What is a Triangle?
Definition: A triangle is essentially a polygon with 3 sides. From the figure we can see that a triangle has 3 vertices and 3 edges. In our daily life we come across various triangular objects such as a clothes hanger, a pizza slice, sandwich and so on.
We use the ‘\( \triangle\)’ symbol to represent a triangle.
Read more about similar triangles
Properties of Triangle
Let us take \( \triangle~ABC\) given in the figure to understand the properties of a triangle.
1. The triangle has three sides and three vertices.
2. The sum of the interior angles of a triangle is 180°.
∠1 + ∠2 + ∠3 = 180°
3. The sum of the exterior angles is 360°.
∠4 + ∠5 + ∠6 = 360°
4. The exterior angle is equal to the sum of its opposite interior angles.
∠4 = ∠2 + ∠3, ∠5 = ∠1 + ∠3, ∠6 = ∠1 + ∠2
5. The adjacent interior and exterior angles of a triangle form a linear pair of angles.
∠1 + ∠4 = 180°
∠2 + ∠5 = 180°
∠3 + ∠6 = 180°
6. For a triangle with two unequal sides the angle opposite to the longer side is larger and the angle opposite to the shorter side is
smaller.
7. The sum of measure of any two sides of a triangle is always greater than the third side.
(AB + BC) > CA
8. The difference between the measure of any two sides of a triangle is always less than the third side.
(AB – BC) < CA
Classification of Triangles Based on Sides
On the basis of sides, triangles are classified into three types:
- Scalene Triangle
- Isosceles Triangle
- Equilateral Triangle.
Scalene Triangle: If all three sides of a triangle are different in measurement then it is known as a scalene triangle.
Isosceles Triangle: If two sides of a triangle are equal in measurement then it is known as an isosceles triangle.
In an isosceles triangle the angles opposite to the equal sides are equal in measure.
Equilateral Triangle: If all three sides of a triangle are equal in measurement then it is known as an equilateral triangle.
In an equilateral triangle all angles are equal in measure. Hence each angle of an equilateral triangle is 60°.
Read more about Classification of Triangles based on sides
Classification of Triangles Based on Angles
On the basis of angles, triangles are classified into three types:
- Acute Triangle
- Right angled Triangle or Right Triangle
- Obtuse Triangle.
Acute Triangle : A triangle in which all three angles are less than 90° in measurement.
Right Angle Triangle: A triangle in which one angle measures 90° (right angle).
Obtuse Triangle: A triangle in which one angle is greater than 90°.
Read more on angles of triangles
Perimeter of Triangle
The perimeter of a triangle is the sum of lengths of all three sides of the triangle. It shows the total length of the outer boundary of the triangle.
The given triangle has sides AB, BC and CA.
So, perimeter of triangle, P = sum of all sides
= AB + BC + CA
Read more about perimeter of a triangle
Area of Triangle
The area of a triangle is the measure of the total space enclosed within its three sides.
If the height and base of triangle are given:
Area of triangle,A = half of the product of base and height
= \(\frac{1}{2}~\times~B~\times~H\)
Read more about area of a triangle, triangle area calculator
Solved Examples on Triangle
Example 1: Identify the type of triangle on the basis of angle.
Solution:
In the given triangle, one angle is 90° (right angle). So, the given triangle is a right angled triangle.
Example 2: Identify the type of triangle on the basis of side.
Solution: The triangle shown has two equal sides. So, this is an isosceles triangle.
Example 3: Jim makes an equilateral triangle from a 165m long wire. Find the side length of the triangle made from the wire.
Solution:
The total length of wire is 165m.
So, the perimeter of the equilateral triangle is 165m.
Perimeter of equilateral triangle = side + side + side
165 = 3 × side [Substitute 165 for perimeter]
Divide both sides of the equation by 3,
55 = side
So, the length of the side of the equilateral triangle is 55m.
Example 4: John was studying geometrical shapes and found a triangle of height 14cm and base 21cm. What is the area of the triangle?
Solution: The height and base of the triangle is 14cm and 21cm respectively.
Area of triangle \(~=\frac{1}{2}~\times~B~\times~H\)
\(~=\frac{1}{2}~\times~21~\times~14\) [Substitute 21 for B and 14 for H]
\(~=147\)
Therefore the area of triangle is 147 square cm.
A triangle is a 2-D closed shape with three sides whose sum of interior angles is 180°.
A polygon is a closed figure with n sides and a triangle is a 3 sided closed figure. So, a triangle is a polygon.
The sum of exterior angles of a triangle is 360°.
As per the angle sum property of a triangle the sum of all interior angles of a triangle is 180°.