What is the height, x, of the equilateral triangle shown? a. 5in. b. 53in. c. 10in. d. 103in.

The equilateral triangle calculator will help you with calculations of standard triangle parameters. Whether you are looking for the equilateral triangle area, its height, perimeter, circumradius, or inradius, this great tool is a safe bet. Scroll down to read more about valuable formulas (such as to calculate the height of an equilateral triangle) and learn what an equilateral triangle is.

The equilateral triangle, also called a regular triangle, is a triangle with all three sides equal. What are the other important properties of that specific regular shape?

• All three internal angles are congruent to each other, and all of them are equal to 60°.
• The altitudes, the angle bisectors, the perpendicular bisectors, and the medians coincide.

The equilateral triangle is a special case of an isosceles triangle, having not just two but all three sides equal. If you would like to learn more about the isosceles triangle, our isosceles triangle calculator is just the tool you need.

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.

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

You can easily find the perimeter of an equilateral triangle by adding all triangles sides together. This regular triangle has all sides equal, so the formula for the perimeter is:

perimeter = 3 × a

How to find the radius of the circle circumscribing the three vertices and the inscribed circle radius?

circumcircle_radius = 2 × h / 3 = a × √3 / 3.

incircle_radius = h / 3 = a × √3 / 6.

Let's take an example from everyday life: we want to find all the parameters of the yield sign.

1. Type the given value into the correct box. Assume we have a sign with a 36-in side length.

2. The equilateral triangle calculator finds the other values in no time. Now we know that:

   • Yield sign height is 31.2 in;
   • Its area equals 561 in²;
   • Perimeter: 108 in;
   • Circumcircle radius is 20.8 in; and
   • Incircle radius 10.4 in.

3. Check out our tool flexibility. Refresh the calculator, and type in the other parameter, e.g., perimeter. It's working this way as well. Isn't that cool?

To find the area of an equilateral triangle, follow the given instructions:

1. Take the square root of 3 and divide it by 4.

2. Multiply the square of the side with the result from step 1.

3. Congratulations! You have calculated the area of an equilateral triangle.

To find the height of an equilateral triangle, proceed as follows:

1. Take the square root of 3 and divide it by 2.

2. Multiply the result from step 1 with the length of the side.

3. You will get the height of the equilateral triangle.

The perimeter of the given triangle is 24 cm.

To calculate the perimeter of an equilateral triangle, we need to multiply its side length by 3. The length of each side of the given triangle is 8 cm. Hence its perimeter will be 3 × 8 cm = 24 cm.

No, a right triangle can't be an equilateral triangle. One of the angles in a right triangle is 90°. Since the sum of all the interior angles in a triangle is 180°, the other two angles in a right triangle are always less than 90°.

According to the definition of equilateral triangles, all the internal angles are equal. Hence, a right triangle can never be equilateral triangle.

ΔABC is an equilateral triangle with side 12.

Find the height of ΔABC (to the nearest tenth).

Possible Answers:

Explanation:

Equilateral triangles have sides of all equal length and angles of 60°. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = 

ΔABC is an equilateral triangle with side of length 8.

Find the height (to the nearest tenth).

Possible Answers:

Explanation:

Equilateral triangles have sides of equal length, with angles of 60°. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = 

What is the height of an equilateral triangle with a side length of 8 in?

Possible Answers:

Correct answer:

Explanation:

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of

The side with length

What is the height of a triangle with side lengths 4, 4, 4?

Possible Answers:

Correct answer:

Explanation:

To solve, it's easiest to first visualize the height's relationship with the rest of the triangle's sides:

The height is one of the legs of a right triangle. The hypotenuse is 4, and the other leg is 2, or half of the base side, 4. To determine the height, use Pythagorean Theorem:

If an equilateral triangle has a length of

Possible Answers:

Correct answer:

Explanation:

The following formula can be used to determine the height of an equilateral triangle when we are given the length of the sides:

The equilateral triangle shown below has a side length of

Possible Answers:

Correct answer:

Explanation:

To solve for the height of an equilateral triangle, we can divide the triangle into two right triangles. In the below image, the bisecting line represents the height, and we can solve for height by applying the Pythagorean Theorem:

Find the height of an equilateral triangle with a side length of

Possible Answers:

Correct answer:

Explanation:

An equilateral triangle has three equal sides, as can be deduced from its name. This also means that as a result, the triangle is also equiangular. That is, all its interior angles are the same. Because the sum of internal angles of a triangle is

The height (the dotted line in the diagram) is the length that bisects the apical angle while bisecting the base of the triangle. Drawing the dotted line splits the equilateral triangle into two congruent right triangles. Because they're the same, we only need to consider one of the two smaller triangles to obtain our answer. Keeping in mind that the base of the equilateral triangle has been bisected, this means that the base of the right triangle has the length of

This problem can also be solved using trigonometric functions or even revisiting 30-60-90 triangle rules. 

Using the Pythagorean Theorem, 

Therefore, the height of the equilateral triangle is 

Ellen
Certified Tutor

St. Catherine University, Bachelor of Science, Public Health. Rasmussen College-Minnesota, Bachelor of Science, Nursing (RN).

Lucy
Certified Tutor

Pukyung National University, Bachelor of Science, Japanese Studies. University of Padova, Master of Science, International Bu...

Ana Maria
Certified Tutor

Florida International University, Bachelor of Science, Mathematics Teacher Education.

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: