# 3 Ways 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 in the problem, it is easy to calculate it based on what you already know! This article will teach you two different ways to find the height of a triangle, depending on what information you have been given.

## Steps

### Method 1 of 3: Using Base and Area to Find Height

#### Step 1. Recall the formula for finding the area of ​​a triangle

She is represented by A = ½ bh.

• THE = area of ​​the triangle.
• B = length of triangle base.
• H = height of the base of the triangle.

#### Step 2. Look at the triangle and determine which variables are known

In this case, you already know the area value, so you can use it to define THE. You must also know the length value of one side; set this value to B. If you don't know the area and length of a side, you will have to use another method.

• Any side of the triangle can be the base, no matter how it is drawn. To visualize this concept, imagine rotating the triangle until the known side length is the bottom.
• For example, if you know that the area of ​​a triangle is equal to 20, and one of its sides is 4, then: A = 20 and b = 4.

#### Step 3. Enter the values ​​into the equation A = ½ bh and do the calculations

First, multiply the base (B) by ½ and then divide the area (THE) for the product. The resulting value will represent the height of the triangle!

• In our example: 20 = ½ (4) h
• 20 = 2 h
• 10 = h

### Method 2 of 3: Finding the Height of an Equilateral Triangle

#### Step 1. Recall the properties of an equilateral triangle

An equilateral triangle has three equal sides and three equal angles, 60 degrees each. If you cut it in half, there are two congruent right triangles left.

### In this example, we will use an equilateral triangle with 8-gauge sides

#### Step 2. Recall the Pythagorean Theorem

Pythagoras' theorem states that for any right triangle with measure legs The and B and a long hypotenuse ç, The2 + b2 = c. We can use this equation to figure out the height of our equilateral triangle.

#### Step 3. Divide the equilateral triangle in half and set values ​​for variables a, b, and c

the hypotenuse ç will be equal to the original side length. the collared The will have a measurement equal to ½ of the side length and the side B represents the height of the triangle we want to discover.

• Using the equilateral triangle from our example, with sides measuring 8, c = 8 and a = 4.

Step 4. Enter the values ​​in Pythagoras' theorem and find the value of b2.

First, raise ç and The, multiplying each number by itself. Then subtract The2 in ç2.

• 42 + b2 = 82
• 16+b2 = 64
• B2 = 48

Step 5. Find the square root of b2 to get the height of the triangle.

Use the square root function in a calculator to find the value of b2. The answer will be the height of the equilateral triangle.

• b = √b(48) = 6, 93

### Method 3 of 3: Determining Height with Angles and Sides

#### Step 1. Determine which variables are known

You can find the height of a triangle when you know the values ​​of the angles and one side if the angle is between the base and the side in question, or all three vertices. We will call the sides of the triangle a, b and c, and the angles A, B and C.

• If you know the value of three sides, you can use Heron's formula and the formula for the area of ​​a triangle.
• If you know the value of two sides and an angle, you should use the formula for the area to find out the values ​​of the two angles and the remaining side. A = ½ ab (sin C).

#### Step 2. Use Heron's formula if you know the value of the three sides

This equation has two parts. First, you must find the variable s, which is equal to half the perimeter of the triangle. This is done using the following formula: s = (a+b+c) / 2.

• Thus, for a triangle with sides a = 4, b = 3 and c = 5, s = (4+3+5) / 2. As a result, we have s = (12) / 2 = 6.
• Then you can use the second part of Heron's formula: Area = √[s(y-a)(y-b)(y-c)]. Replace Area with its equivalent value in the formula for the area of ​​the triangle: ½ bh (or ½ ah or ½ ch).
• Do the calculations to find the value of h. In the triangle in our example, it will look like this: ½ (3) h = √[6(6-4)(6-3)(6-5)]. As a result, we have that 3/2 h = √[6(2)(3)(1)] = √[36]. Use a calculator to find the square root of this value, which in this case is equal to 3/2 h = 6. So the height will have a measure equal to 4 if we take side b as the base.

#### Step 3. If you know the value of one side and an angle, use the equation for an area with two sides and an angle

Replace the area value with its equivalent in the formula for the area of ​​a triangle: ½ bh. This will give you a formula similar to ½ bh = ½ ab (sin C). It can be simplified to h = a (sin C), thus eliminating one of the variables relative to the sides.