A triangle is a three sided polygon formed by the pairwise intersection of three straight lines. An angle is formed at each of this intersection, thus a triangle can also be viewed as a polygon with three interior angles. Indeed the name triangle is derived from Greek word 'tri' meaning three exactly suggests this. Any triangle should have at least two acute angles. Triangles with all acute angles are called Acute Triangles. In the Δ ABC shown above, all the measures of all the three angles A, B and C are less than 90º and hence they are all acute angles.

Acute Triangle Properties

1. All the three angles of an acute triangle are acute or in other words, their measures are less than 90 degrees.
2. The sum of measures of any two angles of an acute triangle is greater than 90 degrees.
3. All the points of concurrency like the Centroid, Circumcenter, Incenter and the Orthocenter fall inside the triangle.

Acute Scalene Triangle

If an acute triangle is scalene, then all the sides of a triangle are different. This also means all the three acute angles forming by the triangle have different measures. In general an acute scalene triangle is the default triangle. If no other information is given, a triangle is generally assumed to be acute scalene.

Acute Isosceles Triangle

An isosceles triangle has two of its sides congruent. The base angles, that is the angles opposite to these congruent sides are also congruent. If we have to term an isosceles triangle to be acute, then the base angles as well as the vertex angles are all acute. In Δ ABC shown above, sides AC and BC are marked congruent. The base angles A and B are also congruent and each measure 75º. The vertex angle C is also acute as it measures 30º.

Acute Equilateral Triangle

An equilateral triangle by definition is a triangle with all three congruent sides. Equilateral triangles are also equiangular, meaning all the three angles are congruent. As the sum of the three angles of a triangle is 180º, the measure of each angle in an equilateral triangle is 60º.

This is true whatever be the equal lengths of the sides. Thus all equilateral triangles are acute triangles. Equilateral triangles are perhaps the most commonly spotted triangles in real life situations.

Acute Right Triangle

A right triangle is a triangle with one of the angle is a right angle. A right angle cannot be classified as an acute angle as the measure of a right angle is 90º and not less than 90º. Hence an acute right triangle is impossible to be formed.

Acute Triangle Formula

A triangle can be checked whether acute or not if the measures of two of its angles are known.If the lengths of three sides of a triangle are known, is there a way to determine whether the triangle is an acute angle?
An inequality in the line of Pythagorean Identity is used to determine this.

Test for finding an acute triangle:
A given triangle is an acute triangle if the sum of the squares of the smaller side is greater than the square of the largest side.

We can write this as a mathematical statement. Suppose a, b and c are the lengths of a triangle and c the largest side. Then Δ ABC is an acute triangle if,
a2 + b2 > c2.

Solved Example

Question: The sides of a triangle were given as 4 cm 5 cm and 7 cm. Bob felt that it was an acute triangle. Check using the Pythagorean inequality whether Bob was right.
Solution:

Sum of the squares of the smaller sides = 42 + 52 = 16 + 25 = 41
Square of the Largest side = 72 = 49.
41 < 49 ⇒ a2 + b2 < c2
Hence the triangle was not an acute triangle and Bob was wrong in his guess.

Acute Triangle in Real Life

1. Many Traffic signs are done on acute angle bases.
2. Tiles in the shape of equilateral triangles are used in forming new designs.(In Tessellations)
3. Acute triangle shapes are used in Jewelry, furniture and decorative pieces.

Acute Triangle Examples

Solved Examples

Question 1: Determine whether each of the following is always, sometimes or never true.
a. Equiangular triangles are also acute.
b. Right triangles are acute.
Solution:

The first statement is always true.
The second statement is never true.

Question 2: In Δ PQR the measures of angles P and Q are given as m $\angle$P = 35º and m $\angle$Q = 47º.  Determine whether PQR is an acute triangle.
Solution:

The sum of the measures of the three angles in a triangle = m $\angle$P + m $\angle$Q + m $\angle$R = 180º
Hence m $\angle$R = 180 - ($\angle$P + $\angle$Q) = 180 - (35 + 47) = 180 - 82 = 98º > 90
$\angle$R is not acute. Hence ΔPQR is not acute.

Question 3: The lengths of the sides of a triangle are given as 8 cm, 10 cm and 12 cm. Determine whether the triangle is acute.
Solution:

The sum of the squares of the smaller sides = a2 + b2 = 82 + 102 = 64 + 100 = 164
The square of the largest side = c2 = 122 = 144
Using the test for acute triangle,
a2 + b2 < c2.
Hence the given triangle is an acute triangle.