What is an Acute Angle?

Acute Angle is a type of angle based on its measure, where the measure is less than 90°. Imagine the angle formed when you bend your elbow, creating a sharp but not wide opening; this is an example of an Acute Angle. In geometry, an acute angle falls within the range of 0° to 90°. These angles play a significant role in shaping various geometric figures, particularly triangles, and find applications in diverse mathematical and scientific scenarios.

In this article, we have covered the various concepts related to acute angles—definition, properties, and real-life examples of acute angle to gain a clearer understanding of their significance.

What is an Acute Angle?

An acute angle is a small angle measuring less than 90°. It is like the angle formed when you bend your elbow, making a sharp but not wide opening. An acute angle can be represented as 0°< θ < 90°. Acute angles are commonly found in triangles and other geometric shapes, and they play a role in various mathematical and scientific applications. A few examples of acute angles are 23°, 45°, and 60°.

Definition of Acute Angle

An acute angle is a small angle that measures less than 90°.

Shape of Acute Angle

The acute angle looks like a small wedge or a slice of pizza. It is less than 90°, and it points in a narrow way. Imagine a corner where two walls meet, and the angle between them is sharp and small. That’s what an acute angle is like—pointed and less than a right angle.

• The vertex is the point where the two arms of the angle meet. It’s like the tip of the slice in the pizza analogy.
• The arms of the angle are the two lines that come out from the vertex, like the two sides of the pizza slice. They spread apart from each other but don’t go too far, staying less than a right angle (90°).

Properties of Acute Angle

The properties of the acute angle are:

• An acute angle has a measurement greater than zero, denoted as 0° < θ.
• The measurement of an acute angle is less than that of a right angle, specifically 90° > θ.
• The sine (sin) of any acute angle falls between 0 and 1.
• The cosine (cos) of any acute angle falls between 1 and 0.
• The tangent (tan) of any acute angle falls between 0 and infinity (∞).

Acute Angle Degree

The degree of an acute angle, which is any angle smaller than 90°, can be represented by the formula:

0° < Measure of Acute Angle < 90°

This means that the measurement of an acute angle falls within the range from 0° to less than 90°. Examples of acute angle° include 63°, 31°, 44°, 68°, 83°, and 85°. The formula provides a guideline for understanding and identifying acute angles based on their degree measurements.

Triangle Properties of Acute Angle

An Acute Angled Triangle possesses angles that are all smaller than 90°. When all three angles in a triangle measure 60°, it becomes a special type known as an equilateral triangle.

Acute triangles can be categorized into acute scalene triangles, acute isosceles triangles, and equilateral triangles. The term “acute triangle” refers to a type of triangle where all interior angles are less than 90°. In the illustrated triangle below, all its angles are smaller than 90°, leading to its classification as an acute triangle.

Formula of Acute Angle

In an acute angle triangle, there’s a rule called the triangle inequality theorem. It’s like a formula which saying that if a triangle has sides named a, b, and c (where c is the longest side), then a squared plus b squared is more than c squared. So, if a squared plus b squared is greater than c squared, you have an acute triangle. This rule helps us understand the relationships between the sides of a triangle when it’s an acute one.

• a² + b² > c²
• c² + b² > a²
• a² + c² > b²

Acute Angle in Various Shapes

Acute angles can be found in various geometric shapes. Here are some shapes that often contain acute angles:

• Triangle
• Quadrilateral
• Pentagon
• Polygons

Let’s discuss acute angles in these shapes in detail.

Acute Angle in Triangle

In a triangle, an acute angle is an angle that measures less than 90°. Imagine a triangle with one corner pointing outward but not too widely. For example, in a right-angled triangle, the angle opposite the right angle is acute. If the angle is less than 90°, it’s an acute angle in a triangle.

Acute Angle in Quadrilateral

In a quadrilateral, which is a four-sided shape, an acute angle is an angle measuring less than 90°. Unlike right angles found in squares or rectangles, acute angles in a quadrilateral are smaller, creating a sharper corner. For example, consider a kite-shaped quadrilateral where the angles formed are acute.

Acute Angle in Polygon

In a regular polygon, where all sides and angles are equal, acute angles are not present. Consider a regular pentagon, hexagon, or any other polygon with equal sides and angles. In these shapes, all interior angles are equal and measure more than 90°, making them obtuse. Acute angles are typically found in irregular polygons, where sides and angles can vary, allowing for angles measuring less than 90°.

Acute Angle in Irregular Shapes

Irregular shapes can also have acute angles. These are angles that are less than 90° but might be found in shapes that don’t follow a regular pattern. For example, in a shape with uneven sides and angles, any angle measuring less than 90° is an acute angle.

Acute Angle in Composite Shapes

Composite shapes, formed by combining two or more simple shapes, can contain acute angles. Picture a shape made by joining a rectangle and a triangle. The angle where the two shapes meet can be acute if it measures less than 90°. In composite shapes, any angle smaller than 90° is an acute angle.

