How to Find the Height of a Triangle

The properties of triangles allow for many methods of calculating height.
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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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