Equilateral Triangle Calculator
Calculate the side length, perimeter, area, height, inradius, and circumradius of an equilateral triangle from one known value.
Complete equilateral triangle solver
An equilateral triangle has three equal sides and three equal angles of 60 degrees. Because every measurement is connected, one known value is enough to calculate all the other main properties.
An equilateral triangle calculator finds side length, height, perimeter, and area using one known value. In an equilateral triangle, all three sides and angles are equal, with each angle measuring 60 degrees.
Use formulas like area = (sqrt(3) / 4) x side^2 to calculate exact dimensions, then compare the result with the diagram and step-by-step work.
Choose whether you know the side, perimeter, area, height, inradius, or circumradius. The calculator returns exact geometry values and a labeled diagram for quick checking.
You can also choose separate input and output units, then control decimal places for cleaner homework answers, construction notes, or quick estimates.
Area
Side length --, perimeter --
Side length
--
All three sides are equal.
Perimeter
--
P = 3s.
Height
--
h = s sqrt(3) / 2.
Radii
--
Inradius and circumradius.
Median and altitude
--
Median, altitude, and angle bisector coincide.
Centroid distances
--
From base and from vertex.
Area formula
--
Height formula
--
Angle measure
60 degrees each
Step-by-step calculation
Note: These formulas assume a true equilateral triangle. If only two sides are equal, use an isosceles triangle method instead.
How to use the equilateral triangle calculator
- Choose the known value: Select side length, perimeter, area, height, inradius, or circumradius.
- Enter the measurement: Use a positive number and choose the input unit that matches your problem.
- Choose output settings: Select a different output unit if you need conversion, and choose how many decimal places to show.
- Calculate the side: The calculator first converts the known value into the side length s.
- Read every result: Area, perimeter, height, inradius, and circumradius are calculated from the same side length.
- Check the diagram: The diagram labels the side, height, center, and 60 degree angle so you can connect the formulas to the shape.
Unit conversion and rounding guide
This calculator can accept a value in one unit and show the answer in another. That is useful when a worksheet gives centimeters but a plan, material list, or drawing needs inches, feet, or meters.
Length conversion
Side length, perimeter, height, inradius, circumradius, median, and altitude use the selected output length unit.
Area conversion
Area is converted with the square of the length conversion. For example, converting meters to centimeters multiplies area by 10,000.
Rounding control
Use fewer decimals for quick estimates and more decimals for exact-looking homework checks or measurement-sensitive drawings.
Equilateral triangle formulas
An equilateral triangle is a regular triangle. Every side has length s, every interior angle is 60 degrees, and the height splits the shape into two 30-60-90 right triangles.
The most common formulas are area = s^2 sqrt(3) / 4, perimeter = 3s, height = s sqrt(3) / 2, inradius = s sqrt(3) / 6, and circumradius = s sqrt(3) / 3.
A = s^2 sqrt(3) / 4
P = 3s
h = s sqrt(3) / 2
r = s sqrt(3) / 6, R = s sqrt(3) / 3
If you know area instead of side length, solve the area formula backward: s = sqrt(4A / sqrt(3)). If you know height, use s = 2h / sqrt(3).
Formula reference table
| Known value | Find side length | Then calculate | Best used for |
|---|---|---|---|
| Side s | s = s | Area, height, perimeter, radii | Most textbook problems |
| Perimeter P | s = P / 3 | Divide first, then use side formulas | Fence, border, or edge length tasks |
| Area A | s = sqrt(4A / sqrt(3)) | Perimeter, height, radii | Floor, tile, and surface problems |
| Height h | s = 2h / sqrt(3) | Area = sh / 2 | Altitude or construction drawings |
| Inradius r | s = 2 sqrt(3) r | Circle touching all three sides | Inscribed circle geometry |
| Circumradius R | s = sqrt(3) R | Circle through all three vertices | Circumscribed circle geometry |
General triangle reference: Encyclopaedia Britannica - Triangle.
Which measurement should you enter?
Start with the value that is measured most directly. This keeps the calculation cleaner and reduces rounding errors before the calculator finds the side length.
Use side length when possible
Side length is the most direct input because every other formula starts from s. It is best for homework, drawings, and physical objects where one edge is easy to measure.
Use perimeter for border problems
If you know the total trim, fence, wire, or edge length, choose perimeter. The calculator divides by 3 to recover one side.
Use area for surface coverage
Choose area when the problem is about paint, fabric, tile, land, or any triangular surface. Make sure the area unit is squared, such as ft^2 or m^2.
Use radii for circle geometry
Choose inradius for a circle touching the sides, or circumradius for a circle passing through the vertices. These appear often in advanced geometry problems.
Exact radicals vs decimal results
Many equilateral triangle answers are exact radical expressions because the height comes from a 30-60-90 triangle. Decimal values are easier to measure, but exact forms are often preferred in algebra and geometry classes.
| Side length | Exact height | Decimal height | Exact area | Decimal area |
|---|---|---|---|---|
| 4 | 2sqrt(3) | 3.464 | 4sqrt(3) | 6.928 |
| 6 | 3sqrt(3) | 5.196 | 9sqrt(3) | 15.588 |
| 10 | 5sqrt(3) | 8.660 | 25sqrt(3) | 43.301 |
Tip: keep the exact radical form until the last step when a teacher asks for an exact answer. Use decimals when measuring, estimating material, or matching a real-world unit.
