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Acute Triangle - Definition, Properties, Examples

TL;DR

An acute triangle is a triangle in which all three interior angles are less than 90°. Its three angles still add to 180°, its orthocenter lies inside it, and it can be equilateral, isosceles, or scalene. This article covers the definition, properties, how to test a triangle from its sides, the area formula, and worked examples.

Last updated on July 21, 2026, 9 min read

What Is an Acute Triangle?

An acute triangle (also called an acute-angled triangle) is a triangle in which all three interior angles measure less than 90°. An acute angle is any angle smaller than a right angle, and an acute triangle is the shape you get when every one of its three corners is acute.

Because a triangle's angles always add to 180° (the triangle sum theorem), an acute triangle spreads that 180° across three angles that each stay below 90°. A set of angles like 60°, 70°, 50° works - all under 90°, summing to 180°. A set like 100°, 40°, 40° does not, because 100° is obtuse.

Look at any slice of pizza cut from a round pie - three corners, every one of them a snug angle under 90°. That is an acute triangle.

What Are the Properties of an Acute Triangle?

An acute triangle carries a handful of defining features. Bold the ones worth remembering:

• All three angles are less than 90°. This is the definition; nothing else qualifies a triangle as acute.
• The angles sum to 180°, like every triangle.
• A triangle cannot be acute and right, or acute and obtuse, at the same time. The three types are mutually exclusive - a triangle is exactly one of them.
• The orthocenter lies inside the triangle. The point where the three altitudes meet, the orthocenter, sits inside for acute triangles (it sits on a vertex for right triangles and outside for obtuse ones).
• The largest angle faces the longest side. The angle opposite the longest side is the biggest; the angle opposite the shortest side is the smallest.

The Side Test For An Acute Triangle

You can decide whether a triangle is acute from its side lengths alone, without measuring any angle. If aaa, bbb, and ccc are the sides with ccc the longest, then the triangle is acute when:

a² + b² > c²

This is the Pythagorean theorem turned into a comparison. When a² + b² = c², the triangle is right; when a² + b² < c², it is obtuse; and when the square of the longest side is less than the sum of the other two squares, all angles stay acute.

What Are the Types of Acute Triangles?

Acute triangles are classified further by their sides, exactly as any triangle is:

• Equilateral acute triangle - all three sides equal, all three angles exactly 60°. Every equilateral triangle is acute.
• Isosceles acute triangle - two sides equal, two base angles equal, and all three angles under 90°.
• Scalene acute triangle - all three sides different, all three angles different, and all three under 90°.

How Do You Find the Area of an Acute Triangle?

The area of an acute triangle uses the standard triangle formula, since being acute changes nothing about how area is computed:

Area = ( \tfrac{1}{2} \times \text{base} \times \text{height} )
Here the base is any one side and the height is the perpendicular distance from that side to the opposite vertex. When you know all three sides but no height, use Heron's formula: with ( s = \tfrac{a + b + c}{2} ) (the semi-perimeter),

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

Examples of the Acute Triangle

Each example is worked step by step. The problem statement is in bold; the working is not.

Example 1

Is a triangle with angles 65°, 75°, and 40° an acute triangle? Check each angle against 90°: 65°<90°, 75°<90°, 40°<90°. All three are under 90°. Confirm they sum to 180°: 65° + 75° + 40° = 180°. Valid triangle. Final answer: Yes, it is an acute triangle.

Example 2

A triangle has two angles of 40° and 50°. A student concludes it is acute because both given angles are under 90°. Are they right? The tempting move is to check only the angles you are handed: 40° and 50° are both acute, so call the triangle acute. Wrong path: "Both angles under 90°, so acute." But a triangle has three angles, and the third is hiding. Find it using the angle sum: third angle = 180°-40°-50° = 90°. The third angle is exactly 90° - a right angle. So the triangle is a right triangle, not acute. Checking only the visible angles missed the one that decides the classification. Final answer: No - the third angle is 90°, making it a right triangle. Always find and test all three angles.

Example 3

Classify the triangle with sides 5, 6, and 8 using the side test. The longest side is c = 8. The other two are a = 5, b = 6. Compare a² + b² with c²: a² + b² = 25 + 36 = 61, c² = 64. Since 61 < 64, we have a² + b² < c². Final answer: The triangle is obtuse, not acute (the square of the longest side exceeds the sum of the other two squares).

Example 4

Classify the triangle with sides 6, 7, and 8 using the side test. Longest side c = 8, others a = 6, b = 7: a² + b² = 36 + 49 = 85, c² = 64. Since 85 > 64, we have a² + b² > c². Final answer: The triangle is acute - all three angles are under 90°.

Example 5

Find the area of an acute triangle with base 10 cm and height 6 cm. Use the base-and-height formula: Area = ( \tfrac{1}{2} \times 10 \times 6 = 30 \text{ cm²} ). Final answer: 30 cm².

Example 6

An acute triangle has sides 5 cm, 5 cm, and 6 cm. Find its area using Heron's formula. Semi-perimeter: s = ( \frac{5 + 5 + 6}{2} = 8 ). Heron's formula: Area = ( \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{8(8 - 5)(8 - 5)(8 - 6)} = \sqrt{144} = 12 \text{ cm²} ). (With two equal sides, this is also an isosceles acute triangle.)

Why Acute Triangles Matter - "The strongest way to hold a load"

Classifying triangles is not busywork. The angles of a triangle decide how it behaves under force, and acute triangles show up wherever a structure needs to stay rigid without any corner buckling.

Where the distinction earns its keep:

• Trusses and frameworks - bridges and roofs are built from triangles because a triangle cannot deform without changing its side lengths. Acute triangles, with no single dominating angle, distribute load evenly across all three members, which is why they appear throughout structural engineering.
• Trigonometry and surveying - an acute triangle keeps every angle inside the "friendly" range where sine and cosine behave simply, which is one reason surveyors prefer to break a plot into acute triangles when triangulating distances.
• Telling the types apart - knowing that the orthocenter sits inside an acute triangle, on a right triangle, and outside an obtuse one is a quick visual check that ties the classification to a concrete construction.

The idea worth carrying forward: the single number that decides a triangle's whole character is its largest angle. Below 90° and it is acute; exactly 90° and it is right; above and it is obtuse. One angle governs the type - and the side test a² + b² versus c² is just that governing angle in disguise.

What Are the Most Common Mistakes With Acute Triangles?

Mistake 1: Checking only some of the angles

Where it slips in: Classifying a triangle from a couple of given angles. Don't do this: Calling a triangle acute because the two angles you were shown are both under 90°. The correct way: A triangle is acute only when all three angles are under 90° - find the missing third angle and test it too.

Mistake 2: Testing the wrong side in the Pythagorean check

Where it slips in: Using a² + b² versus c² to classify a triangle from its sides. Don't do this: Squaring any two sides and comparing to the third without identifying which side is longest. The correct way: The comparison must use the longest side as c.

Mistake 3: Confusing "equilateral" with a separate type from "acute"

Where it slips in: Sorting triangles into type buckets. Don't do this: Treating "equilateral" and "acute" as competing labels. The correct way: Triangles are classified two ways at once - by sides and by angles. Every equilateral triangle is also acute.

Conclusion

• An acute triangle has all three interior angles less than 90°.
• Its angles still sum to 180°, and its orthocenter lies inside the triangle.
• Test from sides with a² + b² > c² (where c is the longest side).
• It can be equilateral, isosceles, or scalene - every equilateral triangle is acute.
• Area uses ( \tfrac{1}{2} \times \text{base} \times \text{height} ) or Heron's formula.