# Acute Triangle Calculator

Table of contents

What is an acute triangle? Definition of acute trianglesHow do I know if a triangle is acute based on side lengths?How to use this acute triangle calculator?Other relevant Omni toolsFAQsNo matter if you're a beginner wondering what an acute triangle actually is or a seasoned explorer of the world of **acute, obtuse, and right triangles**, this acute triangle calculator will serve you well. In what follows we'll:

- Discuss the
**definition of acute triangles**; - Explain what acute
**scalene**and acute**isosceles**triangles are; - Tackle the most intriguing question related to this topic:
*but how do we know if a triangle is acute***based on the side length**?

## What is an acute triangle? Definition of acute triangles

Recall there are three **types of angles** that you can encounter while dealing with *tri*angles:

**Acute angle**: it measures less than`90˚`

;**Right angle**: it measures exactly`90˚`

; and**Obtuse angle**: it measures more than`90˚`

and less than`180˚`

.

Based on that, we distinguish three **types of triangles**:

**Acute triangle**: all three of its angles are acute;**Right triangle**: has a right angle (and two acute angles); and**Obtuse triangle**: has an obtuse angle (and two acute angles).

Acute triangles can be further divided into three categories, based on the side lengths (more precisely, on side length ratio):

- acute
**equilateral**triangle: all three sides are equal; - acute
**isosceles**triangle: two sides are equal; and - acute
**scalene**triangle: all sides have different lengths.

As you see, once we understand what an acute triangle is, it's obvious how to decide if a triangle is acute if you know its angles: just take a look at these angles and make sure they are all strictly less than 90° (π/2 rad). But how do we know if a triangle is acute based on the side length?

## How do I know if a triangle is acute based on side lengths?

If you know the side lengths, you can quickly check if your triangle is acute:

- Compute the
**sum of squares of the two smaller sides**. - Compare it to the
**square of the longest side**.- If the sum is greater, your triangle is
**acute**. - If they are equal, your triangle is
**right**. - If the sum is shorter, your triangle is
**obtuse**.

- If the sum is greater, your triangle is

This method is based on the law of cosines.

Namely, observe that the **biggest angle** (that can potentially be obtuse or right) is the one opposite the longest side. Let's say `a`

and `b`

are the shorter sides and `c`

is the longest side (see the image below). From the **law of cosines**, the biggest angle `γ`

satisfies:

`cos(γ) = (a² + b² - c²)/(2ab)`

The angle `γ`

is acute if `cos(γ) > 0`

. Since `2ab`

is always positive, we need to verify that the numerator is positive as well, i.e., that

`a² + b² > c²`

to be sure that `γ`

is acute, and so that the triangle is acute.

Otherwise, if the numerator is zero, then `cos(γ) = 0`

, i.e., `γ = 90˚`

, and if the numerator is negative, then `cos(γ) < 0`

, which translates into `90˚ < γ < 180˚`

.

In principle, it's not hard to verify if a given triangle is acute or not; however, sometimes it may require quite a lot of daunting calculations. That's exactly the moment when Omni's acute triangle calculator enters the stage!

## How to use this acute triangle calculator?

That's how to use this tool:

- Choose the mode based on
**what you know about the triangle**:- Three angles (AAA);
- Three sides (SSS);
- Two sides and the angle between them (SAS); or
- Two angles and the side between them (ASA).

- Our acute triangle calculator will immediately determine if your triangle is
**acute/obtuse/right**as well as**scalene/isosceles/equilateral**. - Additional data concerning your triangle:
**missing sides or angles, side length ratio, area, perimeter**gets calculated as well.

## Other relevant Omni tools

Done with acute triangles? Dive deeper into the world of triangles with the help of our calculators:

### How do I find the longest side of an acute triangle?

The longest side `c`

of an acute triangle is the **one opposite the largest angle** `γ`

. To determine its length, use the **law of cosines**: `c = √(a²+ b² - 2ab cos(γ))`

, where `a`

and `b`

are the two shorter sides of the triangle.

### How many acute angles are in an acute triangle?

There are **three** acute angles in an acute triangle. In other words, in an acute triangle, **all angles** have to be acute - in fact, this is the definition of an acute triangle.

### Can a right triangle be acute?

**No**, a triangle cannot be at the same time right and acute. If it's acute, it means all of its angles are acute and so none of them can be right.

### Is the triangle 2 3 4 acute?

**No**, the triangle with side lengths `2`

, `3`

, `4`

is **not acute** because the sum of squares of the shorter sides `2²+ 3² = 13`

is strictly less than the square of the longest side `4² = 16`

. In fact, the biggest angle in this triangle is a bit more than `104°`

, so it's an **obtuse angle**.