The Equilateral Triangle Calculator finds the area, height, perimeter, and inscribed and circumscribed circle radii of an equilateral triangle from a single side length. Because all three sides and all three angles of an equilateral triangle are equal, one measurement is enough to determine every other property of the shape.
An equilateral triangle is the most symmetric of all triangles: every side measures the same length, and every interior angle measures exactly 60 degrees. That symmetry is what collapses the general triangle formulas down to simple one-variable expressions.
Calculate the area of an equilateral triangle
The area formula for an equilateral triangle comes from the general triangle area formula, simplified using the fact that all sides and angles are equal.
A = (√3 / 4) × side²Worked example: an equilateral triangle with a side of 8 units.
1. Square the side length. 8² = 64.
2. Multiply by √3 / 4. 64 × (1.7321 / 4) = 64 × 0.4330 ≈ 27.71 square units.
Calculate the height of an equilateral triangle
The height (or altitude) drops from any vertex perpendicular to the opposite side, splitting the equilateral triangle into two congruent 30-60-90 right triangles.
h = (√3 / 2) × sideWorked example: the same triangle with a side of 8 units.
1. Multiply the side by √3 / 2. 8 × (1.7321 / 2) = 8 × 0.8660 ≈ 6.93 units.
Because every side is equal, the height dropped from any vertex is identical, which is not true for a general scalene or even isosceles triangle.
Calculate the perimeter
Perimeter is simply three times the single side length, the most direct of the equilateral formulas.
P = 3 × sideWorked example: the same triangle with a side of 8 units.
1. Multiply the side by 3. 3 × 8 = 24 units.
Find the inscribed and circumscribed circle radii
An equilateral triangle has a particularly clean relationship with its inscribed circle (inside, tangent to all three sides) and circumscribed circle (outside, passing through all three vertices).
r (inscribed) = side / (2√3)
R (circumscribed) = side / √3Worked example: the same triangle with a side of 8 units.
Inscribed radius: 8 / (2 × 1.7321) ≈ 8 / 3.4642 ≈ 2.31 units.
Circumscribed radius: 8 / 1.7321 ≈ 4.62 units.
Notice that the circumscribed radius is exactly twice the inscribed radius for any equilateral triangle, a relationship that does not generally hold for other triangle shapes.
Understand why every angle is 60 degrees
Since all three sides of an equilateral triangle are equal, all three angles opposite those sides must also be equal, a direct consequence of the isosceles triangle theorem applied to all three pairs of sides at once. Because the three angles are equal and must sum to 180 degrees, each one is exactly 180 / 3 = 60 degrees.
Compare equilateral and general triangle formulas
| Property | General triangle | Equilateral triangle |
|---|---|---|
| Area | ½ × base × height, or Heron's formula | (√3/4) × side² |
| Perimeter | a + b + c | 3 × side |
| Angles | vary, sum to 180° | all 60° |
The equilateral formulas are special cases of the general triangle formulas; the symmetry simply removes the need for separate side and angle inputs.
Work through a second full example
An equilateral triangle with a side of 12 units shows every formula together. 1. Area. (√3/4) × 12² = 0.4330 × 144 ≈ 62.35 square units. 2. Height. (√3/2) × 12 = 0.8660 × 12 ≈ 10.39 units.
3. Perimeter. 3 × 12 = 36 units.
4. Inscribed radius. 12 / (2 × 1.7321) ≈ 3.46 units.
5. Circumscribed radius. 12 / 1.7321 ≈ 6.93 units.
As with the 8-unit example earlier, the circumscribed radius here is again exactly double the inscribed radius, confirming the same fixed ratio holds regardless of the side length chosen.
Verify the area using Heron's formula as a cross-check
Because an equilateral triangle is also a valid input to the general Heron's formula, plugging in three equal sides confirms the simplified formula gives the same answer.
For a side of 8, the semi-perimeter is (8+8+8)/2 = 12, and Heron's formula gives √(12 × 4 × 4 × 4) = √768 ≈ 27.71 square units, matching the (√3/4) × side² result found earlier on this page.
This cross-check is a useful way to confirm the simplified equilateral formula whenever the general formula is already familiar.
Frequently asked questions
How do you find the area of an equilateral triangle?
Use A = (√3/4) × side². For a side of 8 units, the area is about 27.71 square units.
How do you find the height of an equilateral triangle?
Use h = (√3/2) × side. For a side of 8 units, the height is about 6.93 units.
Why are all the angles in an equilateral triangle 60 degrees?
Because all three sides are equal, all three opposite angles must also be equal (by the isosceles triangle theorem applied to every pair). Since the three equal angles sum to 180°, each one is 60°.
What is the relationship between the inscribed and circumscribed circle radii?
For an equilateral triangle, the circumscribed radius is always exactly twice the inscribed radius, a relationship unique to this triangle's high symmetry.
Is an equilateral triangle also isosceles?
Yes. Every equilateral triangle satisfies the isosceles triangle condition (at least two equal sides), since it has three equal sides, making it a special case of isosceles rather than a separate category.
How is the height formula derived?
Dropping a perpendicular from one vertex to the midpoint of the opposite side creates two 30-60-90 right triangles. Applying the Pythagorean theorem to one of those halves gives h = (√3/2) × side.
Can two different measurements be given instead of just the side?
Since one measurement fully determines an equilateral triangle, entering a second, independent measurement, such as the area, works backward to confirm or recalculate the side length instead.
Summary
An equilateral triangle's area, height, perimeter, and circle radii all follow from its single side length, thanks to its three equal sides and three 60-degree angles. A triangle with an 8-unit side has an area of about 27.71 square units, a height of about 6.93 units, and a perimeter of 24 units.
The circumscribed circle radius is always exactly twice the inscribed circle radius for this shape.