Properties of Triangle: A Complete Guide

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.

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.

1. Sum of Interior Angles: The sum of the interior angles of a triangle is always 180 degrees.
2. 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.

3. 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.

4. Exterior Angles: The sum of the exterior angles of a triangle is always 360 degrees.
5. Perimeter: The perimeter of a triangle is the sum of the lengths of its three sides.
6. 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

=3²+4²

=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

RELATED BLOGS

What are Vertices, Faces And Edges?	Perimeter of Rectangle: Formula and Examples
Surface Area of Prism: Formula and Steps	Area of Rectangle: Formula and Example
Surface Area of a Cuboid: Definition, Derivation and Formula	Different Maths Shapes for Students and Kids

This was all about “Properties of Triangle”. For more such informative blogs, check out our Study Material Section, or you can learn more about us by visiting our  Indian exams page.