Is a triangle with sides of length 2, 3 and 4 acute?

No. The longest side is 4 and the sum of the squares of 2 and 3 is 4 + 9 = 13 which is less than the square of 4, which is 16. So this triangle is obtuse because the sum of the squares is smaller, with the largest angle being approximately 104.5 degrees, so it's definitely more than 90 degrees.

How many acute angles does an acute triangle have?

Three. All interior angles of an acute triangle are acute angles, which is where its name comes from. Any triangle has at least two acute angles, but only acute triangles have all three interior angles as acute angles.

Can a triangle be both right and acute?

No. A right triangle has one angle that is exactly 90 degrees, while an acute triangle must have all angles less than 90 degrees. A triangle can only be one of three types: acute, right or obtuse, and it cannot belong to two of these categories at the same time.

Is an equilateral triangle an acute triangle?

Yes, definitely. Since all three sides of an equilateral triangle are the same length, then all three interior angles are also equal to each other and measure 60 degrees. Because 60 is less than 90, an equilateral triangle is classified as an acute triangle. An equilateral triangle is also the most symmetrical type of acute triangle.

Acute Triangle Calculator

Q: What is an acute triangle?

In an acute triangle all three interior angles are less than 90 degrees. The sum of the interior angles in a triangle is always 180 degrees, so it follows that for a triangle to be acute, even the largest angle, which is opposite the longest side, must be less than 90 degrees.

Q: How can you tell if a triangle is acute based on its side lengths?

Find the longest side and label it c. Compare the sum of the squares of the two remaining sides to the square of c. If the sum of the squares of a and b is greater than the square of c, then the triangle is acute. If they are equal, it's a right triangle, and if the sum of the squares is smaller, it's an obtuse triangle. This is a method for using the Law of Cosines to determine which angle is largest.

Enter the three sides

Type the three side lengths. Any two sides must add up to more than the third, or the lengths cannot close into a triangle.

Side a

Side b

Side c

Solution

Acute triangle. All three angles - A = 46.57 deg (°), B = 57.91 deg (°), C = 75.52 deg (°) - are below 90 degrees. Equivalently, the squares of the two shorter sides add up to more than the square of the longest side.

By its sides it is scalene: all three sides differ, so all three angles differ.

Angle A (opposite a)

Angle B (opposite b)

Angle C (opposite c)

Area

Perimeter

Semiperimeter s

Show heights, medians and circle radii

The perpendicular height and the median to each side, plus the circumscribed and inscribed circle radii.

Triangle area

m²

Sides a = 6 m, b = 7 m, c = 8 m give angles A = 46.57 deg (°), B = 57.91 deg (°), C = 75.52 deg (°), which sum to 180 degrees. Area 20.33, perimeter 21.

Steps and charts

Show the step-by-step working

Walk through the law of cosines, Heron's formula, and the acuteness test that produce the answers above.

Show the charts

See the three interior angles and the a^2 + b^2 vs c^2 acuteness test at a glance.

An acute triangle is a triangle with all interior angles measuring less than 90°. Enter the lengths of the three sides and this tool will calculate the type of triangle that it makes: acute, right or obtuse and also calculate all three internal angles, area, perimeter, heights, medians, inradius and circumradius.

What is an acute triangle?

All triangles are classified into one of three categories based on the size of their largest angle. If the largest angle is less than 90 degrees then it is an acute triangle, exactly 90 degrees makes it a right triangle and more than 90 degrees makes it an obtuse triangle. Since the sum of all three interior angles in any triangle is always 180 degrees, there cannot be two or more right or obtuse angles in a triangle. Therefore, the type of triangle is determined solely by its largest angle, and even that largest angle in an acute triangle is less than 90 degrees.

Like other triangles, acute triangles are divided into three categories based on their side lengths. An equilateral acute triangle has all three sides the same length and all three interior angles measure 60 degrees. An isosceles acute triangle has two equal sides and two equal interior angles. A scalene acute triangle has three unequal sides and three unequal interior angles, but they are all less than 90 degrees.

How can you tell if a triangle is acute based on its side lengths?

No need to measure angles. Label each side a, b and c and compare the sum of the squares of the two shortest sides with the square of the longest side.

a^2 + b^2 > c^2 \;\Rightarrow\; \text{acute} \qquad a^2 + b^2 = c^2 \;\Rightarrow\; \text{right} \qquad a^2 + b^2 < c^2 \;\Rightarrow\; \text{obtuse}

This method of determination is directly derived from the law of cosines, where the largest angle gamma is opposite the longest side c.

\cos\gamma = \frac{a^2 + b^2 - c^2}{2ab}.

Since the denominator 2ab is always positive, the sign of the numerator determines the type of angle. If the sum of a squared and b squared is greater than c squared then the cosine value will be positive and this angle is less than 90 degrees so the whole triangle is an acute triangle. If the largest angle is less than 90 degrees then obviously the other two smaller angles are also less than 90 degrees.

Example calculation:

Consider the case where the side lengths are 2, 3 and 4. The longest side is 4 so we compare the sum of squares of 2 and 3 which is 4 + 9 = 13 with the square of 4 which is 16. Since 13 is less than 16 this triangle is not acute but obtuse, with the largest angle being about 104.5 degrees. Now consider the case where the side lengths are 6, 7 and 8. The sum of squares of 6 and 7 is 85 which is greater than the square of 8 which is 64. So this triangle is acute with the largest angle being about 75.5 degrees.

