Mathematics
Grade 11
15 min
Area of a triangle: sine formula
Area of a triangle: sine formula
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1
Introduction & Learning Objectives
Learning Objectives
Derive the sine formula for the area of a triangle from the standard A = (1/2)bh formula.
Calculate the area of any triangle given Side-Angle-Side (SAS) information.
Rearrange the sine area formula to solve for an unknown side length when the area is given.
Rearrange the sine area formula to solve for an unknown angle when the area is given.
Determine when to use the sine area formula versus other area formulas.
Apply the sine area formula to solve multi-step problems and real-world scenarios.
Correctly use their calculator in degree mode and apply the inverse sine function to find angles.
How could you find the area of a triangular plot of land if you can only measure two sides and the angle between them, but you can't measure its height? 🗺️
This...
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Key Concepts & Vocabulary
TermDefinitionExample
SAS (Side-Angle-Side)A condition for describing a triangle where two side lengths and the measure of the included angle are known.In triangle ABC, knowing side a, side b, and the angle C between them is SAS information.
Included AngleThe angle formed at the vertex between two specific sides of a triangle.In triangle ABC, angle C is the included angle between sides a and b.
Sine Function (sin)In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The sine area formula uses this ratio to relate the included angle to the triangle's height.In a right triangle with angle θ, sin(θ) = opposite / hypotenuse.
Altitude (Height)A perpendicular line segment from a vertex of a triangle to the opposite...
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Core Formulas
Sine Area Formula (using Angle C)
Area = \frac{1}{2}ab \sin(C)
Use this formula when you know the lengths of sides 'a' and 'b' and the measure of the included angle 'C'.
Sine Area Formula (using Angle A)
Area = \frac{1}{2}bc \sin(A)
Use this formula when you know the lengths of sides 'b' and 'c' and the measure of the included angle 'A'.
Sine Area Formula (using Angle B)
Area = \frac{1}{2}ac \sin(B)
Use this formula when you know the lengths of sides 'a' and 'c' and the measure of the included angle 'B'.
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Challenging
The area of a triangle is A = (1/2)bc sin(A). The Law of Sines can be written as b = (c * sin(B)) / sin(C). By substituting an expression for 'b' from the Law of Sines into the area formula, which of the following is another valid formula for the area of a triangle?
A.Area = (c² sin(A) sin(B)) / (2 sin(C))
B.Area = (2c² sin(A)) / sin(B)
C.Area = (c sin(A) sin(B)) / (2 sin(C))
D.Area = (c² sin(A) sin(C)) / (2 sin(B))
Challenging
In a triangle, sides 'a' and 'b' are constant. The included angle C is increased such that the new area is √2 times the original area. If the original angle was C = 45°, what is the new angle C'?
A.60°
B.90°
C.120°
D.150°
Challenging
For fixed side lengths 'a' and 'b' in a triangle, the area is maximized when the included angle C is what value?
A.45°
B.60°
C.90°
D.180°
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