You can find the height of a triangle with plenty of methods. In all cases, you’ll need to know at least the base, the angle of the triangle, or both. The height will come next, and you’ll be able to determine the area easily if you need to.

Here are three different methods if you want to know how to find the height of a triangle.

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If you have the area and the length of the triangle’s base, you can use them to determine the height. You don’t always need to know any of the angles. Here’s how to do it. 

To get the area of a triangle, you need to use the mathematical formula A=1/2bh. ‘A’ refers to the triangle’s area, while ‘b’ is the base’s length and ‘h’ is the base’s height. Whenever you don’t have the value of any angle, this is the right equation to use.

Here, we need to get ‘h,’ and we supposedly already have ‘A’ and ‘b.’

Step 2: Fill in the Formula With Your Givens

Now that you have your formula ready, the next step is determining the variables that you know. The area is given, so you should assign it to ‘A.’ Afterward, assign the given side length to ‘b.’ 

The value of ‘b’ doesn’t have to be the base. It can be the length of any side of the triangle, as long as you know its value. If you have another length that’s not the base, you can imagine rotating the triangle, and you’ll be able to solve the problem as if the side became the base.

The last step is to do the actual math. Your formula, A=1/2bh, should now have values in the places of ‘A’ and ‘b.’ Multiplying ½ by ‘b’ and dividing A by the resulting number will get you the height you’re looking for.

Method 1: Use the Area Formula

If the variables you have to determine the height are the sides and the angles, you can use either Heron’s formula or the area angle formula. Here’s how to do it.

To determine the height using Heron’s formula, you need to know the lengths of the triangle’s three sides. Alternatively, you can solve the problem using the formula of the area, A = 1/2ab(sin C), but you’ll need to know two sides of the triangle, along with the value of the angle between them. 

Let’s see how to solve it using both methods.

Now, you should have the three sides of the triangle. To use Heron’s formula, you need to determine the value of variable ‘s’ first, which equals half of the triangle’s perimeter. The equation goes as follows: s = (a+b+c)/2.

Then, move on to the second part of Heron’s formula, which dictates that Area = sqr(s(s-a)(s-b)(s-c). 

Firstly, determine the area using the 1/2bh formula, then plug your givens into the equation and do your math.

If you don’t have the three sides but instead have one side and one angle, you can use the area formula: A = 1/2ab(sin C).

All you’ll need to do is replace A with the formula 1/2bh, so your equation should look like 1/2bh = 1/2ab(sin C). After doing the necessary math and eliminating the unneeded variables, you’ll find that it ends up as h = a(sin C).

Now, you have one last step to determine the height. Use your calculator to calculate sin C, and multiply it by ‘a’, and you’re done.

Method 2: Use the Sides & Angles

If you have an equilateral or a right triangle, you can easily determine the height using the Pythagorean Theorem. You’ll cut the equilateral triangle in half, getting two right ones. Meanwhile, if you already have a right triangle, you’ll use it without cutting anything.

Let’s see how to determine the height using the famous mathematical theory.

Each angle equals 60 degrees in all equilateral triangles, so that’s the first given you have. Next, you should have the side length. That way, when cutting the triangle into half to get two right angles, you’ll have the base as half of the given side length.

If you already have a right triangle, make sure you have the length of its hypotenuse and move on.

The Pythagorean Theorem goes as follows: a² + b² = c², with ‘a’ and ‘b’ being the sides and ‘c’ being the triangle's hypotenuse. If you’re working on a right triangle, you’ll use the base and hypotenuse you already have to determine the height.  

If you’re working with an equilateral triangle, you’ll assign values to its sides after cutting it using the height. The hypotenuse(c) will equal the original length of the sides. Meanwhile, the base will equal half the hypotenuse because you’ll cut it in half. The third variable will be the height you need to determine.

Now that you have values for two variables, the hypotenuse and the base, you can plug them into the formula and do the math. Determine the square value of each length by multiplying it by itself, then do the needed subtractions to end up with the squared value of your height. 

For the last step, grab your calculator and use the square root function to determine your exact height.

Method 3: Use The Pythagorean Theorem

Math can be hard a lot of times, but it’s too fun to hate—at least for some people!

Determining the height of the triangle is easy enough if you know all three methods. Regardless of the givens and the triangle’s shape, you’ll be able to plug in the variable into any formula and get your full mark!

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How To Find The Height of a Triangle

Method 1: Use the Area Formula

Method 2: Use the Sides & Angles

Method 3: Use The Pythagorean Theorem

Final Words