Dimensions and traits vary from one triangle to the next, making a straightforward, go-to calculation of the shape’s height difficult. Students should determine the best way to find the height based on what they know about a triangle. For example, when you know the angles of a triangle, trigonometry can help; when you know the area, basic algebra gives the height. Analyze the information you have before developing a game plan for finding a triangle’s height.

## Area Hysteria

Sometimes you know the area and base of a triangle but not its height. In this case, you can manipulate the equation for the area of a triangle to obtain its height. The equation for the area of a triangle is A = (1/2) * b * h, where A is the area, b is the base and h is the height. Using algebra, you can get h alone: Divide both sides by b and then multiply both sides by 2 to get h = 2A / b. Plug in the area and base into this equation to find a triangle’s height. For example, if your triangle has an area of 36 and a base of 9, your equation becomes h = 2 * 36 / 9, which equals 8.

## An Ancient Greek Technique

If you know the base and the length of one other side of the triangle, you can find the height using the Pythagorean theorem. Draw a line straight from the triangle’s vertex to the base. By doing so, you now have a right triangle within your triangle. Set up Pythagorean’s Theorem: a^2 + b^2 = c^2. Plug in the base for “b” and the hypotenuse for “c.” Then solve for a, the height of the triangle. For example, if your base is 3 and hypotenuse is 5, your equation becomes a^2 + 9 = 25. Subtract 9 on both sides to get a^2 = 16. Take the square root of both sides to get a = 4.

## The Height Dangles from an Angle

Because you can draw a right triangle inside any triangle, you can also use trigonometric identities to find the height of a triangle. If you know the angle between the height and the hypotenuse of the triangle, you can set up the equation tan(a) = x / b_, where a is the angle, x is the height and b_ is half the base. Plug in the values. For example, if your angle is 30 degrees and your base is 6, you would have the equation tan(30) = x / 3. Solving for x gives x= 3 * tan(30). Because the tangent of 30 degrees is sqrt(3) / 3, the equation simplifies to give you the height x = sqrt(3).

## One More Formula

Heron’s formula allows you to find the height of a triangle by first computing its half-perimeter. Heron’s formula states that a triangle’s half-perimeter is the sum of the triangle’s sides, divided by 2, or s = (a+b+c) / 2, where a, b and c are the sides of the triangle. It also states that the area of that triangle is equal to the square root of s(s-a) (s-b) (s-c). This calculation leads to the area, which you can use to find the height via an earlier method h = 2A / b. For example, if the sides of your triangle are 6, 8 and 10, s = (6+8+10) / 2 = 12. Then A = sqrt(12_6_4_2) = sqrt(576) = 24. If 10 is the triangle’s base, h = 2_24 / 10 = 4.8.

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