Triangles are fascinating and fundamental shapes in geometry, forming the basis for much of the subject. This guide will take you through the various properties of triangles, such as their types based on sides and angles, the sum of their interior angles, and important concepts like the Pythagorean theorem. Whether you’re a student just starting out or someone needing a refresher, this comprehensive guide to the properties of triangles will help you understand and appreciate the unique characteristics of triangles. Keep reading learn more about the properties of triangles and explore their formulas in a simple and straightforward way.

Table of Contents

## Definition of Triangle

A triangle is a three-sided polygon with three edges and three vertices. It is one of the simplest shapes in geometry and is defined by its three sides and the angles between those sides. Here are some key properties of triangles.

**Sum of Interior Angles**: The sum of the interior angles of a triangle is always 180 degrees.**Types of Triangles by Sides**:

**Equilateral Triangle**: All three sides are equal, and all three interior angles are 60 degrees.**Isosceles Triangle**: Two sides are equal, and the angles opposite these sides are equal.**Scalene Triangle**: All three sides are of different lengths, and all three interior angles are different.

**Types of Triangles by Angles**:

**Acute Triangle**: All three interior angles are less than 90 degrees.**Right Triangle**: One interior angle is exactly 90 degrees.**Obtuse Triangle**: One interior angle is more than 90 degrees.

**Exterior Angles**: The sum of the exterior angles of a triangle is always 360 degrees.**Perimeter**: The perimeter of a triangle is the sum of the lengths of its three sides.**Area**: The area of a triangle can be calculated using various formulas, such as**1/2 x base x height**or Heron’s formula for more complex cases.

**Also Read: ****Find the Area of the Shaded Region: Square, Rectangle**

## Types of Triangle

Triangles can be categorized based on their sides and their angles. Here are the different types of triangles in each category.

### Types of Triangles by Sides

**Equilateral Triangle**:

- All three sides are of equal length.
- All three interior angles are equal, each measuring 60 degrees.

**Isosceles Triangle**:

- Two sides are of equal length.
- The angles opposite the equal sides are also equal.

**Scalene Triangle**:

- All three sides are of different lengths.
- All three interior angles are different.

### Types of Triangles by Angles

**Acute Triangle**:

- All three interior angles are less than 90 degrees.

**Right Triangle**:

- One interior angle is exactly 90 degrees.
- The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle.

**Obtuse Triangle**:

- One interior angle is more than 90 degrees.

Type by Sides | Description | Type by Angles | Description |

Equilateral | All sides and angles are equal | Acute | All angles are less than 90 degrees |

Isosceles | Two sides and two angles are equal | Right | One angle is exactly 90 degrees |

Scalene | All sides and angles are different | Obtuse | One angle is greater than 90 degrees |

## Properties of Triangle with Formulas

Triangles have various properties and associated formulas that help in solving geometric problems. Here’s a detailed look at the properties of triangles along with their relevant formulas.

**1. Sum of Interior Angles**

**Property**: The sum of the interior angles of a triangle is always 180 degrees.**Formula**: ∠A+∠B+∠C=180∘

**2. Sum of Exterior Angles**

**Property**: The sum of the exterior angles of a triangle is always 360 degrees.**Formula**: ∠D+∠E+∠F=360∘

**3. Types of Triangles Based on Angles and Sides**

**Equilateral Triangle**:- All sides are equal.
- All angles are equal to 60 degrees.

**Isosceles Triangle**:- Two sides are equal.
- The angles opposite the equal sides are equal.

**Scalene Triangle**:- All sides and angles are different.

**Right Triangle**:- Has one 90-degree angle.
- The side opposite the right angle is called the hypotenuse.

**4. Perimeter of a Triangle**

**Property**: The perimeter is the sum of the lengths of all three sides.**Formula**: P=a+b+c.- Where a,b, and c are the lengths of the sides.

**5. Area of a Triangle**

**Property**: The area can be calculated using different methods depending on the information available.**Formula (Base and Height)**: Area=1/2×base×height**Heron’s Formula**:- Used when the lengths of all three sides are known.
- s=a+b+c/2 (semi-perimeter)
- Area= √({s(s-a)(s-b)(s-c)}

**6. Pythagorean Theorem**

**Property**: Applies to right triangles.**Formula**: c²=a²+b²- Where ccc is the hypotenuse and aaa and bbb are the other two sides.

