What is an Acute Angle?
An angle whose measure is less than 90 degrees is called acute angle.
The arms of a wall clock form acute angles many times in a day. At 10 o’clock, the hour hand and minute hand of a clock make an acute angle.
What is an Acute Triangle?
A closed polygon with three straight lines and three interior angles is known as a triangle. Triangles can be categorized based on their sides and angles.
According to the measurement of angles, triangles are categorized into acute-angled triangles, right-angled triangles and obtuse-angled triangles. A triangle with three acute interior angles is known as an acute triangle (or acute-angled triangle).
What are the Different Types of Acute Triangles?
An acute triangle can be further divided into the following categories:
- Equilateral Acute Triangle: An equilateral acute triangle has sides that are equal and angles that are all equal to 60°.
- Isosceles Acute Triangle: In an isosceles acute triangle, all interior angles are less than 90°, and two sides are equal in length.
- Scalene Acute Triangle: A scalene acute triangle is a triangle with three non-congruent sides and different interior angles, which are less than 90°.
Properties of an Acute Triangle
We can recognise an acute triangle by a few key characteristics. They are:
- A triangle cannot simultaneously be a right-angled triangle and an acute-angled triangle.
- A triangle cannot simultaneously be an obtuse-angled triangle and an acute-angled triangle.
- The interior angles are always less than 90° or range from 0° to 90°.
- The smallest side of a triangle is opposite to the smallest angle.
Formulas of Acute Triangle
We use two formulas to solve questions related to acute triangle:
- Area of acute triangle, \(A~=~\left ( \frac{1}{2} \right )~\times~b~\times~h \) square units.
Where, ‘b’ denotes the base of the triangle and ‘h’ denotes the altitude on the base of the triangle.
- Perimeter of acute triangle, \(P~=~(a~+~b~+~c) \) units
Where, a, b and c are the side lengths of the triangle.
Solved Acute Angle Examples
Example 1: Rosie observes numerous triangular shapes on a bridge in her city. Some of these shapes are given below. Recognize the acute-angled triangles from the provided images.
a. 
b. 
c. 
Solution:
Option a: It is not an acute triangle because one of its internal angles is equal to 90 degrees.
Option b: It is an acute triangle because all internal angles are less than 90 degrees.
Option c: It is not an acute triangle because one of its internal angles is 110 degrees, which is greater than 90 degrees.
Example 2: The sides of an acute triangle ABC are AB = 9 inches, BC = 10 inches, and CA = 7 inches. Find the perimeter of the triangle.
Solution:
The perimeter of a triangle is, \(P~=~(a~+b~+~c) \) units.
\(P~=~(9~+~10~+~7) \) (Substitute the values)
\(P~=~26 \) inches (Simplify)
Hence, the perimeter of a triangle ABC is 26 inches.
Example 3: The base of a triangle is 6 yards and its height is 3 yards. Calculate the area of the triangle.
Solution:
Area of triangle \(A~=~\left ( \frac{1}{2} \right )~\times~b~\times~h \).
\(A~=~\left ( \frac{1}{2} \right )~\times~6~\times~3 \) (Substitute the values)
\(A~=~9~yd^2 \) (Simplify)
So, the area of the triangle is 9 square yards.
All inner angles of a triangle must be between 0 and 90 degrees for it to be considered an acute-angled triangle. For instance, if a triangle has angles of 85 degrees, 55 degrees, and 40 degrees, all three angles are less than 90 degrees. Therefore, the triangle is an acute triangle.
An equilateral triangle is always an acute triangle.
The sum of all interior angles of any triangle is 180 degrees.