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The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1 ⁄ 4 turn or 90 degrees).
For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
There are also formulas for the area of a triangle in terms of its side lengths for triangles in the sphere or the hyperbolic plane. [ 19 ] For a triangle in the sphere with side lengths a , {\displaystyle a,} b , {\displaystyle b,} and c {\displaystyle c} the semiperimeter s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac ...
The area can be calculated by the standard formula for the area of right triangles (half the product of the two short sides) as . The cosine of the larger of the two non-right angles is the ratio of the adjacent side (the shorter of the two sides) to the hypotenuse, φ {\displaystyle \varphi } , from which it follows that the two non-right ...
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The same area formula can also be derived from Heron's formula for the area of a triangle from its three sides. However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between the semiperimeter and side length in those triangles. [19]
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