Free Equilateral Triangle Calculator

Understanding the Equilateral Triangle

An equilateral triangle, often called a regular triangle, is a foundational geometric figure in which all three sides are identical in length. This perfect symmetry gives rise to several important properties: every interior angle is exactly 60°, and each altitude, angle bisector, perpendicular bisector, and median coincide into the same line. Because it shares features with isosceles triangles while going one step further in equality, the equilateral triangle stands as a special case of that family.

For students, engineers, or anyone working with such a shape, a Regular Triangle Calculator (also known as an equilateral triangle calculator) can rapidly compute every relevant parameter. This free online regular triangle area and height finder delivers instant values for the Equilateral Triangle Height, Equilateral Triangle Area, side length, perimeter, circumradius, and inradius—all from a single known measurement.

Area and Height Formulas

The two most frequently requested measurements—area and height—can be expressed with straightforward formulas that rely only on the side length $a$.

$$\text{Area} = \frac{a^{2} \sqrt{3}}{4}$$ $$\text{Height} = \frac{a \sqrt{3}}{2}$$

Both expressions originate from either the Pythagorean theorem or trigonometric relationships.

Derivation via the Pythagorean Theorem

The universal area formula for a triangle is $\dfrac{1}{2} \times \text{base} \times \text{height}$. When we draw one altitude in an equilateral triangle, we split it into two congruent right triangles. In each right triangle, the hypotenuse equals the side $a$, one leg is the height $h$, and the other leg is half a side, $a/2$. Applying the Pythagorean theorem:

$$\left(\frac{a}{2}\right)^{2} + h^{2} = a^{2}$$

Solving for $h$ yields $h = a \cdot \dfrac{\sqrt{3}}{2}$, which is the Equilateral Triangle Height. Substituting this into the area formula gives the Equilateral Triangle Area: $\dfrac{a^{2} \sqrt{3}}{4}$.

Derivation Using Trigonometry

The trigonometric area formula states $\text{Area} = \dfrac{1}{2} \times a \times b \times \sin(\gamma)$, where $\gamma$ is the angle between the two sides. For an equilateral triangle, all sides are equal and all angles are 60°, so the formula simplifies:

$$\text{Area} = \frac{1}{2} \times a \times a \times \sin 60^\circ = \frac{1}{2} a^{2} \times \frac{\sqrt{3}}{2} = \frac{a^{2} \sqrt{3}}{4}$$

Height follows from the definition of sine: $\sin 60^\circ = \dfrac{h}{a}$, so $h = a \sin 60^\circ = a \cdot \dfrac{\sqrt{3}}{2}$.

Perimeter, Circumcircle, and Incircle Radius

Because all sides are equal, the perimeter is simply three times the side length:

$$\text{Perimeter} = 3a$$

The circumscribed circle (circumcircle) radius and the inscribed circle (incircle) radius also depend only on the side $a$:

$$R_{\text{circum}} = \frac{a \sqrt{3}}{3}, \qquad r_{\text{in}} = \frac{a \sqrt{3}}{6}$$

Equivalently, in terms of the height $h$: $R_{\text{circum}} = \dfrac{2}{3}h$ and $r_{\text{in}} = \dfrac{1}{3}h$.

Using the Equilateral Triangle Calculator in Practice

This geometry tool is designed for maximum flexibility. As an example, consider a standard yield sign whose Triangle Side Length is 36 inches. By entering that single number, the calculator instantly returns all other parameters:

• Height: 31.2 in
• Area: 561 in²
• Perimeter: 108 in
• Circumradius: 20.8 in
• Inradius: 10.4 in

The same process works in reverse—you can start with the height, perimeter, or any other known value, and the tool will compute the rest. Whether you need the Equilateral Triangle Height for a construction project or the area for material estimation, this calculator supplies exact results in seconds.