Area of Triangles

Pre-Calculus · Axiom Academy

The base-times-height formula needs a height. For a slanted triangle, the sine of the included angle hands you that height for free. Pick two sides of a triangle, a and b , meeting at the included angle C . Treat side a as the base. The area is still — but the height is the perpendicular distance from the far vertex down to that base, and nobody handed it to us. Base is side a — but where is the height? In that little right triangle, , so . Substitute it into the base–height formula and the height disappears entirely — replaced by something we already know: The factor is doing real work. Pin the two side lengths a and b and only open or close the angle between them: the area is largest when the sides are perpendicular and vanishes as the triangle flattens. — the sides are perpendicular and the area is as large as those two side lengths allow. — the triangle collapses toward a line segment and its area shrinks to nothing. A triangle has sides a = 8 and b = 10 with an included angle of . We have side–angle–side, so the formula applies directly — no height to hunt for. You've turned the familiar base–height rule into a formula that works on any triangle — using nothing but two sides and the angle between them. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.