How to Find the Sides of an Equilateral Triangle if the Area of an Equilateral Triangle is known?

Image credit: Desmos

Before we begin, let’s review what an equilateral triangle is — a triangle with three equal side lengths and three equal internal angles of 60° each. Now, let’s get one thing straight: The area of an equilateral triangle is not the perimeter of an equilateral triangle. It's the total space of the triangle’s surface.

As you know, there are many different types of triangles: right triangles, scalene triangles, and isosceles triangles. Again, in an equilateral triangle, the length of the sides of an equilateral triangle are equal.

To determine the area of an equilateral triangle, you must know its side lengths. So, before diving into the equilateral triangle area formula, let's look at how to find the side lengths.

How to Find the Side Lengths of an Equilateral Triangle

When you know the height of the triangle, you can determine the side lengths. Once you've found the side length, you can then determine the area of an equilateral triangle.

When you divide an equilateral triangle into two right triangles, you see the height of an equilateral triangle. This is done by slicing an equilateral triangle in half from the tip of a vertex to the midpoint of one side to form an angle bisector.

Image credit: Desmos

The perpendicular bisector, the straight line that forms two 90° angles, represents the height of the equilateral triangle, as marked by height h. By creating this bisector, we’ve divided this equilateral into two right triangles. To find the height, you can use the Pythagorean theorem:

Since all sides of an equilateral triangle are the same, side A=side C. And since the base of the right triangle is half the side length of the equilateral triangle, side A=side C/2. Now let’s plug in the height, base, and side length of C for the hypotenuse to isolate the value of h:

If you only know the height of an equilateral triangle’s perpendicular bisector, you can use this formula to determine the length of each equal side:

![area of equilateral triangle: formula to determine the length

Let’s apply this formula to a triangle in which h = 9 to find the side lengths:

Now that we know how to use the height of an equilateral triangle to determine its missing side length, let's learn how to solve for the area.

Formula for the Area of Equilateral Triangles

Image credit: Desmos

The figure above is an equilateral triangle. The following formula is used to determine the area of the triangle:

Let's use this formula to determine the area of the triangle above:

Mastering the Area of Equilateral Triangles

Equilaterals are triangles with three equal sides and angles that all measure 60°. When you create a perpendicular bisector line through the vertex of an equilateral, you form two right triangles. You can use the Pythagorean theorem and height of the right triangles within the equilateral to determine the missing side lengths of an equilateral triangle.

Then, you can use the formula A = √3/4 (a²) to determine the area of an equilateral triangle. Knowing how to find the height and area of a triangle with equal sides makes it so much easier to learn other trigonometry formulas.

More Math Homework Help

h = altitude

*Length units are for your reference only since the value of the resulting lengths will always be the same no matter what the units are.

Calculator Use

An equilateral triangle is a special case of a triangle where all 3 sides have equal length and all 3 angles are equal to 60 degrees. The altitude shown h is hb or, the altitude of b. For equilateral triangles h = ha = hb = hc.

If you have any 1 known you can find the other 4 unknowns. So if you know the length of a side = a, or the perimeter = P, or the semiperimeter = s, or the area = K, or the altitude = h, you can calculate the other values. Below are the 5 different choices of calculations you can make with this equilateral triangle calculator. Let us know if you have any other suggestions!

Formulas and Calculations for a equilateral triangle:

• Perimeter of Equilateral Triangle: P = 3a
• Semiperimeter of Equilateral Triangle: s = 3a / 2
• Area of Equilateral Triangle: K = (1/4) * √3 * a2
• Altitude of Equilateral Triangle h = (1/2) * √3 * a
• Angles of Equilateral Triangle: A = B = C = 60°
• Sides of Equilateral Triangle: a = b = c

1. Given the side find the perimeter, semiperimeter, area and altitude

• a is known; find P, s, K and h
• P = 3a
• s = 3a / 2
• K = (1/4) * √3 * a2
• h = (1/2) * √3 * a

