Triangles in Geometry

A triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional region.

Triangles are fundamental geometric shapes that play a crucial role in various fields, from mathematics and architecture to engineering and art. They can be classified based on their angles and sides:

• By Angles:
  • Acute-angled: All angles are less than 90°.
  • Obtuse-angled: One angle is greater than 90°.
  • Right-angled: One angle is exactly 90°.
• By Sides:
  • Equilateral: All sides (edges) are equal.
  • Isosceles: Two sides (edges) are equal.
  • Scalene: All sides (edges) are different.

Triangles are fundamental in Euclidean geometry, and their properties are essential for various real-world applications.
The triangle shape is one of the most common shapes that we observe in our daily lives. We observe traffic signals, snacks, cloth hangers, etc. which are shaped like triangles.

Parts of a Triangle

A triangle as the name suggests has three angles thus it is called a "tri" angle. It contains various parts. A triangle has 3 angles, 3 vertices, and 3 sides. 

In the triangle, △PQR given in the image below shows the vertices, sides, and interior angles of a triangle.

• Three Angles of the triangle are, ∠PQR, ∠QRP, and ∠RPQ.
• Three sides of the triangles are side PR, side PQ, and side RQ.
• There are three vertices of a triangle, which are P, Q, and R.

Angles in a Triangle

A triangle has three angles, an angle is formed when two sides of the triangle meet at a common point, this common point is known as the vertex. The sum of the three interior angles is equal to 180 degrees.

Let's take the triangle with the interior angles at ∠1, ∠2, and ∠3 and their respective exterior angle at, ∠4, ∠5 and ∠6

Now, using the triangle sum property
∠1 + ∠2 + ∠3 = 180°

When the sides are extended outwards in a triangle, then it forms three exterior angles. The sum of the interior and exterior angles pair of a triangle is always supplementary. Also, the sum of all three exterior angles of a triangle is 360 degrees.

Now,
∠4 + ∠5 + ∠6 = 360°

Types of Triangles

Classification of triangles is done based on the following characteristics:

• Types of Triangles Based on Sides
• Types of Triangles Based on Angles

Types of Triangles Based on Sides

Based on sides, there are 3 types of triangles:

1. Scalene Triangle
2. Isosceles Triangle
3. Equilateral Triangle

Equilateral Triangle

In an Equilateral triangle, all three sides are equal to each other as well as all three interior angles of the equilateral triangle are equal.
Since all the interior angles are equal and the sum of all the interior angles of a triangle is 180° (one of the Properties of the Triangle). We can calculate the individual angles of an equilateral triangle.

∠A+ ∠B+ ∠C = 180°
∠A = ∠B = ∠C

Therefore, 3∠A = 180°

∠A= 180/3 = 60°
Hence, ∠A = ∠B = ∠C = 60°

Properties of Equilateral Triangle

• All sides are equal.
• All angles are equal and are equal to 60°
• There exist three lines of symmetry in an equilateral triangle
• The angular bisector, altitude, median, and perpendicular line are all the same and here it is AE.
• The orthocentre and centroid are the same.

Equilateral Triangle Formulas

The basic formulas for equilateral triangles are:

• Area of Equilateral Triangle = √3/4 × a²
• Perimeter of Equilateral Triangle = 3a

where, a is Side of Triangle

Isosceles Triangle

In an Isosceles triangle, two sides are equal and the two angles opposite to the sides are also equal. It can be said that any two sides are always congruent. The area of the Isosceles triangle is calculated by using the formula for the area of the triangle as discussed above.

Properties of Isosceles Triangle

• Two sides of the isosceles triangle are always equal
• The third side is referred to as the base of the triangle and the height is calculated from the base to the opposite vertex
• Opposite angles corresponding to the two equal sides are also equal to each other.

Scalene Triangle

In a Scalene triangle, all sides and all angles are unequal. Imagine drawing a triangle randomly and none of its sides is equal, all angles differ from each other too.

Properties of Scalene Triangle

• None of the sides are equal to each other.
• The interior angles of the scalene triangle are all different.
• No line of symmetry exists.
• No point of symmetry can be seen.
• Interior angles may be acute, obtuse, or right angles in nature (this is the classification based on angles).
• The smallest side is opposite the smallest angle and the largest side is opposite the largest angle (general property).

Types of Triangles Based on Angles

Based on angles, there are 3 types of triangles:

1. Acute Angled Triangle
2. Obtuse Angled Triangle
3. Right Angled Triangle

Acute Angled Triangle

In Acute angle triangles, all the angles are greater than 0° and less than 90°. So, it can be said that all 3 angles are acute (angles are less than 90°)

Properties of Acute Angled Triangles

• All the interior angles are always less than 90° with different lengths of their sides.
• The line that goes from the base to the opposite vertex is always perpendicular.

Obtuse Angled Triangle

In an obtuse angle Triangle, one of the 3 sides will always be greater than 90°, and since the sum of all three sides is 180°, the rest of the two sides will be less than 90° (angle sum property).

Properties of Obtuse Angled Triangle

• One of the three angles is always greater than 90°.
• The sum of the remaining two angles is always less than 90° (angle sum property).
• Circumference and the orthocentre of the obtuse angle lie outside the triangle.
• The incentre and centroid lie inside the triangle.

Right Angled Triangle

When one angle of a triangle is exactly 90°, then the triangle is known as the Right Angle Triangle. 

