Triangle Area with Half ab sin C · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Half, times, times sine

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. Triangle PQR has sides PQ = 5, QR = 7, and PR = 9. You want to find the area using angle Q. Which two sides do you multiply by sin(Q)?

    • PQ and QR
    • QR and PR
    • PQ and PR
    • All three sides

  2. A triangle has two sides of length 8 cm and 10 cm, with an angle of 30° between them. What is the area of the triangle?

    • 20 cm²
    • 40 cm²
    • 80 cm²
    • 10 cm²

  3. Triangle PQR has sides PQ, QR, and PR. You want the area using angle Q. Which two sides go with sin Q?

    • PQ and QR
    • QR and PR
    • PQ and PR

  4. A diagonal splits a quadrilateral into triangles of area 14 cm squared and 9 cm squared. What is the total area, in cm squared?

    Answer: ______________

  5. Two sides are 5 cm and 8 cm and the area is 10 cm squared. What is the angle between those sides, in degrees?

    Answer: ______________

  6. A triangle has two sides of length 5 cm and 8 cm, and its area is 10 cm². What is the angle between those two sides, in degrees?

    Answer: ______________

  7. A triangle has two sides of length 12 cm and 5 cm, with a 30° angle between them. What is the area of the triangle, in cm²?

    Answer: ______________

  8. Two sides are 12 cm and 5 cm with a 30 degree angle between them. What is the area, in cm squared?

    Answer: ______________

  9. A triangle has two sides of length 10 cm and 10 cm, and its area is 25 cm². What is the smaller of the two possible angles between those sides, in degrees?

    Answer: ______________

  10. Two sides are both 10 cm and the area is 25 cm squared. What is the smaller of the two possible angles between them?

    • 150 degrees
    • 90 degrees
    • 30 degrees
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Half, times, times sine W1-mt_PiwizT4D2b-s1

  1. PQ and QR · Angle Q sits between sides PQ and QR, so those are the two sides that belong in the formula.
  2. 20 cm² · Area = 1/2 × 8 × 10 × sin(30°) = 1/2 × 80 × 0.5 = 20 cm².
  3. PQ and QR · Angle Q sits between PQ and QR, so those two belong in the formula.
  4. 23 · Add the two triangle areas: 14 plus 9 is 23.
  5. 30 · Sin C equals 20 over 40, which is 0.5, and inverse sine gives 30.
  6. 30 · sin(C) = (2 × 10) / (5 × 8) = 20/40 = 0.5, and sin⁻¹(0.5) = 30°.
  7. 15 · Area = 1/2 × 12 × 5 × sin(30°) = 30 × 0.5 = 15 cm².
  8. 15 · One half times 12 times 5 times sin 30 is 30 times 0.5.
  9. 30 · sin(C) = (2 × 25) / (10 × 10) = 50/100 = 0.5. sin⁻¹(0.5) gives 30° or 150°, so the smaller angle is 30°.
  10. 30 degrees · Sin C is 0.5, shared by 30 and 150, and the smaller is 30.
Worksheet · LightMySky