Area of a Triangle using trigonometry
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Area of a Triangle using trigonometry — Free MYP5 Mathematics (Standard) 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]
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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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3QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
5 marks~8 minCriterion B
A city surveyor maps a triangular park with vertices at A(0,0)A(0,0), B(7,0)B(7,0), and C(3,5)C(3,5), where units are in metres.
a
Show that AB=7AB = 7 m, AC=34AC = \sqrt{34} m, and deduce that angle A≈59.03°A \approx 59.03°. [2]
b
A fence post is to be placed at the point on ABAB directly below CC. Calculate the area of the park using Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}, identifying the perpendicular height clearly. [2]
c
The surveyor claims the trigonometric formula Area=12 absin⁡A\text{Area} = \frac{1}{2}\,ab\sin A always gives the same area as the base–height method. Justify this claim algebraically, and explain what this means for calculating land areas when a perpendicular height cannot be measured directly. [1]
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4QuestionFinding 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]
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Solutions

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