Area of a Triangle using trigonometry
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Area of a Triangle using trigonometry — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionProblem Solving with Non-Right TrianglesConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a triangular garden plot. The plot is modelled as triangle ABCABC, where AB=8AB = 8 m, AC=5AC = 5 m, and the included angle at AA is 30°30°.

A local regulation states that any garden plot smaller than 12 m212 \text{ m}^2 requires an additional permit.
a
Calculate the area of triangle ABCABC. [1]
b
Justify whether the architect must apply for an additional permit. [1]
Question diagram

Solutions

2QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
4 marks~6 minCriterion C
A landscape architect is designing a triangular garden bed with two fixed side lengths a=9.2a = 9.2 cm and b=12.5b = 12.5 cm (on a scale drawing). The graph below shows how the area of the triangle varies with the included angle CC, using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C.
a
Show that the area formula simplifies to Area=57.5sin⁡C\text{Area} = 57.5\sin C. [1]
b
Explain how the shape of the graph shows that the maximum area occurs at C=90°C = 90°. [2]
c
The architect claims a right-angled triangular bed gives the most efficient use of the fixed side lengths. Calculate the maximum area correct to 3 significant figures, and advise the architect whether this right-angle design should be adopted to maximise the garden bed area. [1]
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Solutions

3QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
8 marks~12 minCriterion D
A homeowner measures two sides of a triangular roof section for solar panel installation: a=6.8 ma = 6.8 \text{ m} and b=5.2 mb = 5.2 \text{ m}, with included angle C=72°C = 72°. The tape measure has accuracy ±0.05 m\pm 0.05 \text{ m} and the protractor has accuracy ±1°\pm 1°.

Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C
a
Calculate the area of the triangular roof section. [2]
b
Analyse how the measurement uncertainties affect the estimated area by calculating the maximum and minimum possible areas. [3]
c
Advise the homeowner whether the calculated area alone is sufficient to determine how many solar panels to order, identifying at least two limitations of the triangular model. [3]
Question diagram

Solutions

4QuestionProblem Solving with Non-Right TrianglesAssessment Practice
6 marks~9 minCriterion B
A triangular solar panel frame is being designed with two fixed struts of length 10 cm and 14 cm. The included angle CC between the struts can be adjusted during assembly.
a
Using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C, calculate the area of the triangle for each angle below. [2]

Angle CC (degrees): 20, 40, 60, 80, 100

Area (cm²): ___, ___, ___, ___, ___
b
Describe the pattern in the areas as CC increases from 20° to 100°, and deduce the angle that produces the maximum area. [2]
c
Justify the engineer's claim that C=90°C = 90° produces the greatest area, using the behaviour of sin⁡C\sin C, and advise the assembly team on how to set the struts and what happens to the enclosed area if they deviate from this angle. [2]

Solutions

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