Acute Triangle Calculator

What Is an Acute Triangle?

An acute triangle (or acute-angled triangle) is a planar geometric polygon with three interior angles, where every single angle is strictly less than $90^\circ$ ($\pi/2$ radians): $$0^\circ < \alpha < 90^\circ, \quad 0^\circ < \beta < 90^\circ, \quad 0^\circ < \gamma < 90^\circ$$ Because all three angles are acute, no angle is a right angle ($90^\circ$) or an obtuse angle ($> 90^\circ$). All equilateral triangles are acute triangles because each angle measures exactly $60^\circ$.

How to Test if a Triangle is Acute Using Side Lengths

If you know the three side lengths $a$, $b$, and $c$ where $c$ is the longest side ($c \ge a$ and $c \ge b$), you can verify whether the triangle is acute using the Law of Cosines test:

1. Check the Triangle Inequality: Confirm that $a + b > c$. If not, the sides cannot connect to form a valid closed triangle.
2. Compare the Sum of Squares:
   • If $a^2 + b^2 > c^2$, the triangle is Acute.
   • If $a^2 + b^2 = c^2$, the triangle is Right (Pythagorean Theorem).
   • If $a^2 + b^2 < c^2$, the triangle is Obtuse.

Formulas for Solving Acute Triangles

Depending on the given inputs, acute triangles can be solved using standard trigonometric laws:

• Law of Cosines (finding angles from sides): $$\cos\alpha = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos\beta = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos\gamma = \frac{a^2 + b^2 - c^2}{2ab}$$
• Area via Heron's Formula: $$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$$ where $s = \frac{a + b + c}{2}$ is the semi-perimeter.
• Inradius ($r$) and Circumradius ($R$): $$r = \frac{\text{Area}}{s}, \quad R = \frac{a \cdot b \cdot c}{4 \cdot \text{Area}}$$
• Altitudes ($h_a, h_b, h_c$): $$h_a = \frac{2 \cdot \text{Area}}{a}, \quad h_b = \frac{2 \cdot \text{Area}}{b}, \quad h_c = \frac{2 \cdot \text{Area}}{c}$$

Comparison: Acute vs. Right vs. Obtuse Triangles

Every triangle falls into exactly one of three angle classifications based on its largest angle:

• Acute Triangle: All 3 angles $< 90^\circ$. For specific angle-side solvers, explore our AAS Triangle Calculator and AAA Triangle Calculator.
• Right Triangle: Exactly 1 angle $= 90^\circ$. See our Right Triangle Calculator.
• Obtuse Triangle: Exactly 1 angle $> 90^\circ$ and 2 acute angles.