Free Triangle Height Calculator

What Is the Triangle Height Calculator?

The Triangle Height Calculator (also referred to as a Triangle Altitude Calculator) is a free online tool that allows you to quickly calculate the height of any triangle. Supporting right, equilateral, isosceles, and scalene triangles, it provides not only altitudes but also side lengths, angles, perimeter, and area. This makes it an invaluable resource for students, teachers, and professionals seeking a free triangle height calculator.

Defining Triangle Altitude

In geometry, the altitude (or height) of a triangle is the perpendicular distance from a vertex to the line containing the opposite side, which is called the base. Every triangle has three such altitudes. Their intersection point is known as the orthocenter. Knowing how to find the height of a triangle is key to solving area problems and many other geometric questions.

General Methods for Calculating Height

Using Area and Base

The most direct method to find the altitude when the area ($A$) and the length of a base ($b$) are known is to rearrange the standard area formula:

$$A = \frac{1}{2} \times b \times h \quad \Rightarrow \quad h = \frac{2A}{b}$$

This equation works for any triangle and is often the fastest approach if the area is available.

Using Three Sides (Heron's Formula)

If only the side lengths $a, b, c$ are known, you can compute the area first with Heron's formula. The semi-perimeter is:

$$s = \frac{a + b + c}{2}$$

The area then becomes:

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

After obtaining the area, the altitude for a specific base (say $a$) is:

$$h_a = \frac{2A}{a},\quad h_b = \frac{2A}{b},\quad h_c = \frac{2A}{c}$$

The Triangle Height Calculator performs these calculations instantly, delivering all three altitudes without manual effort.

Using Two Sides and the Included Angle

When two sides and the angle between them are given, the area can be expressed trigonometrically:

$$A = \frac{1}{2} ab \sin(\gamma)$$

where $\gamma$ is the angle between sides $a$ and $b$. The altitude to the third side is then $h = 2A / \text{opposite side}$.

Special Triangle Types and Their Height Formulas

Equilateral Triangle

All sides are equal ($s$) and every interior angle is $60^\circ$. The three altitudes are identical in length. The height is given by:

$$h = \frac{s \sqrt{3}}{2}$$

For example, with a side length of 10 cm, the height is $10 \times \sqrt{3} / 2 \approx 8.66\ \text{cm}$. This formula, derived from the Pythagorean theorem, is one of the most common ways to calculate triangle height in equilateral shapes.

Isosceles Triangle

An isosceles triangle has two equal legs ($a$) and a base ($b$). The altitude from the apex to the base is perpendicular and bisects the base. Applying the Pythagorean theorem:

$$h_{\text{apex}} = \sqrt{a^{2} - \left(\frac{b}{2}\right)^{2}}$$

For instance, legs of 13 cm and a base of 10 cm give $h_{\text{apex}} = \sqrt{169 - 25} = \sqrt{144} = 12\ \text{cm}$. The other two altitudes can be found using area or trigonometric formulas once the area is known.

Right Triangle

In a right triangle, the two legs are perpendicular and each can be considered an altitude. The third altitude (from the right angle to the hypotenuse) is obtained from the area:

$$h_{\text{hypotenuse}} = \frac{p \times q}{\text{hypotenuse}}$$

where $p$ and $q$ are the legs. The hypotenuse is $\sqrt{p^{2} + q^{2}}$. For a 3-4-5 triangle, this yields $h = (3 \times 4) / 5 = 2.4\ \text{units}$. This method shows an alternative way to find the height of a triangle without explicit area input.

Practical Example Using the Calculator

Consider a scalene triangle with sides $a = 6\ \text{in}$, $b = 14\ \text{in}$, and $c = 17\ \text{in}$. Choose “scalene” in the Triangle Height Calculator, enter the three side lengths, and the tool instantly returns the three altitudes: approximately $13.17\ \text{in}$, $5.644\ \text{in}$, and $4.648\ \text{in}$. Along with these altitudes, you get the angles, perimeter ($37\ \text{in}$), and area. This example demonstrates how the Calculator for Triangle Height can handle complex inputs with ease.

Conclusion

The Free Triangle Height Calculator is an efficient tool for anyone needing to calculate triangle height quickly. By supporting a variety of input methods (sides, area, angle-based) and covering all triangle types, it eliminates tedious manual work. Whether you want to compute altitude from area, side lengths, or special triangle formulas, this calculator provides immediate and accurate results—making it an indispensable geometry companion.