30-60-90 triangle reference: Math Open Reference - 30-60-90 Triangle.
How to verify the triangle is really equilateral
The formulas on this page work only when the triangle is equilateral. Use these checks before relying on the result for a worksheet, layout, cutting plan, or design sketch.
Side check
Measure all three sides. If all three side lengths match, the triangle is equilateral and each angle is 60 degrees.
Angle check
If all three interior angles are 60 degrees, the triangle is equilateral. This is useful when a problem gives angles instead of side lengths.
Height check
The height should land at the midpoint of the opposite side. It should also equal about 0.866 times the side length.
Angle proof reference: ProofWiki - Internal Angle of Equilateral Triangle.
Worked example: side length 6 cm
Suppose an equilateral triangle has side length s = 6 cm. Since all sides are equal, the perimeter is 3 times the side length.
Perimeter
P = 3s
P = 3 x 6 = 18 cm.
Height
h = s sqrt(3) / 2
h = 6 sqrt(3) / 2 = 5.196 cm.
Area
A = s^2 sqrt(3) / 4
A = 36 sqrt(3) / 4 = 15.588 cm^2.
Radii
r = h / 3, R = 2h / 3
r = 1.732 cm and R = 3.464 cm.
Common mistakes with equilateral triangles
Most mistakes come from mixing triangle types, using inconsistent units, or treating height as if it were the same as side length.
Confusing height and side
The height is shorter than the side. For side s, height is about 0.866s.
Forgetting square units
Area uses square units such as cm^2 or m^2. Length results use the original unit.
Using the wrong triangle
A triangle with only two equal sides is isosceles, not equilateral, so these formulas may not apply.
Interesting fact
An equilateral triangle with side length 1 has an area of sqrt(3) / 4, or about 0.433 square units. Because each interior angle is 60 degrees, six identical equilateral triangles can meet around one point and fill a full 360 degrees. Wolfram MathWorld lists the 60-degree angle property and the equilateral triangle area formula; source: Wolfram MathWorld - Equilateral Triangle.
Frequently Asked Questions
What is an equilateral triangle in geometry?
An equilateral triangle is a triangle with three equal side lengths. Because the sides are equal, the three interior angles are also equal, and each angle measures 60 degrees. In geometry, each vertex follows the same pattern, so the shape is useful for a student, teacher, or worksheet that needs a predictable triangle diagram.
How does the calculator find the area of an equilateral triangle?
The calculator uses the formula A = s^2 sqrt(3) / 4, where s is the side length. For example, if s = 6, the area is 6^2 sqrt(3) / 4, or about 15.588 square units. This gives the same solution as using base x height / 2 after finding the altitude.
How do I find the height or altitude from the side length?
The height, also called the altitude, is h = s sqrt(3) / 2. It runs from a vertex to the midpoint of the opposite base and divides the equilateral triangle into two 30-60-90 right triangles, which is why sqrt(3) appears in the formula.
Can I find the side length from an area measurement?
Yes. Rearrange the area formula to get s = sqrt(4A / sqrt(3)). This math tool uses that conversion when you choose area as the known measurement, then uses the recovered side length to calculate perimeter, height, and radius values.
What is the difference between inradius, circumradius, and radius?
The inradius is the radius of the circle that touches all three sides from inside the triangle. The circumradius is the radius of the circle that passes through all three vertices. In an equilateral triangle, the incenter, circumcenter, and centroid meet at the same point, and the circumradius is twice the inradius.
Does the calculator work with different measurement units?
Yes. Enter the known value in the input measurement unit, then choose the output unit for the results. The calculator converts side length, height, altitude, perimeter, radius values, median, and centroid distances into the output unit, while area uses the square of that output unit.
Is an equilateral triangle always an acute triangle?
Yes. An acute triangle has all angles less than 90 degrees, and every equilateral triangle has three 60 degree angles. That means an equilateral triangle is always acute, always equiangular, and always regular. On a diagram, the three angle marks should match.
How do I find the perimeter if I only know the area?
First find the side length from the area using s = sqrt(4A / sqrt(3)). Then multiply that side length by 3 to get the perimeter. The calculator does both steps automatically when you choose area as the known value, which is helpful when a worksheet gives surface area but asks for the outside edge.
Why is the circumradius twice the inradius in this triangle?
In an equilateral triangle, the centroid sits on every median and altitude. The inradius reaches from the centroid to a side, while the circumradius reaches from the centroid to a vertex. Along the same height, those distances are h / 3 and 2h / 3, so the circumradius is twice the inradius.
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Disclaimer: This equilateral triangle calculator is for general educational and informational use only. It provides geometry calculations based on user-entered values and standard equilateral triangle formulas. Always verify measurements, units, rounding, and formatting requirements for schoolwork, construction sketches, design layouts, tests, or professional projects.
Last updated: May 18, 2026