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 **acute angle triangle **is one of them.

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

Scalene Triangle |
Isosceles Triangle |
Equilateral Triangle |

A triangle with no equal sides or a triangle in which all the sides of a triangle are of different length. | A triangle with two equal sides and two equal angles is called an isosceles triangle. Also, two angles of the triangle are of the same measure | A triangle in which all three sides are equal and each interior angle of a triangle measure 60 degrees is called the equilateral triangle |

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

Acute Angle Triangle |
Right Angle Triangle |
Obtuse Angle Triangle |

A triangle which consists of three acute angles in which all the angles are less than 90 degrees | A triangle in which one angle measures 90 degrees and other two angles are less than 90 degrees (acute angles) | A triangle in which one angle measures above 90 degrees and the other two angles that measure less than 90 degrees. |

## Acute Angle Triangle Definition

An acute angle triangle (or acute angled triangle) is a triangle that has acute angles as all of its interior 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

Formulas for Acute Triangle | |
---|---|

Area of Acute Angle | (½) × b × h |

Perimeter of Acute Triangle | a + b + c |

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** = \(A = \sqrt{S (S-a)(S-b)(S-c)}\) square units

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 AB, BC, and CA 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** - A triangle has three vertices
- The interior angles of a triangle are formed when two edges of a triangle meet.

## 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 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.

### Square of the Longest Side

For an acute angle triangle, the square of the longest side will always be less than the sum of the squares of the other two sides.

## Practice Questions

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

## Frequently Asked Questions From Acute Angle Triangle

### Can an Equilateral Triangle be an Acute Angle Triangle?

Yes, all equilateral triangles are acute angle triangles. It is because an equilateral triangle has three equal angles i.e. 60° each which are acute angles.

### How to Find Third Angle in Acute Angle Triangle?

To find the third angle of an acute triangle, add the other two sides and then subtract the sum from 180°. Thus, the formula to find the third angle is ∠a + ∠b + ∠c = 180°.

### Is an Acute Scalene Triangle Possible?

Yes, an acute scalene triangle is possible if the interior angles of the scalene triangles are acute. Not only scalene, but an acute triangle can also be an isosceles triangle if it satisfies its condition.

### Can a Triangle Have Only One Acute Angle?

A triangle can never have only one acute angle. If a triangle has 1 acute angle, the other angles will be either right angles or obtuse angles which is not possible as the sum of interior angles of a triangle is always 180°. So, every triangle needs to have at least 2 acute angles.

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