Acute Triangle Calculator

No 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?

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

• 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?

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

1. Compute the sum of squares of the two smaller sides.
2. 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.

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!

That's how to use this tool:

1. 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).
2. Our acute triangle calculator will immediately determine if your triangle is acute/obtuse/right as well as scalene/isosceles/equilateral.
3. Additional data concerning your triangle: missing sides or angles, side length ratio, area, perimeter gets calculated as well.

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

• Triangle area;
• AAA triangle;
• Midsegment of a triangle;
• Circumcenter of a triangle;
• Triangle congruence;
• Obtuse triangle;
• Oblique triangle;
• Base of a triangle;
• AAS triangle;
• SAS triangle;
• SSS triangle; and
• ASA 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.

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.

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.

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.