Acute Angle Triangle

An acute-angle triangle is one in which all three angles measure less than 90°. Its corners appear sharp and narrow, giving it a distinctive look. Beyond geometry textbooks, acute triangles show up in real life—think of engineering blueprints, architectural patterns, and even the shapes of certain rooftops and road signs. Because every angle is acute, it falls into one of the three main angle-based triangle categories: acute, right, and obtuse.

Let’s dive into the types, properties, and formulas of acute triangles.

1.0Types of Triangles

Before zeroing in on the acute triangle, let’s get into the types of triangles based on sides and angles:

Based on Sides

Based on Angles:

2.0What Is an Acute Angle Triangle?

An acute-angle triangle is a type of triangle in which each angle is less than 90°.

To see this in action, picture a triangle named ΔABC. Its angles are:

∠A = 80°, ∠B = 65°, and ∠C = 35°. Every angle comes in under 90°, so ΔABC earns the title of an acute triangle.

3.0Acute Angle Triangle Formulas

Follow this to calculate the area of the acute triangle.

1. Area (When Base and Height Are Known):

Area = ½ × base × height

2. Area Using Heron’s Formula (All Sides Known):

s = a + b + c / 2

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

Where a, b, and c are the sides of the triangle, and s is the semi-perimeter.

3. Area Using Two Sides and Included Angle:

If you know two sides and the angle between them, here is the formula you can use:

Area = ½ × a × b × sin⁡(C)

4. Perimeter of an Acute Triangle

Perimeter = a + b + c, where a, b, and c are the sides of the triangle.

4.0Properties of an Acute Angle Triangle

Certain properties of acute-angle triangles help you quickly identify them. This ensures that you are using the right formulas. Let’s take a look at the properties of an acute triangle:

• Every interior angle is less than 90°.
• The altitude from any vertex will be inside the triangle.
• It can be equilateral, isosceles, or scalene.
• The triangle centres lie within the triangle.

5.0Important Triangle Centres in Acute Triangles

There are four points of intersection in a triangle that are important. In an acute triangle, all of these are present inside the triangle. Let’s take a look at them:

Note: In an acute triangle, the distance between the circumcenter and orthocenter is less than the radius of the circumscribed circle of the triangle.

Practice Problems

Example 1:

If two angles of a triangle are 85° and 30°, what is the third angle?

Solution:

Sum of interior angles = 180°

Third angle = 180° - (85° + 30°) = 65°

All angles are < 90°, which means it is an acute triangle.

Example 2:

Find the area of an acute triangle with a base = 8 cm and a height = 6 cm.

Solution:

Area = (½) x b x h

= (½) x 8 x 6

= 24 cm²

Example 3:

Construct a triangle with a base of 7 cm, and angles of 65° and 75°. Is it acute?

Solution: 

Third angle = 180° - (65° + 75°) = 40°

All angles are acute. Yes, it is an acute triangle.