opposite side from two sides and their included angle · Retention check
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Calculate an acute or obtuse angle from three triangle sides · Retention check
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Calculate triangle area from two sides and their included angle · Retention check
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Calculate the requested distance and bearing in a two-leg journey · Retention check
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Calculate the angle between a line and a plane in a cuboid or right pyramid · Retention check
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Practise: Calculate a hypotenuse from two perpendicular lengths
Practise questions on the skills you are learning.
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Practise: Calculate a shorter side from a hypotenuse and another side
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Practise: Test the Pythagorean equality using the longest side
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Practise: Choose a right-triangle ratio and calculate a side
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Practise: Choose a right-triangle ratio and calculate an acute angle
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Practise: Calculate a trigonometric height difference and the height from its reference
Practise questions on the skills you are learning.
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Practise: Calculate a side from opposite angle-side pairs using the sine rule
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Practise: Find every possible triangle angle from sine-rule givens
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Practise: Calculate the opposite side from two sides and their included angle
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Practise: Calculate an acute or obtuse angle from three triangle sides
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Practise: Calculate triangle area from two sides and their included angle
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Practise: Calculate the requested distance and bearing in a two-leg journey
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Practise: Calculate the angle between a line and a plane in a cuboid or right pyramid
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Next step
Revision notes
Choose trigonometric rules from the given triangle data in G3 Mathematics, then interpret bearings and three-dimensional lengths correctly.
Key ideas
- Pythagoras and its converse
Use the side opposite a right angle and test whether three lengths form a right triangle.
- Right-triangle sine, cosine and tangent
Name sides relative to the chosen acute angle and choose a ratio using the available quantities.
- Sine rule, cosine rule and triangle area
Choose a rule from the given sides and included angles, and check ambiguous sine-rule solutions.
- Bearings and three-dimensional triangles
Draw the relevant plane triangle and keep horizontal projections distinct from sloping lengths.
From the G3 Mathematics 2027 syllabus
- Use of Pythagoras’ theorem
- Determining whether a triangle is right-angled given the lengths of three sides
- Use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angled triangles
- Extending sine and cosine to obtuse angles
- Use of the formula ½ab sin C for the area of a triangle
- Use of sine rule and cosine rule for any triangle
- Problems in two and three dimensions including those involving angles of elevation and depression and bearings