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Triangle Area with Trig

A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

When you know two sides and the angle between them (SAS), the area of a triangle is (1/2)*a*b*sin(C), where C is the angle included between sides a and b. When you know all three sides instead (SSS), Heron's formula gives the area: with semi-perimeter s = (a+b+c)/2, area = sqrt(s(s-a)(s-b)(s-c)). Both formulas extend the familiar (1/2)*base*height idea to triangles that are not conveniently right-angled.

What you'll learn

Why it matters: Surveyors and land-mapping software compute the area of irregular plots by splitting them into triangles and applying these exact formulas from measured sides and angles, since you rarely get a clean base-and-height to measure directly.

Worked example

Problem. Find the area of a triangle with sides a = 10 and b = 8 and included angle C = 30 degrees.

  1. Use the SAS area formula: Area = (1/2)*a*b*sin(C).
  2. Substitute: Area = (1/2)*10*8*sin(30) = 40*sin(30).
  3. Since sin(30) = 1/2, Area = 40*(1/2) = 20.

Answer: 20

Practice problems

1. Find the area of a triangle with sides a = 6 and b = 9 and included angle C = 90 degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the area of a triangle with sides a = 6 and b = 9 and included angle C = 90 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use Area = (1/2)*a*b*sin(C).
  4. Substitute: (1/2)*6*9*sin(90).
  5. Since sin(90) = 1, Area = 27.
  6. Check the result by substituting or estimating: the response should match 27 and make sense in the original problem.

Answer: 27

2. Find the area of a triangle with sides a = 10 and b = 8 and included angle C = 30 degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the area of a triangle with sides a = 10 and b = 8 and included angle C = 30 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use Area = (1/2)*a*b*sin(C).
  4. Substitute: (1/2)*10*8*sin(30) = 40*sin(30).
  5. Since sin(30) = 1/2, Area = 20.
  6. Check the result by substituting or estimating: the response should match 20 and make sense in the original problem.

Answer: 20

3. In the formula Area = (1/2)*a*b*sin(C), what must angle C be?

Choices: The angle included between sides a and b · The largest angle in the triangle · The angle opposite the longest side · Any angle in the triangle

Show solution
  1. The formula uses the two sides a and b and the angle between them.
  2. C must sit between sides a and b.
  3. It is the included angle, not just any angle.

Answer: The angle included between sides a and b

4. Find the exact area of a triangle with sides a = 12 and b = 5 and included angle C = 60 degrees. Give the exact value.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the exact area of a triangle with sides a = 12 and b = 5 and included angle C = 60 degrees. Give the exact value.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use Area = (1/2)*12*5*sin(60) = 30*sin(60).
  4. Since sin(60) = sqrt(3)/2, Area = 30*(sqrt(3)/2).
  5. Area = 15sqrt(3).
  6. Check the result by substituting or estimating: the response should match 15sqrt(3) and make sense in the original problem.

Answer: 15sqrt(3)

5. Find the area of a triangle with sides a = 7 and b = 10 and included angle C = 45 degrees, to the nearest tenth.

Show solution
  1. Use Area = (1/2)*7*10*sin(45) = 35*sin(45).
  2. sin(45) = sqrt(2)/2 is about 0.7071.
  3. Area = 35*0.7071 = 24.7 (to the nearest tenth).

Answer: 24.7

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