Classifying Triangles Calculator

How to Classify Triangles

In Euclidean geometry, every triangle is classified based on two fundamental geometric attributes: the relative lengths of its three sides and the measures of its three internal angles. Because internal angles always sum to exactly 180 degrees ($\alpha + \beta + \gamma = 180^\circ$), side relationships and angle relationships are directly linked.

Classification by Side Lengths

Comparing the side lengths $a$, $b$, and $c$ produces three categories:

• Equilateral Triangle: All three sides have equal length ($a = b = c$). As a consequence, all three internal angles are also equal, measuring exactly 60 degrees each. Every equilateral triangle is equiangular and acute.
• Isosceles Triangle: At least two sides are of equal length (for example, $a = b \neq c$). The angles opposite these equal sides are also congruent (equal).
• Scalene Triangle: All three sides have different lengths ($a \neq b \neq c$). All three internal angles have distinct measures.

Classification by Internal Angles

Evaluating the internal angles ($\alpha, \beta, \gamma$) or using the Pythagorean relationship ($a^2 + b^2$ compared with $c^2$ for the longest side $c$) yields three categories:

• Acute Triangle: All three angles are strictly less than 90 degrees ($\alpha, \beta, \gamma < 90^\circ$). In terms of side lengths, $a^2 + b^2 > c^2$.
• Right Triangle: Exactly one internal angle equals 90 degrees ($\gamma = 90^\circ$). The side lengths satisfy the Pythagorean Theorem: $a^2 + b^2 = c^2$.
• Obtuse Triangle: Exactly one internal angle is strictly greater than 90 degrees ($\gamma > 90^\circ$). In terms of side lengths, $a^2 + b^2 < c^2$.

The Triangle Inequality Theorem

Before classifying any triangle from three side lengths, the Triangle Inequality Theorem must hold true:

$$a + b > c, \quad a + c > b, \quad b + c > a$$

If the sum of any two sides is less than or equal to the third side, the segments cannot meet to form a closed triangle.

Formulas for Complete Triangle Metrics

This calculator computes all derived geometric properties:

• Internal Angles (Law of Cosines): $$\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$$
• Area (Heron's Formula): $$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}, \quad s = \frac{a + b + c}{2}$$
• Inradius: $$r = \frac{\text{Area}}{s}$$
• Circumradius: $$R = \frac{a \cdot b \cdot c}{4 \cdot \text{Area}}$$
• Altitudes: $$h_a = \frac{2 \cdot \text{Area}}{a}, \quad h_b = \frac{2 \cdot \text{Area}}{b}, \quad h_c = \frac{2 \cdot \text{Area}}{c}$$

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