Equilateral triangle calculator
An equilateral triangle with sides of 6 has an area of 15.5885, a height of 5.1962 and a perimeter of 18 in the equilateral triangle calculator.
Area
- Area Heron's formula from the three sides: s = 9, area = sqrt(s(s-a)(s-b)(s-c)).
- 15.5885
- Sides
- a = 6, b = 6, c = 6
- Angles Summing to 180 degrees, as every plane triangle must.
- A = 60deg, B = 60deg, C = 60deg
- Area
- 15.5885
- Perimeter
- 18
- Type
- equilateral, acute
- Heights The perpendicular from each side to the opposite vertex. Area is half the base times the height for any of the three.
- to a: 5.1962, to b: 5.1962, to c: 5.1962
- Circumradius The radius of the circle through all three vertices.
- 3.4641
- Inradius The radius of the largest circle that fits inside.
- 1.7321
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 30-40-50 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | 40-42-58 |
| 9 | 40 | 41 | 18-80-82 |
- The Pythagorean theorem holds only for right triangles. For any other triangle the law of cosines generalizes it: c squared equals a squared plus b squared minus 2ab cos(C), which reduces to Pythagoras when C is 90 degrees and cos(C) is zero.
- Any two sides must add to more than the third, or the shape does not close. That triangle inequality is checked here before anything else is computed.
How the Triangle Calculator works
All three sides of an equilateral triangle are equal and all three angles are 60°, so one side length a fixes everything. The height is h = (√3 ÷ 2) × a, the area is (√3 ÷ 4) × a², and the perimeter is 3a.
For a = 6: h = 0.8660 × 6 = 5.1962, the area is 0.4330 × 36 = 15.5885, which is exactly 9√3, and the perimeter is 18.
The height formula is the Pythagorean theorem. The height splits the triangle into two 30-60-90 triangles, each with a hypotenuse of a and a short leg of a ÷ 2, so h² = a² - (a ÷ 2)² = ¾a² and h = (√3 ÷ 2)a. Half the base times that height gives the area, (√3 ÷ 4)a².
The circumradius is a ÷ √3 = 3.4641 and the inradius is a ÷ (2√3) = 1.7321, exactly half of it. Both circles share a center, a third of the way up the height from the base.
Working backward, the side from the area is a = √(4 × area ÷ √3), and from the height it is a = 2h ÷ √3.
Triangle Calculator FAQ
What is the area of an equilateral triangle with side 10?
(√3 ÷ 4) × 100 = 43.30, exactly 25√3.
What are the angles of an equilateral triangle?
60° each, since the three are equal and a triangle's angles add up to 180°.
Is an equilateral triangle also isosceles?
Yes, by the usual definition: an isosceles triangle has at least two equal sides, and an equilateral one has three.
More from the Triangle Calculator
- Right triangle calculator
- Law of cosines calculator
- Law of sines calculator
- Isosceles triangle solver
- Pythagorean theorem calculator
- Triangle area calculator