Curriculum placement
Precalculus. This describes the skill, not a learner estimate.
Area from two sides and the included angle
Derive and apply the two-sides-and-included-angle triangle area formula
Problem types
- Area from two sides and the included angle
- Triangle area from two sides and the angle
- Triangle area from two sides and the included angle
- Show the work: triangle area from two sides and the included angle
- Place the foot of the height for the area formula
- Complete the derivation of the area formula
- Judge a derivation of the area formula
Builds on
- Sine, cosine, and tangent
- Area of triangles and other polygons
- Why the circle and solid formulas hold · Earlier idea
The area formula one half base times height becomes one half ab sin C when the height is written with a sine.
- Arc length and sector area · Useful idea
The area of a circular segment is the sector area minus the triangle area, which combines sector and trigonometric area.
- Regular polygons inscribed in a circle · Useful idea
A regular polygon inscribed in a circle splits into congruent triangles whose area uses the sine of the central angle.
Leads to
- Deriving the laws of sines and cosines
- Modeling with geometry · See it applied
The area of a regular polygon is found by adding the areas of congruent triangles with sine area.
Related skills
- The laws of sines and cosines · Related idea
The height used to count triangles in the ambiguous case is b sin A, the same product that appears in the area formula one half ab sin C.
- Vectors and their components · Related idea
The area of a triangle with sides a and b and included angle C is half the magnitude of the cross product, one half ab sin C.
- Sine and cosine of complementary angles · Shared foundations
- Perimeter and area from coordinates · Shared foundations
- Side ratios of special right triangles · Shared foundations
Open the interactive map to explore lessons and skill connections.