Trigonometry
Grade 6-7Area of a Triangle
When the height of a triangle is not given directly, the trig area formula, ½ ab sin C, finds the area from any two sides and the angle between them. This lesson covers the area formula, finding a missing side or angle when the area is already known, and finding the area of a composite shape split into triangles.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The area formula using two sides and the included angle
The area of a triangle can be found using Area = ½ ab sin C, where a and b are two sides of the triangle, and C is the angle between them, the included angle. This works for any triangle, not just right-angled ones.
With sides of 8 cm and 10 cm and an included angle of 50°: Area = ½ × 8 × 10 × sin 50° = 30.6 cm² (3 s.f.).
Finding a missing side or angle from a given area
When the area is already known, substitute the known values into the formula and rearrange to find the missing side or angle. If finding an angle, use the inverse sine function, sin⁻¹, at the end.
A triangle has AB = 8 cm, angle A = 45°, and an area of 24 cm². Substituting: 24 = ½ × 8 × AC × sin 45°, so AC = 48 ÷ (8 × sin 45°) = 8.49 cm (3 s.f.).
Finding a missing angle from a given area
When both sides are known along with the area, rearrange to find sin C, then use sin⁻¹ to find the angle.
A triangle has AB = 7 cm, AC = 8 cm, and an area of 20 cm². Substituting: 20 = ½ × 7 × 8 × sin A, so sin A = 40 ÷ 56 = 0.714, giving A = sin⁻¹(0.714) = 45.6° (1 d.p.).
Finding the area of a shape split into triangles
A quadrilateral split into two triangles by a diagonal can have its total area found by applying the area formula to each triangle separately, then adding the two results together.
A quadrilateral is split into a triangle with sides 6 cm and 8 cm and included angle 50°, and a triangle with sides 7 cm and 9 cm and included angle 60°. The total area is (½ × 6 × 8 × sin 50°) + (½ × 7 × 9 × sin 60°) = 18.4 + 27.3 = 45.7 cm² (3 s.f.).
Worked Examples
Three exam-style questions, fully solved.
A triangle has two sides of length 10 cm and 8 cm, with an included angle of 50°. Find the area of the triangle. Give your answer correct to 3 significant figures.
Easy- 1.Substitute into the formula: Area = ½ × 8 × 10 × sin 50°
- 2.Calculate: Area = 30.64...
Answer: 30.6 cm²
Triangle ABC has AB = 8 cm and angle A = 45°. The area of the triangle is 24 cm². Find the length of AC. Give your answer correct to 3 significant figures.
Medium- 1.Substitute into the formula: 24 = ½ × 8 × AC × sin 45°
- 2.Rearrange to make AC the subject: AC = 48 ÷ (8 × sin 45°)
Answer: 8.49 cm
A quadrilateral ABCD is split into two triangles by the diagonal BD. Triangle ABD has AB = 6 cm, AD = 8 cm and angle A = 50°. Triangle CBD has CB = 7 cm, CD = 9 cm and angle C = 60°. Find the total area of the quadrilateral. Give your answer correct to 3 significant figures.
Hard- 1.Find the area of triangle ABD: ½ × 6 × 8 × sin 50° = 18.39...
- 2.Find the area of triangle CBD: ½ × 7 × 9 × sin 60° = 27.28...
- 3.Add the two areas together: 18.39... + 27.28...
Answer: 45.7 cm²
Avoid These
The most common mistakes students make.
Using two sides that do not enclose the given angle, instead of the two sides that meet at that angle.
Forgetting the ½ in the formula, or using the angle itself instead of its sine.
Making an algebraic error when rearranging the formula to find a missing side or angle.
Forgetting to apply sin⁻¹ at the end when finding a missing angle from a known area.
When finding the area of a shape split into two triangles, calculating only one triangle's area and forgetting to add the second.
FAQ
Questions parents and students ask.
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