Types of Triangles and Their Properties
The concept of triangle definition in maths is a foundational part of geometry, appearing throughout school textbooks and real-life scenarios. Understanding triangles is essential for solving shape problems, calculating area and perimeter, and preparing for exams like Olympiads, NTSE, and CBSE boards.
What Is Triangle Definition in Maths?
A triangle in maths is a closed two-dimensional shape or polygon with exactly three straight sides and three angles. Each point where two sides meet is called a vertex, and the sum of the interior angles in a triangle is always 180°. You’ll find triangle definition in maths used in topics like classification of shapes, measurement of areas, trigonometry, and coordinate geometry.
Properties of Triangle in Maths
Every triangle shares a basic set of features useful for classifying, measuring, and solving geometric problems. Here are the key properties:
- It has 3 sides, 3 angles, and 3 vertices.
- The sum of all interior angles is always exactly 180°.
- The sum of any two sides is always greater than the third side.
- The difference of any two sides is less than the third side.
- The side opposite the largest angle is the longest side.
Types of Triangle in Maths
Triangles are classified based on their sides or angles. Here’s a quick table to help you remember the different types:
| Classification | Types | Properties |
|---|---|---|
| By Sides | Equilateral, Isosceles, Scalene |
Equilateral: All sides equal, all angles 60°
Isosceles: 2 sides equal, 2 angles equal
Scalene: All sides and all angles different
|
| By Angles | Acute, Obtuse, Right-angled |
Acute: All angles < 90°
Right-angled: One angle is 90°
Obtuse: One angle > 90°
|
Key Formula for Triangle Definition in Maths
Here are the standard formulas every student should know for triangles:
- Angle Sum Property: ∠A + ∠B + ∠C = 180°
- Perimeter: a + b + c (sum of all sides)
- Area: \( \frac{1}{2} \times \text{base} \times \text{height} \)
- Heron's Formula (all side lengths known): Area = \( \sqrt{s(s - a)(s - b)(s - c)} \) where s = semi-perimeter = (a+b+c)/2
- Pythagoras’ Theorem (Right triangle): hypotenuse² = base² + height²
Triangle Examples with Solutions
Let’s see two examples to understand how triangle formulas are used:
Example 1 (Area using Base and Height):
2. Use area formula: \( \frac{1}{2} \times 5 \times 4 = 10 \) cm2
3. Final Answer: The area of the triangle is 10 cm2
Example 2 (Missing Side using Perimeter):
2. Third side = 19 − (7 + 6) = 6 cm
3. Final Answer: The third side is 6 cm
Speed Trick or Vedic Shortcut
To quickly check if three lengths can form a triangle, use this shortcut:
- Add any two sides, the sum must be greater than the third side:
6+8 = 14 > 13 ✔
8+13 = 21 > 6 ✔
13+6 = 19 > 8 ✔
All checks passed – these can form a triangle!
This triangle inequality trick saves time in MCQs and mental maths rounds. Vedantu’s live tutors show many such exam tips in their online sessions.
Triangle in Real Life and Other Subjects
Triangles are everywhere – in traffic signs, ramps, roofs, art, and bridges. In maths, they form the basis for trigonometry and coordinate geometry. Triangle concepts are also important in physics (statics, mechanics) and computer graphics. Students preparing for JEE, NEET, or board exams will always find problems based on triangles and their properties.
Try These Yourself
- Classify a triangle with sides 5 cm, 5 cm, 5 cm (by sides and angles).
- Calculate the perimeter of a triangle with sides 7 cm, 4 cm, 3 cm.
- Find the area of a triangle with base 8 cm and height 3 cm.
- Check whether 2 cm, 4 cm, 7 cm can form a triangle.
Common Mistakes with Triangles
- Forgetting the sum of angles is always 180°.
- Not checking if three sides satisfy the triangle inequality before drawing.
- Using incorrect units (mixing cm and mm).
- Mixing up types (calling an equilateral triangle “isosceles” and vice versa).
Related Important Math Concepts
The triangle definition in maths is closely related to polygons, quadrilaterals, and angle concepts. Mastering triangles helps with understanding congruence of triangles, similar triangles, area and perimeter, and more advanced topics in geometry.
Quick Classroom Tip
Remember: “The three angles of any triangle add up to a straight angle—180°.” Drawing triangles with different tools and checking their angle sum using a protractor helps make this rule natural. Vedantu class teachers often assign drawing and measuring real triangle objects at home for extra practice.
We explored the triangle definition in maths—from basic meaning, properties, types, important formulae, and solved examples, to real-life use and exam tips. To learn more or ask doubts, explore interactive lessons and more triangle resources with Vedantu’s online maths programs.
For more on triangles, check out these helpful resources:
FAQs on What is a Triangle? Definition, Properties & Examples
1. What is the definition of a triangle in maths?
In mathematics, a triangle is a closed two-dimensional geometric shape with three straight sides and three angles. It is also known as a polygon with three sides. The sum of the interior angles of any triangle always equals 180 degrees.
2. What are the different types of triangles based on their sides?
Triangles are classified into three main types based on the lengths of their sides:
- Equilateral Triangle: All three sides are equal in length.
- Isosceles Triangle: Two sides are equal in length.
- Scalene Triangle: All three sides have different lengths.
3. How are triangles classified based on their angles?
Triangles are also categorized based on the measure of their angles:
- Acute Triangle: All three angles are less than 90 degrees.
- Right-angled Triangle: One angle is exactly 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.
4. What is the formula for the area of a triangle?
The area (A) of a triangle is calculated using the formula: A = (1/2) * base * height, where 'base' is the length of one side and 'height' is the perpendicular distance from that side to the opposite vertex.
5. What is the perimeter of a triangle?
The perimeter of a triangle is the total length of its three sides. It is found by adding the lengths of all three sides together: Perimeter = side1 + side2 + side3
6. What is the Pythagorean theorem, and how does it apply to triangles?
The Pythagorean theorem applies only to right-angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs or cathetus). The formula is: hypotenuse² = side1² + side2²
7. What is Heron's formula, and when is it used?
Heron's formula is used to calculate the area of any triangle when you know the lengths of all three sides. Let 'a', 'b', and 'c' be the lengths of the sides, and 's' be the semi-perimeter (s = (a+b+c)/2). The formula is: Area = √[s(s-a)(s-b)(s-c)]
8. What is the sum of the exterior angles of a triangle?
The sum of the exterior angles of any triangle is always 360 degrees.
9. Give a real-life example of a triangle.
Many everyday objects are triangular! Examples include the faces of a pyramid, a traffic sign, and the support structure of a bridge.
10. How are triangles used in construction?
Triangles are very strong shapes, making them ideal for use in construction. They provide stability to structures like roofs, bridges, and towers. The triangular shape distributes weight effectively.
11. What is the difference between congruent and similar triangles?
Congruent triangles are triangles that have the same size and shape; all corresponding sides and angles are equal. Similar triangles have the same shape but may be different sizes; corresponding angles are equal, and corresponding sides are proportional.
