Scalene Triangle Calculator

What Is a Scalene Triangle?

A scalene triangle is a triangle in which all three sides have different lengths ($a \neq b \neq c$) and consequently all three internal angles have different measurements ($\alpha \neq \beta \neq \gamma$). Unlike an equilateral triangle (where all sides and angles are identical) or an isosceles triangle (which possesses two equal sides), a scalene triangle has no lines of symmetry.

Scalene triangles can also be classified by their interior angles:

• Acute Scalene Triangle: All three angles are less than $90^\circ$ ($\alpha, \beta, \gamma < 90^\circ$). Explore more acute geometry with our Acute Triangle Calculator.
• Right Scalene Triangle: One angle measures exactly $90^\circ$, and the side lengths satisfy the Pythagorean theorem $a^2 + b^2 = c^2$. Calculate right triangles using our Right Triangle Calculator.
• Obtuse Scalene Triangle: One angle is greater than $90^\circ$. You can explore obtuse properties with our Obtuse Triangle Calculator.

Scalene Triangle Formulas

Given three side lengths $a$, $b$, and $c$, several core geometric formulas apply:

1. Perimeter and Semiperimeter

The perimeter $P$ is the total boundary distance around the triangle: $$P = a + b + c$$ The semiperimeter $s$ is half of the perimeter: $$s = \frac{P}{2} = \frac{a + b + c}{2}$$

2. Area Using Heron's Formula

When all three sides $a$, $b$, and $c$ are known, Heron's formula calculates the area $A$ without requiring an altitude: $$A = \sqrt{s(s - a)(s - b)(s - c)}$$ Alternatively, if an angle $\gamma$ between sides $a$ and $b$ is known (SAS): $$A = \frac{1}{2}ab \sin(\gamma)$$ For specific two-side problems, compare with our SAS Triangle Area Calculator.

3. Internal Angles via Law of Cosines

The three interior angles are determined directly from the side lengths: $$\cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc}$$ $$\cos(\beta) = \frac{a^2 + c^2 - b^2}{2ac}$$ $$\cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab}$$ The sum of interior angles in Euclidean geometry always equals $180^\circ$ ($\alpha + \beta + \gamma = 180^\circ$).

4. Altitudes (Heights)

The perpendicular height dropped to each side is given by: $$h_a = \frac{2A}{a}, \quad h_b = \frac{2A}{b}, \quad h_c = \frac{2A}{c}$$

5. Inradius and Circumradius

The radius $r$ of the inscribed circle (incircle) and radius $R$ of the circumscribed circle (circumcircle) are: $$r = \frac{A}{s}$$ $$R = \frac{abc}{4A}$$

Step-by-Step Calculation Example

Consider a scalene triangle with side lengths $a = 5$, $b = 6$, and $c = 7$:

• Verify Triangle Inequality: $5 + 6 = 11 > 7$, $5 + 7 = 12 > 6$, and $6 + 7 = 13 > 5$. Valid triangle.
• Perimeter: $P = 5 + 6 + 7 = 18$
• Semiperimeter: $s = 18 / 2 = 9$
• Area: $A = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} \approx 14.6969$
• Angle $\alpha$: $\cos(\alpha) = \frac{36 + 49 - 25}{2 \times 6 \times 7} = \frac{60}{84} \approx 0.7143 \implies \alpha \approx 44.4153^\circ$
• Angle $\beta$: $\cos(\beta) = \frac{25 + 49 - 36}{2 \times 5 \times 7} = \frac{38}{70} \approx 0.5429 \implies \beta \approx 57.1217^\circ$
• Angle $\gamma$: $\gamma = 180^\circ - 44.4153^\circ - 57.1217^\circ \approx 78.4630^\circ$
• Inradius: $r = 14.6969 / 9 \approx 1.6330$
• Circumradius: $R = (5 \times 6 \times 7) / (4 \times 14.6969) \approx 3.5722$