Real Life Example of Acute Angle

Some examples of Real-life scenarios which looks like Acute angle are:

• Clock Hands: When the minute hand on a clock points at 3 and the hour hand is at 4, the angle between them forms an acute angle. It’s less than 90°.

• Roof Design: In the roof of a house, the intersection of two inclined sides often forms acute angles, especially at the peak of the roof.

• Triangle in a Playground Slide: The shape of a playground slide often involves acute angles, particularly at the top where the two sides of the slide come together.

• Pizza Slices: When you cut a pizza into slices, each slice is like a triangle with acute angles, meeting at the center of the pizza.

• Road Signs: The shape of some road signs, like yield signs or diamond-shaped warning signs, may have acute angles formed by their edges.

Acute Angle, Obtuse Angle and Right Angle

• Acute Angle: An acute angle is an angle whose measure is less than 90°.
  • In an acute-angled triangle, all three angles are acute angles. An example of an acute angle is an angle measuring 60°.
• Obtuse Angle: An obtuse angle is an angle whose measure is greater than 90° but less than 180°.
  • In an obtuse-angled triangle, one of the angles is an obtuse angle. An example of an obtuse angle is an angle measuring 120°.
• Right Angle: A right angle is an angle whose measure is exactly 90°.
  • In a right-angled triangle, one of the angles is a right angle.

Acute and Obtuse Angle

Key difference between acute and obtuse angles are listed in the following table:

Acute vs Obtuse Angle
Characteristics	Acute Angle	Obtuse Angle
Definition	An angle whose measure is less than 90°.	An angle whose measure is greater than 90° but less than 180°.
Angle Measure	Less than 90°	Between 90° and 180°
Triangle Type	Can be found in an acute-angled triangle	Can be found in an obtuse-angled triangle
Appearance	Often appears sharper or narrower	Often appears wider or more open
Relationship with right angle	Smaller than a right angle	Larger than a right angle
Example	An angle measuring 60°.	An angle measuring 120°.
Symbol	∠ABC, where m∠ABC < 90°	∠PQR, where 180°>∠PQR > 90°

Read More,

• Types of Angles
• Adjacent Angles in Geometry
• Obtuse Angle

Examples on Acute Angle

Example 1: Which of the following angles given in the figure 1 are acute angles? Give reason for why and why not.

Solution:

a. 45°: Yes, it is an acute angle. An acute angle is any angle smaller than 90°, and 45° falls within this range.

b. 90°: No, it is not an acute angle. An acute angle must be less than 90°, and a 90° angle is called a right angle.

c. 120°: No, it is not an acute angle. Acute angles are smaller than 90°, and 120° exceeds this limit.

d. 60°: Yes, it is an acute angle. A 60° angle is smaller than 90°, fitting the criteria for an acute angle.

Example 2: What is the sum of the interior angles in an acute triangle if one angle measures 30°, another angle measures 45°, and the third angle is 60°?

Solution:

To find the sum of the interior angles in the given acute triangle, add the measures of the three angles:

30°+45°+60°=135°.

Therefore, the sum of the interior angles in the acute triangle is 135°

Acute Angle: Practice Questions

Question 1: Determine if the following sets of angles form an acute triangle.

• 30°, 50°, 80°
• 60°, 70°, 80°
• 45°, 45°, 90°
• 100°, 40°, 40°

Question 2: Find the sum of the acute angles in the triangles.

• Triangle ABC has angles 30°, 45°, 60°.
• Triangle XYZ has angles 20°, 70°, 80°.
• Triangle PQR has angles 50°, 30°, 60°.

Question 3: If an angle is 25°, what is its complement, and is it an acute angle?

Question 4: Classify each triangle based on its angles.

• Triangle LMN with angles 60°, 70°, 50°.
• Triangle DEF with angles 80°, 60°, 40°.
• Triangle UVW with angles 30°, 60°, 90°.

Acute Angle: FAQs

What is an Acute Angle?

An acute angle is a small angle, measuring less than 90°. It’s like when you bend your elbow, creating a sharp but not wide opening.

Is an Acute Angle 90 Degrees?

No, an acute angle is less than 90 degrees.

Is 45 degree Angle is Acute?

Yes, a 45-degree angle is considered acute.

What is the Sum of all Acute Angles in Triangle?

The sum of all acute angles in a triangle is 180 degrees.

What are Some Real Life Examples of Acute Angles?

Real-life examples of acute angles include a corner of a book, the hands of a clock at 3:00, and a slice of pizza.

What is a Sharp Angle?

A sharp angle is another term for an acute angle; both refer to angles less than 90 degrees.

What Shape Does Acute Angle Has?

An acute angle can be found in various shapes, such as triangles, rectangles, and pentagons.