Acute Angle Triangle

In geometry, a triangle is a closed two-dimensional plane figure with three sides and three angles. A triangle is considered as a three-sided polygon. Based on the sides and the interior angles of a triangle, there can be various types of triangles, and the acute angle triangle is one of them.

According to the sides of the triangle, the triangle can be classified into three types, namely.

According to the interior angles of the triangle, it can be classified as three types, namely

Acute Angle Triangle Definition

An acute angle triangle (or acute-angled triangle) is a triangle in which all the interior angles are acute angles. To recall, an acute angle is an angle that is less than 90°.

Example: Consider ΔABC in the figure below. The angles formed by the intersection of lines AB, BC and CA are ∠ABC, ∠BCA, and ∠CAB, respectively. We can see that,

∠ABC = ∠B = 75°

∠BCA = ∠C = 65°

∠BAC = ∠A = 40°

Since all the three angles are less than 90°, we can infer that ΔABC is an acute angle triangle or acute-angled triangle.

Acute Angle Triangle Formula

The formulas to find the area and perimeter of an acute triangle is given and explained below.

The area of acute angle triangle = (½) × b × h square units

Where,

“b” refers to the base of the triangle

“h” refers to the height of a triangle

If the sides of the triangle are given, then apply the Heron’s formula

The area of the acute triangle =

Where S is the semi perimeter of a triangle

It can be found using the formula

S = (a + b + c)/2

The perimeter of an acute triangle is equal to the sum of the length of the sides of a triangle, and it is given as

Perimeter = a + b + c units

Here,

a, b, and c denotes the sides of the triangle.

If two sides and an interior angle is given then,

Area = (½) × ab × Sin B or,

= (½) × bc × Sin C or,

= (½) × ac × Sin A

Here, ∠A, ∠B, ∠C are the three interior angles at vertices A, B, and C, respectively. Also, a, b, and c are the lengths of sides BC, CA and AB, respectively.

Acute Angle Triangle Properties

The important properties of an acute triangle are as follows:

• The interior angles of a triangle are always less than 90° with different side measures
• In an acute triangle, the line drawn from the base of the triangle to the opposite vertex is always perpendicular

Important Terminologies

Circumcenter

A perpendicular bisector is a segment that divides any side of a triangle into two equal parts. The intersection of perpendicular bisectors of all the three sides of an acute-angled triangle form the circumcenter, and it always lies inside the triangle.

Incenter

An angular bisector is a segment that divides any angle of a triangle into two equal parts. The intersection of angular bisectors of all the three angles of an acute angle forms the incenter, and it always lies inside the triangle.

Centroid

A median of a triangle is the line that connects an apex with the midpoint of the opposite side. In acute angle, the medians intersect at the centroid of the triangle, and it always lies inside the triangle.

Orthocenter

An altitude of a triangle is a line that passes through an apex of a triangle and is perpendicular to the opposite side. The three altitudes of an acute angle intersect at the orthocenter, and it always lies inside the triangle.

Distance Between Orthocenter and Circumcenter

For an acute angle triangle, the distance between orthocenter and circumcenter is always less than the circumradius.

Video Lesson on Types of Triangles

Practice Questions

1. If two angles of an acute-angled triangle are 85^o and 30^o, what is the angular measurement of the third angle?
2. Find the area of the triangle if the length of one side is 8 cm and the corresponding altitude is 6 cm.
3. Construct an acute angle triangle which has a base of 7 cm and base angles 65^o and 75^o. Find the circumcenter and orthocenter.