The height of a triangle can be found in a few different ways, depending on the type of triangle and the information that is known or measured. Right triangles, which include one angle of 90 degrees, are the easiest to measure using the Pythagorean theorem (if the lengths of two sides are known) or the area formula (if the area and base are known). Equilateral triangles, in which all sides are of equal length, and isosceles triangles, in which three sides are of equal length, can be cut in half, creating two right triangles. Oblique triangles, which have no internal angle equal to 90 degrees, are more difficult, and require trigonometry to find their height.
This article shows three different methods of finding the height of triangles. Examples follow each step in brackets.
Write down the Pythagorean theorem, c^2 = a^2 + b^2, where c is the hypotenuse (the diagonal line).
Rearrange the theorem to solve for a^2, so a^2 = c^2 - b^2.
Plug in the two known side values, c and b. [a^2 = 19^2 - 18^2]
Perform the math. [a^2 = 361 - 324 = 37]
Take the square root of both sides to find the height or a^2. [a = 6.1]
Draw the triangle and label the sides and known values.
Write down the area formula, A = 1/2 x bh, where A = area, b = base and h = height.
Solve for h, the height. h = A / (.5b)
Plug in the known values. [h = 72 / (.5 x 18)]
Perform the math to find the height. [h = 72 / (.5 x 18) = h = 72 / 9 = 8]
Draw the triangle and label the sides and known values. [A, B and C are the angles. a, b and c are the sides, c being the base. h is the height. In this example, A = 60 degrees and b = 5.]
Write down the area formula, A = 1/2 bh (A = area, b = base, h = height). All values do not need to be known, but the formula helps keep everything oriented correctly.
Find the side adjacent to the base. [side b = 5]
Find the angle adjacent to the base and side in Step 3. If it is not known, a protractor will be required to measure the angle. [angle A = 60]
Write down the formula for the height, which is the side adjacent to the base multiplied by the sine of the side's adjacent angle. [h = 5sin60]
Perform the math to find the height. [h = 5 x 0.87 = 4.33]
The base can be whatever side the triangle is turned on. The trigonometry method (using sine) can be applied to right triangles as well. The three angles of any triangle must add up to 180 degrees.