Area of a Triangle using trigonometry
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Area of a Triangle using trigonometry — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionFinding the Area of Any Triangle Using TrigonometryConcept Practice
2 marks~3 minCriterion B
Triangle ABCABC has sides a=BCa = BC and b=ACb = AC, with a perpendicular height hh drawn from vertex BB to side ACAC. The angle at vertex CC is denoted CC.

Show that the area of triangle ABCABC can be written as

Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C

by starting from Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} and substituting h=asin⁡Ch = a\sin C. [2]
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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 TrigonometryConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a triangular garden plot. Two boundary fences meet at a corner, with lengths AB=10AB = 10 m and AC=7AC = 7 m enclosing an angle of 42°42° at vertex AA.
a
Calculate the area of the triangular plot. [1]
b
Turf costs 18 dollars per m². The architect has a budget of 2 900 dollars. Advise the architect whether the budget is sufficient to turf the entire plot, justifying your answer with a cost calculation. [1]
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Solutions

4QuestionProblem Solving with Non-Right TrianglesAssessment Practice
7 marks~11 minCriterion D
A surveyor records the following measurements for a triangular plot of land: side a=120 ma = 120\text{ m} (opposite angle AA), side b=150 mb = 150\text{ m} (opposite angle BB), and angle A=35°A = 35°. The measurements carry known tolerances: distances ±2%\pm 2\%, angles ±1°\pm 1°.
a
Deduce how many distinct triangles can be formed with these measurements, justifying your reasoning using the sine rule. [2]
b
Calculate the area of each possible triangle using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C. [2]
c
Advise the surveyor whether the area estimate is sufficient for legal land registration, taking into account the effect of measurement tolerances on both the number of possible triangles and the range of possible areas. [3]
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Solutions

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