Acute Triangle

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Acute Triangle: Definition, Properties, Types, and Formulas​

Meta description: Discover what an acute triangle is, its key properties, three main types — scalene, isosceles, and equilateral — plus the essential area and perimeter formulas with clear examples.

What Is an Acute Triangle?​

An acute triangle is a triangle in which all three interior angles measure less than $90°$. In other words, every angle in the triangle is an acute angle.
Consider triangle $\triangle ABC$ — it is an acute triangle if and only if:
$\angle A < 90°, \quad \angle B < 90°, \quad \angle C < 90°$
Since the sum of all interior angles in any triangle equals $180°$:
$\angle A + \angle B + \angle C = 180°$
All three angles must simultaneously be less than $90°$ for the triangle to qualify as acute.

Properties of an Acute Triangle​

The key property of an acute triangle is:

• In $\triangle ABC$, all three interior angles satisfy: $\angle ABC < 90°$, $\angle BAC < 90°$, and $\angle ACB < 90°$

This single condition distinguishes an acute triangle from a right triangle (one angle $= 90°$) and an obtuse triangle (one angle $> 90°$).

Types of Acute Triangles​

Acute triangles are classified into three types based on their side lengths:

1. Acute Scalene Triangle​

An acute scalene triangle has no sides equal and all three angles are different, yet every angle remains less than $90°$.
$a \neq b \neq c, \quad \angle A < 90°, \quad \angle B < 90°, \quad \angle C < 90°$

2. Acute Isosceles Triangle​

An acute isosceles triangle has two sides equal and two equal base angles, with all three angles less than $90°$.
$a = b, \quad \angle A = \angle B < 90°, \quad \angle C < 90°$

3. Acute Equilateral Triangle​

An acute equilateral triangle has all sides equal and all three angles equal to exactly $60°$ — making every equilateral triangle automatically acute.
$a = b = c, \quad \angle A = \angle B = \angle C = 60°$

Formulas for an Acute Triangle​

Area​

The area of an acute triangle is calculated using the standard triangle area formula:
$A = \frac{1}{2} \times b \times h$
where $b$ is the base and $h$ is the perpendicular height corresponding to that base.

Perimeter​

The perimeter of an acute triangle is the sum of all three sides:
$P = a + b + c$
where $a$, $b$, and $c$ are the three side lengths of the triangle.

Conclusion​

An acute triangle is defined by one simple yet powerful condition — all three interior angles must be less than $90°$. It can take three forms: scalene (no equal sides), isosceles (two equal sides), or equilateral (all sides equal, all angles $= 60°$). Whether calculating its area with $A = \frac{1}{2} \times b \times h$ or its perimeter with $P = a + b + c$, the acute triangle remains one of the most fundamental shapes in geometry.