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wikiHow to Find the Height of a Triangle

To calculate the area of a triangle you need to know its height. If this information is not given to you, you can easily calculate it based on what you do know! This article will teach you two different ways to find the height of a triangle, depending on what information you have been given.

Method 1 Using Base and Area to Find Height

1. 1
Recall the formula for the area of a triangle. The formula for the area of a triangle is A=1/2bh.[1]
• A = Area of the triangle
• b = Length of the base of the triangle
• h = Height of the base of the triangle
2. 2
Look at your triangle and determine which variables you know. In this case, you already know the area, so assign that value to A. You should also know the value of one side length; assign that value to "'b'". If you do not know both the Area and the length of one side, you will need to try a different method.
• Any side of a triangle can be the base, regardless of how the triangle is drawn. To visualize this, just imagine rotating the triangle until the known side length is at the bottom.
• For example, if you know that the area of a triangle is 20, and one side is 4, then: A = 20 and b = 4.
3. 3
Plug your values into the equation A=1/2bh and do the math. First multiply the base (b) by 1/2, then divide the area (A) by the product. The resulting value will be the height of your triangle!
• In our example: 20 = 1/2(4)h
• 20 = 2h
• 10 = h

Method 2 Finding an Equilateral Triangle's Height

1. 1
Recall the properties of an equilateral triangle. An equilateral triangle has three equal sides, and three equal angels that are each 60 degrees. If you cut an equilateral triangle in half, you will end up with two congruent right triangles. [2]
• In this example, we will be using an equilateral triangle with side lengths of 8.
2. 2
Recall the Pythagorean Theorem. The Pythagorean Theorem states that for any right triangle with sides of length a and b, and hypotenuse of length c: a2 + b2 = c2. We can use this theorem to find the height of our equilateral triangle![3]
3. 3
Break the equilateral triangle in half, and assign values to variables a, b, and c. The hypotenuse c will be equal to the original side length. Side a will be equal to 1/2 the side length, and side b is the height of the triangle that we need to solve.
• Using our example equilateral triangle with sides of 8, c = 8 and a = 4.
4. 4
Plug the values into the Pythagorean Theorem and solve for b2. First square c and a by multiplying each number by itself. Then subtract a2 from c2.
• 42 + b2 = 82
• 16 + b2 = 64
• b2 = 48
5. 5
Find the square root of b2 to get the height of your triangle! Use the square root function on your calculator to find Sqrt(2. The answer is the height of your equilateral triangle!
• b = Sqrt (48) = 6.93

Method 3 Determining Height With Angles and Sides

1. 1
Determine what variables you know. The height of a triangle can be found if you have an angles and a side, if the angle is between the base and the side you have, or all three sides. We'll call the sides of the triangle a, b, and c, and the angles, A, B, and C.
• If you have all three sides, you'll use Heron's formula, and the formula for the area of a triangle.
• If you have two sides and an angle, you'll use the formula for the area given two angles and a side. A = 1/2ab(sin C).[4]
2. 2
Use Heron's formula if you have all three sides. Heron's formula has two parts. First, you must find the variable s, which is equal to half of the perimeter of the triangle. This is done with this formula: s = (a+b+c)/2.[5]
• So for a triangle with sides a = 4, b = 3, and c = 5, s = (4+3+5)/2. Which is s = (12)/2. Which is s = 6.
• Then you can use the second part of Heron's formula, Area = sqr(s(s-a)(s-b)(s-c). Replace Area in the equation with it's equivalent in the area formula: 1/2bh (or 1/2ah or 1/2ch).
• Do the math to solve for h. For our example triangle this looks like 1/2(3)h = sqr(6(6-4)(6-3)(6-5). Which is 3/2h = sqr(6(2)(3)(1). Which is 3/2h = sqr(36). Use a calculator to calculate the square root, which in this case makes it 3/2h = 6. Therefore, height is equal to 4, using side b as the base.
3. 3
Use the area given two sides and an angle formula if you have a side and an angle. Replace area in the formula with it's equivalent in the area of a triangle formula: 1/2bh. This gives you a formula that looks like 1/2bh = 1/2ab(sin C). This can be simplified to h = a(sin C), thereby eliminating one of the side variables.[6]
• Solve the equation with your given variables. For example, with a = 3, and C = 40 degrees, the equation looks like this: h = 3(sin 40). Use your calculator to finish the equation, which in our example, makes h roughly 1.928.

Community Q&A

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• How do I determine the height of a triangle when I know the length of all three sides?
wikiHow Contributor
You already know the base, so calculate the area by Heron's formula. Then, substitute the values you know in the formula. Area=1/2 * base * height or height=2 * Area/base and find your answer.
• How do I calculate the height of a right triangle, given only the length of the base and the interior angle at the base?
Look up the tangent of the angle in a trigonometry table. Multiply the tangent by the length of the base.
• How do I find the height if the area and base of the quadrilateral are given?
If it's a regular quadrilateral, divide the area by the base.
• How do I find the height of a triangle with the length of the three sides?
wikiHow Contributor
Follow Method 3. "If you have all three sides, you'll use Heron's formula, and the formula for the area of a triangle." Use Heron's formula to determine the area of the triangle. Set the result equal to 1/2bh, and solve for h.
• How do I find the height of a triangle?
You need to know both the length of the base of the triangle and its area. Divide the base into the area, then double that.
• How do I find the area of an equilateral triangle when only the height is given?
wikiHow Contributor
H = height, S = side, A = area, B = base. You know that each angle is 60 degrees because it is an equilateral triangle. If you look at one of the triangle halves, H/S = sin 60 degrees because S is the longest side (the hypotenuse) and H is across from the 60 degree angle, so now you can find S. The base of the triangle is S because all the sides are the same, so B = S. Using A = (1/2)*BH, you get A = (1/2)*SH, which you can now find.
• Can I determine the base of a triangle if I know its height is eight feet?
You would also need to know other information, such as the area or some of the sides or angles.
• What is height of a triangle if the base is 5 and the two sides are 3?
This is a trigonometry question. Draw the height from the obtuse angle to the "5" side. This forms two right triangles inside the main triangle, each of whose hypotenuses are "3". The cosine of either of the original acute angles equals 2½÷3, or 0.833. Look up that angle in a trig table. Find the sine of that angle, and multiply that by 3 to get the height.
• How can I determine the height of an isosceles triangle?
wikiHow Contributor
Since the two opposite sides on an isosceles triangle are equal, you can use trigonometry to figure out the height. You can find it by having a known angle and using SohCahToa. For example, say you had an angle connecting a side and a base that was 30 degrees and the sides of the triangle are 3 inches long and 5.196 for the base side. In order to find the height, you would need to set it up as this: S=o/h, S=sine, o=opposite (the height), h=hypotenuse (the side), S30=o/3 because it's the opposite divided by the three, you can multiply by the reciprocal on both sides so the three gets cancelled on one side and the other is multiplied by 3, 3sin30=o and 3sin30=1.5=height.
• How do I find the area of a triangle if the sides are five, four and six?
wikiHow Contributor
You multiply all of the numbers and divide by two, since it is a triangle.
• How do I find the sides of a right-angled triangle if I only know the area?
• How can I find altitude of a triangle when length of two sides and median is given?
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