2. Given the perimeter find the side, semiperimeter, area and altitude

• P is known; find a, s, K and h
• a = P/3
• s = 3a / 2
• K = (1/4) * √3 * a2
• h = (1/2) * √3 * a

3. Given the semiperimeter find the side, perimeter, area and altitude

• s is known; find a, P, K and h
• a = 2s / 3
• P = 3a
• K = (1/4) * √3 * a2
• h = (1/2) * √3 * a

4. Given the area find the side, perimeter, semiperimeter and altitude

• K is known; find a, P, s and h
• a = √ [ (4 / √3) * K ] = 2 * √ [ K / √3 ]
• P = 3a
• s = 3a / 2
• h = (1/2) * √3 * a

5. Given the altitude find the side, perimeter, semiperimeter and area

• h is known; find a, P, s and K
• a = (2 / √3) * h
• P = 3a
• s = 3a / 2
• K = (1/4) * √3 * a2

For more information on triangles see:

Weisstein, Eric W. "Equilateral Triangle." From MathWorld--A Wolfram Web Resource. Equilateral Triangle.

Weisstein, Eric W. "Altitude." From MathWorld--A Wolfram Web Resource. Altitude.

The area of an equilateral triangle is

Possible Answers:

Correct answer:

Explanation:

An equilateral triangle can be broken down into 2 30-60-90 right triangles (see image). The length of a side (the base) is 2x while the length of the height is

Therefore, if 

A length of the side equals 2x:

What is the area of this triangle if 

Possible Answers:

Correct answer:

Explanation:

We know the formula for the area of an equilateral triangle is:

if 

So, since we are told that 

Find 

Possible Answers:

Correct answer:

Explanation:

This triangle is equilateral; we can tell because each of its sides are the same length, 

What is side 

Possible Answers:

Correct answer:

Explanation:

Since each of this triangle's sides is equal in length, it is equilateral. To find the length of one side of an equilateral triangle, we need to divide the perimeter by 

The height of the triangle is

What is the length of the base of the triangle to the nearest tenth?

Possible Answers:

Correct answer:

Explanation:

Since it is an equilateral triangle, the line that represents the height bisects it into a 30-60-90 triangle.

Here you may use

Use the definition of an equilateral triangle to know that the answer of the hypotenuse also applies to the base of the triangle.

Therefore,

The height of an equilateral triangle is 5. How long are its sides?

Possible Answers:

Correct answer:

Explanation:

The height of an equilateral triangle, shown by the dotted line, is also one of the legs of a right triangle:

The hypotenuse is x, the length of each side in this equilateral triangle, and then the other leg is half of that, 0.5x. 

To solve for x, use Pythagorean Theorem:

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

Correct answer:

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into 

We can then use the height to find the length of the side of the triangle.

Recall that a 

Thus, we can use the ratio and the length of the height to set up the following equation:

Plug in the given height to find the length of the side.

Now, since the perimeter of the shape consists of 

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

Correct answer:

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into 

We can then use the height to find the length of the side of the triangle.

Recall that a 

Thus, we can use the ratio and the length of the height to set up the following equation:

Plug in the given height to find the length of the side.

Now, since the perimeter of the shape consists of 

An equilateral triangle is placed on top of a square, as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

Correct answer:

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into 

We can then use the height to find the length of the side of the triangle.

Recall that a 

Thus, we can use the ratio and the length of the height to set up the following equation:

Plug in the given height to find the length of the side.

Now, since the perimeter of the shape consists of 

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Possible Answers:

Correct answer:

Explanation:

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into 

We can then use the height to find the length of the side of the triangle.

Recall that a 

Thus, we can use the ratio and the length of the height to set up the following equation:

Plug in the given height to find the length of the side.

Now, since the perimeter of the shape consists of 

Jovany
Certified Tutor

Illinois Institute of Technology, Engineer, Biomedical Engineering.

Kelly
Certified Tutor

Ohio State University-Main Campus, Bachelor of Fine Arts, Photography. Ohio State University, Master of Arts, German.

If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources.

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105

Or fill out the form below: