Non-right-triangle area | Year 11

01 / LEARN

The idea, step by step

From base and height

A perpendicular height can be written $b sin C$ . Substitution into half-base-times-height gives a formula for acute or obtuse included angles. The selected angle must lie between the two selected sides.

K = \frac{1}{2} a b sin C

Bounds and inverse questions

For fixed a,b the area cannot exceed ab/2 because sine cannot exceed 1. When area is known, solving for sine may give two possible included angles. A condition such as “obtuse” selects one; otherwise both may be valid. Keep square units in the answer.

A common trap

Using any known angle in the area formula.

Instead: The angle must be included between the two side lengths in the product.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Sides 8 cm and 12 cm enclose 120°. Find the exact area.

$K = 48 sin 120^{\circ}$ .
$sin 120^{\circ} = \sqrt{3} / 2$ .
Multiply the exact values.

24 \sqrt{3} {cm}^{2}

Example 2

Worked example 2

Sides 10 m and 12 m enclose angle C. The area is 30 m². Find possible C.

$30 = 60 sin C$ .
$sin C = 1 / 2$ .
Both 30° and 150° are possible included angles.

30° or 150°

Check your understanding

What is the greatest area between sides 5 cm and 9 cm?

Show answer

22.5 cm².

$K \leq a b / 2 = 22.5$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find area from sides 7 cm,10 cm and included angle 30°.
3 marks
Find area from sides 6 cm,8 cm and included angle 60° exactly.
3 marks
Find exact area for sides 9 m,14 m and included angle 45°.
3 marks
Sides 4 cm and 10 cm enclose an obtuse angle. Area is 10 cm². Find the angle.
3 marks
A sail has sides 3.5 m and 4 m meeting at 75°. Find area to two decimal places.
3 marks
Can sides 6 cm and 8 cm enclose area 25 cm²? Explain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find area from sides 7 cm,10 cm and included angle 30°.
3 marks
$K = 35 (1 / 2) = 17.5$ .
17.5 cm²
Find area from sides 6 cm,8 cm and included angle 60° exactly.
3 marks
$K = 24 (\sqrt{3} / 2)$ .
$12 \sqrt{3} {cm}^{2}$
Find exact area for sides 9 m,14 m and included angle 45°.
3 marks
$K = 63 sin 45^{\circ} = 63 \sqrt{2} / 2$ .
$63 \sqrt{2} / 2 m^{2}$
Sides 4 cm and 10 cm enclose an obtuse angle. Area is 10 cm². Find the angle.
3 marks
$sin C = 10 / 20 = 1 / 2$ .
Select the obtuse supplementary solution.
150°.
A sail has sides 3.5 m and 4 m meeting at 75°. Find area to two decimal places.
3 marks
$K = 7 sin 75^{\circ} \approx 6.76148$ .
6.76 m²
Can sides 6 cm and 8 cm enclose area 25 cm²? Explain.
3 marks
$K \leq \frac{1}{2} (6) (8) = 24$ .
No; maximum area is 24 cm².

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

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