Area of a triangle: ½ab sin C

About this lesson

Precalculus › Trigonometric functions

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Area of a triangle: ½ab sin C

Warm-up

Three quick questions from earlier lessons.

Discover

A triangle has sides 6 and 8 with a right angle between them. What is its area?

  1. 24
  2. 48

Key idea

Area = ½ab sin C.

If two sides aa and bb make an angle CC, the height on side aa is bsin⁡Cb \sin C, so the area is 12absin⁡C\tfrac12 ab \sin C. Use the angle between the two sides. For C=90°C = 90° it gives 12ab\tfrac12 ab.

Worked example

Sides 10 and 7 with a 30° angle between them.

7sin⁡30°=3.57 \sin 30° = 3.5
Height.
12⋅10⋅7⋅12=17.5\tfrac12 \cdot 10 \cdot 7 \cdot \tfrac12 = 17.5
Area.

Your turn, guided

Your turn: sides 4 and 6, angle 30°. Area?

66
½ · 4 · 6 · ½.

Practice ★

  1. A triangle has sides 5 cm and 12 cm with an angle of 30° between them. What is its area, in cm²?
  2. A triangle has sides 12 cm and 6 cm with an angle of 30° between them. What is its area, in cm²?
  1. In triangle PQR, PQ = 9, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
    1. The angle at P, between the two sides
    2. The angle at Q
    3. Any angle of the triangle

Watch out!

Practice ★★

  1. A triangle has sides 5 m and 3 m with an angle of 60° between them. What is its exact area, in m²?
  2. A triangle has sides 8 m and 11 m with an angle of 60° between them. What is its exact area, in m²?
  1. A triangle has sides 5 cm and 12 cm with an angle of 30° between them. What is its area, in cm²?

Exit ticket

Three questions to check you have got it.

  1. A triangle has sides 5 cm and 12 cm with an angle of 30° between them. What is its area, in cm²?
  2. A triangle has sides 12 cm and 6 cm with an angle of 30° between them. What is its area, in cm²?
  3. A triangle has sides 3 cm and 11 cm with an angle of 30° between them. What is its area, in cm²?

Summary card

Area of any triangle

½ab sin C, with C between a and b.

Watch out: use the included angle.

End of the lesson

Next lesson: The law of sines Back to the chapter

Answers: Area of a triangle: ½ab sin C

A

  1. 1515
  2. 1818

B

  1. The angle at P, between the two sides

C

  1. 1534\frac{15 \sqrt{3}}{4}
  2. 22322 \sqrt{3}

D

  1. 1515

E

  1. 1515
  2. 1818
  3. 334\frac{33}{4}

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