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Triangles

Triangles are basic geometric shapes that appear in nature, art, engineering, and architecture everywhere. Since triangles are the foundation of polygonal geometry and help in our understanding of angles, structures, and shapes, studying them is essential. Triangles are used on everything from road signs to rooftops.

Table of Contents

Definition of Triangle
Properties of a Triangle
Types of Triangle
Area of Triangle
Perimeter of Triangle
Common Misconceptions About Triangles
Solved Example
Conclusion
FAQs on Triangles

Definition of Triangle

A triangle is a closed two-dimensional shape formed by three line segments. It has three sides, three vertices, and three angles. The sum of the internal angles of any triangle is always 180 degrees.

Example:

A triangle with vertices A, B, and C is denoted as △ABC.

Properties of a Triangle

A triangle's interior angles add up to 180°.
The sum of the two opposing interior angles determines the exterior angle.
Any two sides added together are larger than the third side.
In a right angle triangle, (Hypotenuse)² = (Base)² + (Height)²
Area = 1/2 × base × height
Perimeter = sum of all three sides

Types of Triangle

There are two major ways to classify triangles:

Based on sides:

Scalene Triangle
Isosceles Triangle
Equilateral Triangle

Based on angles:

Acute Angle Triangle
Right Angle Triangle
Obtuse Angle Triangle

Scalene Triangle

A Scalene Triangle is a triangle in which all three sides and all three angles are different.

Properties of a Scalene Triangle:

No equal sides
No equal angles.
The largest angle lies opposite the longest side.

Example:

If the sides of a triangle are 5 cm, 6 cm, and 7 cm, then it is a Scalene Triangle.

Scalene Triangle is unique in its irregularity, and it lacks symmetry, unlike other types of triangle.

Isosceles Triangle

An Isosceles Triangle has two equal sides and two equal angles.

Properties of an Isosceles Triangle:

Two sides are equal in length.
The angles opposite to equal sides are also equal.
It has one line of symmetry.

Example:

If a triangle has sides of 5 cm, 5 cm, and 3 cm, then it is an Isosceles Triangle.

In real life, the front view of a sail on a boat often forms an Isosceles Triangle.

Equilateral Triangle

An Equilateral Triangle has three equal sides and three equal angles, each measuring 60°.

Properties of an Equilateral Triangle:

All sides are of equal length.
All angles are equal.
Has three lines of symmetry.
It is a regular polygon.

Example:

If each side of a triangle is 6 cm, and all angles are 60°, it is an Equilateral Triangle.

The Equilateral Triangle is the most symmetric triangle and often used in tiling patterns and design.

Acute Angle Triangle

An Acute Angle Triangle is a triangle where all three interior angles are less than 90°.

Properties of an Acute Angle Triangle:

Each angle < 90°.
Can be Scalene, Isosceles, or Equilateral.
The smallest angle lies opposite the shortest side.

Example:

A triangle with angles 60°, 50°, and 70° is an Acute Angle Triangle.

Most Equilateral Triangles are also Acute Angle Triangles since all angles are 60°.

Right Angle Triangle

A Right Angle Triangle has one angle exactly equal to 90°.

Properties of a Right Angle Triangle:

One angle is 90°.
The side opposite the right angle is called the hypotenuse.
Follows the Pythagorean Theorem.

Example:

A triangle with sides 3 cm, 4 cm, and 5 cm forms a Right Angle Triangle.

Right-angled triangles are used in construction, stairs, and measuring slopes.

Obtuse Angle Triangle

An Obtuse Angle Triangle has one angle greater than 90° but less than 180°.

Properties of an Obtuse-Angle Triangle:

Has one obtuse angle (> 90°).
The other two angles are always acute.
Can be Scalene or Isosceles, but never Equilateral.

Example:

A triangle with angles 120°, 30°, and 30° is an Obtuse Angle Triangle.

Among all types of triangles, the Obtuse Angle Triangle is the one that bulges outward.

Area of Triangle

The area of triangle is the space enclosed by its three sides.

Formula 1: Area = 1/2 × base × height

Formula 2: Heron’s formula when all sides are known

s = (a + b + c)/2

Area = √[s(s - a)(s - b)(s - c)]

For an Equilateral Triangle:

Area = (√3 / 4) × side²

Examples:

Equilateral Triangle: Area = (√3 / 4) × a²

Right Angle Triangle: Area = 1/2 × base × height

Area is always measured in square units.

Perimeter of Triangle

The perimeter of a triangle is the total length of its sides.

Formula:

Perimeter = a + b + c

Where a, b, and c are the three sides of the triangle.

Example:

If the sides are 5 cm, 5 cm, and 3 cm

Then Perimeter = 5 + 5 + 3 = 13 cm

Common Misconceptions About Triangles

A right angle is impossible for an equilateral triangle. Its angles are all 60 degrees.
It is impossible for a triangle to have two obtuse angles or two right angles. All angles must add up to 180°.
In a triangle, the longest side is always on the other side of the largest angle.
Not every triangle that lacks a right angle is scalene. They may also be equilateral or isosceles.
The formula for area is ½ × base × height; however, base and height must be perpendicular to one another.
An equilateral has three lines of symmetry, an isosceles has one, and a scalene triangle has none.
No three side lengths can be used to create a triangle. Any two sides added together must be greater than the third.

Solved Example

Example 1: Identify the Type of Triangle by Sides

Question:

The sides of a triangle are 7 cm, 7 cm, and 5 cm. Identify the type of triangle.

Solution:

Two sides are equal (7 cm and 7 cm), and one side is different.

So, this is an Isosceles Triangle.

Example 2: Identify the Type of Triangle by Angles

Question:

A triangle has angles 60°, 60°, and 60°. What type of triangle is it?

Solution:

All three angles are equal and less than 90°.

So, it is both an Equilateral Triangle and an Acute Angle Triangle.

Example 3: Area of Triangle using Base and Height

Question:

Find the area of a triangle with base 10 cm and height 8 cm.

Solution:

Area = 1/2 × base × height

Area = 1/2 × 10 × 8

Area = 40 cm²

Answer: 40 cm²

Example 4: Area of an Equilateral Triangle

Question:

Find the area of an equilateral triangle with each side equal to 6 cm.

Solution:

Area = (√3 / 4) × side²

Area = (√3 / 4) × 6²

Area = (√3 / 4) × 36

Area = 9√3 cm² ≈ 15.59 cm²

Answer: 15.59 cm² (approx)

Example 5: Using Heron’s Formula

Question:

Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.

Solution:

First, find the semi-perimeter:

= (7 + 8 + 9)/2 = 24/2 = 12 cm

Now use Heron’s Formula:

Area = √[s(s - a)(s - b)(s - c)]

Area = √[12(12 - 7)(12 - 8)(12 - 9)]

Area = √[12 × 5 × 4 × 3] = √720 ≈ 26.83 cm²

Answer: 26.83 cm² (approx)

Conclusion

One of the most basic geometric shapes is the triangle. One can effectively classify and solve geometric problems by knowing the different types of triangles based on their sides (Scalene, Isosceles, Equilateral)

and angles (Acute Angle, Right Angle, Obtuse Angle). Understanding a triangle's characteristics and how to determine its area and perimeter are crucial for advanced mathematics as well as practical uses in

fields like engineering, design, and construction. A solid foundation in triangles fosters logical thinking and increases geometric confidence.

Related Links

Congruence of Triangles - Learn the rules of triangle congruence with clear definitions and solved examples.

Parallelogram - Understand the properties of parallelograms through simple explanations and practical problems.