How this calculator works
The area remains half base times perpendicular height even when the altitude lies outside the triangle. SSS, SAS and ASA modes recover a complete triangle and require its largest angle to exceed 90 degrees. The altitude table measures a along B to C, b along C to A and c along A to B. A negative foot coordinate is before the start; a coordinate greater than the base length is beyond the end.
Formula
K = bh/2; h_c = 2K/c; foot along A to B = (b²+c²-a²)/(2c). The obtuse test is max(A,B,C) > π/2.
Worked example
Take A = (0,0), B = (2,0), C = (3,4). Then a = √17, b = 5 and c = 2. Area is 4 square units. The altitude to c has height 4 and foot coordinate 3, one unit beyond B.
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Frequently asked questions
Is an outside altitude negative?
Its perpendicular length is positive. The signed foot coordinate records where it lands along the base line.
Will base and height prove a triangle is obtuse?
No. Many triangles can share the same base and height. Use enough sides or angles to establish the angle class.
Why are right triangles rejected in sided modes?
This page checks the specific obtuse case. An angle of exactly 90 degrees belongs to the right triangle area tool.