Acute Triangle

An acute triangle is a triangle in which all the three interior angles are less than 90º. Although the three interior angles of the acute triangle lie within 0° to 90°, their sum is always 180 degrees.

Triangles can be classified on the basis of angles and sides. An acute triangle is one that is classified on the basis of the measurement of angles. If all the interior angles of a triangle are less than 90°, then the triangle is said to be an acute triangle.

Acute Triangle Definition

The definition of acute triangle states that it is a type of triangle in which all three interior angles are acute angles or less than 90°. The sides of an acute-angled triangle can be equal or unequal depending on whether the triangle is equilateral, isosceles, or scalene. Let us learn about the types of acute triangles in the next section.

As we know that triangles can be classified on the basis of sides and angles, an acute triangle can also be further classified as:

• Equilateral Acute Triangle: In an equilateral acute triangle, all three angles are equal to 60° and all sides are equal.
• Isosceles Acute Triangle: In an isosceles acute triangle, two sides and two angles are equal, and all interior angles are less than 90°.
• Scalene Acute Triangle: In a scalene acute triangle, all the 3 sides are of different lengths and the 3 interior angles are of different measures but all the interior angles measure less than 90°.

Observe the figure given above which shows a acute scalene triangle representing 3 unequal sides and unequal angles. It can be seen that the value of all three angles is less than 90° but they add up to 180°.

There are a few important properties that help us identify an acute triangle. The properties of an acute-angled triangle are listed below:

• According to the angle sum property, all the three interior angles of an acute triangle add up to 180°.
• A triangle cannot be a right-angled triangle and an acute-angled triangle at the same time.
• A triangle cannot be an acute-angled triangle and an obtuse-angled triangle at the same time.
• The angle property of the acute triangle says the interior angles of an acute triangle are always less than 90° or lie between (0° to 90°).
• The side opposite to the smallest angle is the smallest side of the triangle.

There are two basic formulas related to an acute triangle:

• Area of an acute triangle
• Perimeter of an acute triangle

Let us learn about these two formulas of an acute-angled triangle in detail.

Area of Acute Triangle

The area of an acute triangle can be calculated using the formula, Area of triangle = (1/2) × b × h. Here, 'b' denotes the base, and 'h' denotes the height of an acute triangle.

Note: If all the sides of the acute triangle are given then the acute triangle area can be easily calculated using Heron's formula given below.

Area of an acute triangle using Heron's formula = \(\sqrt{S(S-a)(S-b)(S-c)}\). Here, S denotes the semi perimeter which can be calculated with the formula, Semi-perimeter (S) = (a + b + c)/2, where a, b, and c are the sides of the given triangle.

Example: Find the area of an acute triangle whose sides are 4 units, 8 units and 6 units.

Solution: The sides of the triangle are given as, a = 4 units, b = 8 units and c = 6 units
Thus, Semi-perimeter, S = (a + b + c)/2 = (4 + 8 + 6)/2 = 9 units
Area of triangle = √[S(S-a)(S-b)(S-c)] = √[9(9-4)(9-8)(9-6)]
⇒ Area of triangle = √(9 × 5 × 1 × 3) = √135 = 11.61 unit²
∴ The area of the acute triangle is 11.61 unit²

Perimeter of Acute Triangle

The perimeter of an acute triangle is defined as the sum of the three sides and it can be calculated using the formula, Perimeter of triangle = (a + b + c). Here, a, b, and c are the sides of the acute-angled triangle.

Example: Find the perimeter of an acute triangle whose sides are 12 units, 10 units and 6 units.

Solution: The sides of the triangle are given as, a = 12 units, b = 10 units and c = 6 units. Perimeter of an acute triangle = a + b + c. After substituting the values in the formula, we get, Perimeter of an acute triangle = a + b + c = 12 + 10 + 6 = 28. Therefore, the perimeter of the acute triangle = 28 units.

☛ Related Articles

• Types of Triangles
• Scalene Triangle
• Equilateral Triangle
• Obtuse Angles

What is an Acute Triangle?

An acute-angled triangle is a type of triangle in which all three interior angles are less than 90°. For example, if the angles of a triangle are 65°, 75°, and 40°, then it is an acute triangle because all the 3 angles are less than 90°. However, their sum should always be 180°.

Are Isosceles Triangles Always Acute Triangles?

No, an isosceles triangle may not necessarily be an acute triangle. It can be a right-angled triangle with the angles as 90°, 45°, and 45°. It can even be an obtuse triangle with angles as, 30°, 30°, and 120°. It totally depends upon the measure of the angles it has. To be an acute triangle, all three interior angles should measure less than 90 degrees.

How do you know if a Triangle is an Acute Triangle?

A triangle can be acute if all its interior angles are less than 90°, which means all angles should be between 0° to 90°. For example, if the angles of a triangle are 85°, 55°, and 40°, then it is an acute triangle because all the 3 angles are less than 90°.

What are the Types of an Acute Triangle?

There are three types of acute triangles given below:

• Equilateral Acute Triangle: In an equilateral acute triangle, all three angles are equal to 60° and all sides are equal.
• Isosceles Acute Triangle: In an isosceles acute triangle, two sides and two angles are equal, and all interior angles are less than 90°.
• Scalene Acute Triangle: In a scalene acute triangle, all the 3 sides are of different lengths and the 3 interior angles are of different measures but all the interior angles measure less than 90°.

Do Angles of Acute Triangles Add up to 180?

The angles of any triangle add up to 180°. An acute triangle is a type of triangle, so, the sum of its interior angles is 180 degrees, and each individual angle measures less than 90 degrees.

Can a Triangle be Right and Acute?

No, a triangle can either be acute or be right-angled. It cannot be both at the same time. If the value of any one angle of the triangle crosses 90 degrees then it is no more considered to be an acute triangle.

How to Find the Area of an Acute Triangle?

The area of an acute triangle can be calculated if the base and height is given. The formula that is used to find the area is, Area = (1/2) × base × height. In the case where the length of all 3 sides is given, we can use Heron's formula for calculating the acute triangle's area, that is, (Area of triangle = \(\sqrt{S(S-a)(S-b)(S-c)}\); where 'a', 'b', and 'c' are the 3 sides of the triangle.

What does an Acute Triangle Look Like?

An acute triangle is a closed shape made up of three straight lines and three interior angles. All the vertices of the acute triangle are pointed outwards, so it is a convex 2D shape.