The formula for a regular triangle area is equal to the squared side times the square root of 3 divided by 4:
area = (a² × √3)/ 4
and the equation for the height of an equilateral triangle looks as follows:
h = a × √3 / 2, where a is a side of the triangle.
But do you know where the formulas come from? You can find them in at least two ways: deriving from the Pythagorean theorem (discussed in our Pythagorean theorem calculator) or using trigonometry.
1. Using Pythagorean theorem
The basic formula for triangle area is side a (base) times the height h, divided by 2:
area = (a × h) / 2
Height of the equilateral triangle is derived by splitting the equilateral triangle into two right triangles. See our right triangle calculator to learn more about right triangles.
One leg of that right triangle is equal to height, another leg is half of the side, and the hypotenuse is the equilateral triangle side.
`(a/2)² + h² = a²` After simple transformations, we get a formula for the height of the equilateral triangle:h = a × √3 / 2
Substituting h into the first area formula, we obtain the equation for the equilateral triangle area:
area = a² × √3 / 4
2. Using trigonometry
Let's start with the trigonometric triangle area formula:
area = (1/2) × a × b × sin(γ), where γ is the angle between the sides.
We remember that all sides and all angles are equal in the equilateral triangle, so the formula simplifies to:
area = 0.5 × a × a × sin(60°)
What is more, we know that the sine of 60° is √3/2, so the formula for equilateral triangle area is:
area = (1/2) × a² × (√3 / 2) = a² × √3 / 4
The height of the equilateral comes from the sine definition:
h / a = sin(60°) so h = a × sin(60°) = a × √3 / 2
Instructor: Yuanxin (Amy) Yang Alcocer Show bio
Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
An equilateral triangle is a triangle that has three sides of the same length. Explore the steps in finding the area of an equilateral triangle and learn to calculate the area through solved examples. Updated: 01/05/2022
Steps to Solve
To find the area of an equilateral triangle, or a triangle with 3 equal sides, all you have to do is to follow these steps.
Step 1
Use this formula for the area of an equilateral triangle:
Area FormulaIn this case, the upper case A is the area and the lower case a is the length of the sides, which again, are all equal, hence the same letter being used.
Step 2
Plug in your value for the length of a side of your equilateral triangle and evaluate.
That is it! With a simple formula like this, calculating the area is really easy. Just plug in to the formula and evaluate!
Let's see this in action.
- Video
- Quiz
- Course
Finding the Area
Find the area of this equilateral triangle.
ProblemStep 1
The first step is to remember the formula, which is the square root of 3 divided by 4 multiplied by the square of the side of the equilateral triangle.
Area FormulaStep 2
The next step is to plug in the value for the side of the equilateral triangle. For this triangle, the side is 3 inches long. So, you plug in 3 for the variable s. Square that and you get 9. Multiplying the 9 by the square root of 3 and dividing by 4, you get 3.9 inches squared. And you are done!
Another Example
Let's try another one.
Find the area of an equilateral triangle with sides that measure 6 centimeters.
Step 1
First, write down the formula. The square root of 3 divided by 4 multiplied by the square of the side of the equilateral triangle.
Step 2
Your side measures 6 centimeters, so you plug in 6 to the formula. 6 squared is 36. Multiplying that by the square root of 3 and dividing by 4, you get 15.6 centimeters squared. And you are done.
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