Free Scalene Triangle Calculator

Understanding the Scalene Triangle

A scalene triangle (often called an irregular triangle) is defined by having three sides of completely different lengths — no two sides are equal. Because of this asymmetry, calculations involving its area, perimeter, or height require a flexible approach. The scalene triangle area calculator and scalene triangle perimeter calculator described here handle all essential measurements: side lengths, angles, heights, and area, using the most common formulas.

Perimeter of a Scalene Triangle

Finding the perimeter is straightforward once you know the three side lengths. Simply add them up:

$$\text{Perimeter} = a + b + c$$

where $a$, $b$, and $c$ are the lengths of the sides. This is the simplest part of any irregular triangle calculation.

Area of a Scalene Triangle

There are several reliable formulas to compute the area, depending on which measurements you have available. All produce the same result — choose the one that fits your data:

• Base and height:

  $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

• Two sides and the included angle:

  $$\text{Area} = \frac{1}{2} \times a \times b \times \sin(\gamma)$$

  where $\gamma$ is the angle between sides $a$ and $b$.

• Heron's formula (all three sides known):
  First compute the semiperimeter $s = \dfrac{a+b+c}{2}$, then

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

  This is equivalent to the expanded form $\dfrac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}$.

• Two angles and one side:

  $$\text{Area} = \frac{a^{2} \sin(\beta) \sin(\gamma)}{2 \sin(\beta + \gamma)}$$

  where $\beta$ and $\gamma$ are the angles adjacent to side $a$.

The irregular triangle calculator automates these formulas, so you only need to enter the values you have; it selects the appropriate method.

Height of a Scalene Triangle

You can compute the height corresponding to any base using the area:

$$\text{Height} = \frac{2 \times \text{Area}}{\text{base}}$$

If the area is not yet known, the height can also be derived directly from the side lengths using Heron's expression:

$$\text{Height} = \frac{1}{2b} \sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}$$

For a scalene right triangle (a triangle with three unequal sides but one right angle), the height to the hypotenuse $c$ is given by:

$$\text{Height}_{c} = \frac{a \times b}{c}$$

where $a$ and $b$ are the two legs (perpendicular sides).

Scalene Right Triangle — Area Calculation

A scalene right triangle still follows the right triangle area formula:

$$\text{Area} = \frac{a \times b}{2}$$

or equivalently

$$\text{Area} = \frac{b \times \sqrt{c^{2} - b^{2}}}{2}$$

This matches the general triangle formulas, but takes advantage of the known right angle for simpler computation.

What This Calculator Offers

This triangle with unequal sides tool is designed to solve for any missing values when you have enough input. Whether you know sides, angles, base, and height, or a combination, the calculator returns area, perimeter, heights, and all angles. It supports the same equations used by professional surveyors and engineers, making it practical for homework, construction, or design projects that involve irregular triangles.