**7. Altitude, Median, and Angle Bisector**

**Altitude**:- A perpendicular segment from a vertex to the line containing the opposite side.

**Median**:- A segment connecting a vertex to the midpoint of the opposite side.

**Angle Bisector**:- A segment that divides an angle into two equal parts.

Property | Description | Formula |

Sum of Interior Angles | Sum of all interior angles is 180 degrees | ∠A+∠B+∠C=180 degree |

Sum of Exterior Angles | Sum of all exterior angles is 360 degrees | ∠D+∠E+∠F=360 degree |

Perimeter | Sum of the lengths of all sides | P = a+b+c |

Area (Base and Height) | Half the product of base and height | Area=1/2×base×height |

Area (Heron’s Formula) | Square root of product of semi-perimeter and differences | Area= √({s(s−a)(s−b)(s−c)} |

Pythagorean Theorem (Right Triangle) | Hypotenuse squared equals sum of squares of other sides | c² = a²+b² |

## Properties of Triangle Solved Examples

Here are 10 solved examples related to different properties and formulas of triangles.

**Q1**: **Given a triangle with angles ∠A=50∘ and ∠B=60∘, find ∠C. **

**Solution**:

∠A+∠B+∠C=180∘

50∘+60∘+∠C=180∘

∠C=180∘−110∘=70∘

**Q2: Find the perimeter of a triangle with sides a=5 cm, b=7 cm, and c=10cm.**

**Solution: **P=a+b+c

=5cm+7cm+10cm

=22cm

**Q3: Calculate the area of a triangle with a base of 8 cm and height of 5 cm. **

**Solution**:

Area=1/2×base×height

=1/2×8 cm×5 cm=20 cm

Area=1/2×base×height

=1/2×8cm×5cm=20cm²

**Q4**: Find the area of a triangle with sides a=6 cm, b=8 cm, and c=10 c.

**Solution**: s=2a+b+c=26+8+10=12

**Area= √({s(s−a)(s−b)(s−c)}**

√12(12-6)(12-8)(12-10)

=√12x6x4x2

=√576

=24cm²

**Q5**: In a right triangle, if a=3 cm and b=4 cm, find the hypotenuse c.

**Solution**:

**c² = a²+b²**

=3**²**+4**²**

=9+16

=√25

=5cm

**Q6**: What is the measure of each angle in an equilateral triangle?

**Solution**: In an equilateral triangle, all angles are equal.

∠A=∠B=∠C=60∘

**Q7**: Find the sum of the exterior angles of any triangle.

**Solution**: The sum of the exterior angles of any triangle is always 360∘.

**Q8**: Find the altitude of an equilateral triangle with side length a=6 cm.

**Solution**: Altitude= √3/2 x a

= √3/2 x 6cm

=3√3 cm

**Q9**: In a right triangle, if the hypotenuse c=13 cm and one side a=5 cm, find the other side b.

**Solution**: **c² = a²+b²**

= 13² = 5² + b²

=169 = 25 + b²

= b² =144

= b = √144

=12cm

**Q10**: Given an isosceles triangle with equal sides a=5 cm and base b=6 cm, find the perimeter and area.

**Solution:**

**Perimeter:**

P=2a+b

=2×5 cm+6 cm

=16 cm

**Area**:

Height = √ a² – (b/2)²

= √5² – 3²

=√25-9

=√16 = 4cm

Area=1/2×b×height

=1/2×6cm×4cm

=12cm

**Also Read: ****Formula of Profit and Loss: Percentages**

## FAQs

**What is the triangle property?**

The sum of all the angles in any triangle is 180°.

**What is the triangle theorem?**First, all three of the inside sides of a triangle add up to 180 degrees. We know that if ABC is a triangle, then ∠A + ∠B + ∠C = 180°. 2. The angles at the base of an isosceles triangle are the same.

**What is a reflex angle?**An angle that is between 180 and 360 degrees is called a reflex angle. Like, 270 degrees is an example of a reflex angle. An acute angle is less than 180 degrees, an obtuse angle is more than 180 degrees, and a right angle is above 180 degrees.

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