Shape and Space 10
4 marks · HardExplain why the formula can be used to calculate the area of any triangle, not just right-angled triangles.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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Must show: Explanation that for any triangle, height can be written in terms of sine of the included angle, Assume sides a, b, and included angle C, Area = (1/2) × base × height, where height is b × sin C
Common mistakes: Assuming the formula only works for right triangles
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Marking points
M1 · 1 mark · States or refers to the general area formula for any triangle as
M2 · 1 mark · Identifies that the height can be expressed as (or similar for other sides/angles)
A1 · 1 mark · Correctly substitutes for height and derives formula
B1 · 1 mark · Explains or demonstrates the formula works for any triangle, not just right-angled triangles
Worked solution
In any triangle with sides a, b, and included angle C, draw height from one vertex to the opposite side.
Let 'a' and 'b' be two sides, and angle C the included angle. Drop a perpendicular (height, h) from the vertex between a and b to the side opposite angle C.
From trigonometry, the height h = b sin C.
Therefore, this formula can be used for any triangle, because the height can always be written as a side multiplied by the sine of the included angle, not just when the angle is 90°.