Instructions for using the tool to calculate acute triangles:

Enter the three side lengths into fields a, b and c. The unit can be freely chosen but all three sides must have the same unit. The unit can be switched between millimeters, centimeters, meters, inches, feet and yards. When valid side lengths are entered, the result is immediately displayed. The color indicates whether the triangle is acute, right or obtuse. In addition, based on the side lengths it is determined whether the triangle is equilateral, isosceles or scalene, and all measurement results are shown in a list. If no triangle can be formed with the three sides, the calculation tool clearly informs about this and does not generate invalid shapes.

The formulas used for each calculation are as follows:

If all three sides are known, the interior angles can be calculated using the law of cosines. Letting the angle opposite side a be A, then

A = \cos^{-1}\!\left(\frac{b^2 + c^2 - a^2}{2bc}\right),

Angles B and C can be calculated in the same way by just swapping the letters around. The area can then be found using Heron's formula which only requires the side lengths. If half the perimeter is s = (a+b+c)/2, then:

\text{Area} = \sqrt{s\,(s-a)(s-b)(s-c)}.

Once the area is known, the remaining values can be directly calculated. The altitude to each side is twice the area divided by the corresponding base length. The radius of the inscribed circle is the area divided by half the perimeter, while the radius of the circumscribed circle is the product of the sides divided by four times the area.

Quantity	Formula	Meaning
Angle A	cos^-1((b^2 + c^2 - a^2) / 2bc)	Angle opposite side a
Area	sqrt(s(s-a)(s-b)(s-c))	Heron's formula, s = (a+b+c)/2
Height to a	2 x Area / a	Altitude from the opposite vertex
Median to a	half sqrt(2b^2 + 2c^2 - a^2)	Vertex A to the midpoint of a
Inradius r	Area / s	Largest circle that fits inside
Circumradius R	abc / (4 x Area)	Circle through the three vertices

Comparison of acute, right and obtuse triangles:

The results of the side-length determinations will always agree with those of the angle determinations because they are both statements of the same relationship, just from different perspectives. The following table compares the three types of triangles.

Largest angle	Side test (c longest)	Type	Example sides
Below 90 deg	a^2 + b^2 > c^2	Acute	6, 7, 8
Exactly 90 deg	a^2 + b^2 = c^2	Right	3, 4, 5
Above 90 deg	a^2 + b^2 < c^2	Obtuse	2, 3, 4

Features you should know about:

An equilateral triangle is always an acute triangle. This is because all three interior angles are each 60 degrees, making it the most balanced and typical example of an acute triangle. In any acute triangle, the circumcenter (the point that is equidistant from all three vertices) lies inside the triangle. For a right triangle, the circumcenter lies on the hypotenuse, while for an obtuse triangle, it lies outside the triangle. The orthocenter (the intersection of all three altitudes) also follows this pattern: the orthocenter of an acute triangle is located inside the triangle, that of a right triangle at the vertex of the right angle, and that of an obtuse triangle outside the triangle. Thus, the property of being an acute triangle not only affects the angles themselves but also the positions of important centers within the triangle.

Applications of acute triangles:

Acute triangles are an important basic element of rigid structures. This shape is often used in roof constructions, bridge frames and geodesic domes. Since a triangle cannot be deformed without changing the side lengths and an acute triangle with a suitable ratio can distribute the load evenly, it is ideal for such applications. The relationship between angles and sides that is used in this tool also finds application in surveying to determine distances, in sign design and in navigation to determine position.

Explanation of symbols:

Symbol	Meaning	Example
a, b, c	The three side lengths	6, 7, 8
A, B, C	Angles opposite a, b, c	46.57, 57.91, 75.52 deg
s	Semiperimeter (a + b + c) / 2	10.5
R	Circumradius	4.13
r	Inradius	1.94

This tool is for general learning and problem solving only. It assumes that the triangles are planar (Euclidean). If the lengths of the three sides do not satisfy the triangle inequality then it will assume no triangle can be formed.

Frequently asked questions

What is an acute triangle?: In an acute triangle all three interior angles are less than 90 degrees. The sum of the interior angles in a triangle is always 180 degrees, so it follows that for a triangle to be acute, even the largest angle, which is opposite the longest side, must be less than 90 degrees.
How can you tell if a triangle is acute based on its side lengths?: Find the longest side and label it c. Compare the sum of the squares of the two remaining sides to the square of c. If the sum of the squares of a and b is greater than the square of c, then the triangle is acute. If they are equal, it's a right triangle, and if the sum of the squares is smaller, it's an obtuse triangle. This is a method for using the Law of Cosines to determine which angle is largest.
Is a triangle with sides of length 2, 3 and 4 acute?: No. The longest side is 4 and the sum of the squares of 2 and 3 is 4 + 9 = 13 which is less than the square of 4, which is 16. So this triangle is obtuse because the sum of the squares is smaller, with the largest angle being approximately 104.5 degrees, so it's definitely more than 90 degrees.
How many acute angles does an acute triangle have?: Three. All interior angles of an acute triangle are acute angles, which is where its name comes from. Any triangle has at least two acute angles, but only acute triangles have all three interior angles as acute angles.
Can a triangle be both right and acute?: No. A right triangle has one angle that is exactly 90 degrees, while an acute triangle must have all angles less than 90 degrees. A triangle can only be one of three types: acute, right or obtuse, and it cannot belong to two of these categories at the same time.
Is an equilateral triangle an acute triangle?: Yes, definitely. Since all three sides of an equilateral triangle are the same length, then all three interior angles are also equal to each other and measure 60 degrees. Because 60 is less than 90, an equilateral triangle is classified as an acute triangle. An equilateral triangle is also the most symmetrical type of acute triangle.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Math is Fun: Triangles
Classification of triangles by angle and by side.
Wikipedia: Acute and obtuse triangles
Properties of acute triangles and the law of cosines classification.