Two sides and the angle between them: When you know two sides and the angle between them (the included angle), the area is ½ × (one side) × (other side) × sin(included angle).
Interactive: tap Area to highlight the two sides (a, b) and the included angle (C) that the formula ½ab·sin C uses.
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C must be the included angle: The angle in the formula must sit between the two sides you use — not just any angle of the triangle.
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Substitute and evaluate: Identify the two sides and the angle between them, drop them into ½ab·sinC, and evaluate.
IB-style question — straight area
A triangle has sides 6 and 8 with an included angle of 30°. Find its area.
Step by step
- Substitute.
- Evaluate (sin 30° = ½).
Final answer
Area = 12.
Area = ½ × (the two sides) × sin(the angle BETWEEN them). Sides 6 and 8 meet at 30°: Area = ½(6)(8)sin 30° = 12.
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Pick the right two sides: Use the two sides that enclose the given angle — if a different angle is given, use the sides next to it.
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Given the area, find a side or the angle: Set ½ab·sinC equal to the given area and solve for the unknown — a side (rearrange) or the included angle (then use sin⁻¹, watching for the obtuse possibility).
IB-style question — find the angle
A triangle with sides 10 and 12 has area 30. Find the included angle (acute).
Step by step
- Set up.
- Solve.
Final answer
C = 30° (the acute solution).
Working backwards from the area: ½(10)(12)sin C = 30 → sin C = 0.5 → C = 30° (sin C = 0.5 also gives 150° — choose from the context).
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GDC walkthrough
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Two possible angles: sin C = 0.5 also gives C = 150°. Use the context (or 'acute/obtuse') to choose — both can give the same area.
Find a missing side/angle first, then the area: Many questions need a two-step approach: use the sine or cosine rule to find a missing side or the included angle, then apply ½ab·sinC.
IB-style question — cosine rule then area
A triangle has sides 5 and 7 with included angle 80°. Find its area, then the third side.
Step by step
- Area directly (included angle given).
- Third side by the cosine rule.
Final answer
Area ≈ 17.2; third side ≈ 7.86.
Sides 5 and 7 meet at 80°. Area = ½(5)(7)sin 80° ≈ 17.2; the third side a (opposite the 80°) comes from the cosine rule, a² = 5² + 7² − 2(5)(7)cos 80° → a ≈ 7.86.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Read what you're given: If the included angle is missing, find it first (cosine rule from SSS, or sine rule); the area formula needs it.