Triangle Area with Half ab sin C
Find the area of any triangle from two sides and the angle between them, and run the same rule backwards to find an angle from a known area.
What a learner can do afterwards
- Use the two sides either side of the given angle, not any two sides
- Find an angle when the area and two sides are known
- Split a quadrilateral into two triangles to find its area
1 · Read
Half base times height always works for area, but you need the height. When nobody hands you a height, build one. Take a triangle with sides a and b and the angle C between them. Use side a as the base. Drop a perpendicular from the far corner to the base, making a right triangle whose hypotenuse is b and whose height is h. In that right triangle, h equals b sin C. Substitute into the base-height formula and the area is one half times a times b times sin C. The rule is that a and b must touch angle C: grab a far side and the answer breaks.
When angle C is obtuse, the perpendicular meets the extension of the base instead of the base itself. The auxiliary triangle still has hypotenuse b and an acute angle of 180 minus C, and sin of 180 minus C equals sin C, so the height is unchanged. At exactly 90 degrees the perpendicular is side b itself, which is the right-angle case of the same formula.
The formula also runs backwards. When you know the area and two sides, solve for the sine of the angle between them: sin C equals 2 times area over a times b, then take the inverse sine. Sine only reaches 1, so the area can never exceed half of a times b, and that maximum happens only at 90 degrees. When sin C is exactly 1 both candidates coincide at 90 degrees, counted once. For any smaller sine, both C and 180 minus C are possible, check which one fits your triangle.
A four-sided shape needs one extra move. Draw a diagonal that stays inside the shape to split the quadrilateral into two triangles, find each triangle's area with the same formula, then add the two areas together. A split giving 12 and 7 makes 19 for the whole shape, and the same adding works whatever each half measures. In a concave quadrilateral one diagonal can fall outside, so always choose the internal one.
Multiply the two touching sides by the sine between them, halve it, and split quadrilaterals before adding.
Two readings, eight practice questions were written for this triangle area topic. The video is linked from its original source.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.