Properties of Right-angled Triangle

• A Right-angled Triangle must have one angle exactly equal to 90°, it may be scalene or isosceles but since one angle has to be 90°, hence, it can never be an equilateral triangle.
• The side opposite 90° is called Hypotenuse.
• Sides are adjacent to the 90° are base and perpendicular.
• Pythagoras Theorem: It is a special property for Right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the base and perpendicular i.e. AC² = AB² + BC²

Triangle Formulas

In geometry, for every two-dimensional shape (2D shape), there are always two basic measurements that we need to find out, i.e., the area and perimeter of that shape. Therefore, the triangle has two basic formulas which help us to determine its area and perimeter. Let us discuss the formulas in detail.

Perimeter of Triangle

The Perimeter of a triangle is the total length of all its sides. It can be calculated by adding the lengths of the three sides of the triangle, suppose a triangle with sides a, b, and c is given then its perimeter is given by:

Perimeter of triangle = a + b + c

Area of a Triangle

The area of a triangle is the total area covered by the triangle boundary. It is equal to half the product of its base and height. It is measured in square units. If the base of a triangle is b and its height is h then its area is given by:

Area of Triangle = 1/2 × b × h

Read More: Finding the Area of the Triangle using Heron's Formula.
Also check: How to Find Area of Triangle, Formulas, Examples.

Properties of Triangles

Various properties of triangles are,

• A Triangle has 3 sides, 3 vertices, and 3 angles.

• Area of Triangle: 1/2× base × height.

• For similar triangles, the angles of the two triangles have to be congruent to each other and the respective sides should be proportional.

• The difference between the length of any two sides is always lesser than the third side. For example, AB - BC < AC or BC - AC < AB

• The side opposite the largest angle is the largest side of the triangle. For instance, in a right-angled triangle, the side opposite 90° is the longest.

• The sum of the length of any two sides of a triangle is always greater than the third side. For example, AB + BC > AC or BC + AC > AB.

• Angle Sum Property: The sum of all three interior angles is always 180°. Therefore. In the Triangle ΔPQR shown above, ∠P + ∠Q + ∠R = 180°, the interior angles of a triangle will be greater than 0° and less than 180°. 

• The perimeter of a figure is defined by the overall length the figure is covering. Hence, the perimeter of a triangle is equal to the sum of lengths on all three sides of the triangle. Perimeter of ΔABC= (AB + BC + AC)

• The exterior angle of a triangle is equal to the sum of the interior opposite and non-adjacent angles (also referred to as remote interior angles). In the triangle ΔPQR, if we extend side QR to form an exterior angle ∠PRS, then: ∠PRS = ∠PQR + ∠PRQ

Read More about the Properties of Triangles.

Important Concepts of Triangle

Triangle is an important chapter, let's learn the important concepts of Triangle.

• Median of Triangle
  A median is a line segment that joins the vertex of the triangle with the midpoint of opposite sides of the triangle. A median of a triangle bisects the side which it joins.

• Altitude of Triangle
  The altitude of the Triangle is the perpendicular distance from the base of the triangle to its apex vertex.

• Centroid of Triangle
  A centroid is the point inside a triangle where all the medians of a triangle meet each other. The Centroid of a Triangle divides the median into 2:1.

• Circumcentre of a Triangle
  The Circumcentre of a Triangle is the point where all the perpendicular bisectors of the sides of the triangle.

• Orthocentre of a Triangle
  The Orthocentre of a Triangle is the point where all the altitudes of a triangle meet each other.

• Incentre of a Triangle
  The center of a Triangle is a point where all the angle bisectors of a triangle meet each other.

Fun Facts about Triangles

Below are 10 interesting facts about Triangles that find real-life significance:

• Strong and Stable: Triangles are incredibly strong for their size because they distribute stress evenly throughout their shape. This is why bridges, trusses, and even airplane wings are often built using triangular frameworks.

• Minimal Materials: Because triangles are so strong, they can be built using less material compared to other shapes for the same level of stability. This makes them a lightweight and efficient choice in construction.

• Stacking Efficiency: Triangles, especially equilateral ones, can be packed together very efficiently. This is useful for things like creating stable and space-saving containers for fruits and vegetables.

• Direction Indicators: Triangles are universally recognized as pointing arrows. This makes them ideal for road signs, warning labels, and directional markers, ensuring clear communication.

• Musical Harmony: The basic principles of harmony in music rely on the perfect fifth, which has a frequency ratio of 3:2. This ratio can be visualized as a 30-60-90 degree triangle, making triangles a foundational concept in musical theory.

• Tooth Shape Efficiency: Our premolar teeth have triangular cusps that are perfect for grinding and tearing food. The triangular shape allows for maximum surface area and efficient chewing.

• Aerodynamic Design: The triangular shape plays a role in aerodynamics. The delta wing, a triangular airplane wing design, is known for its stability and maneuverability at high speeds.

• Facial Recognition: Our brains use triangles to recognize faces. The triangular arrangement of eyes, nose, and mouth helps us quickly identify and differentiate faces.

• Fracture Lines: Even in breaking, triangles can be helpful! Cracks in glass or other materials often propagate in triangular patterns, which can help predict how something might break and potentially prevent accidents.

• Building Blocks of Life: The basic building block of DNA, the double helix, can be visualized as two intertwined triangles representing the sugar-phosphate backbones. This triangular structure is essential for storing and transmitting genetic information.

Triangles Solved Examples

Example 1: In a triangle ∠ACD = 120°, and ∠ABC = 60°. Find the type of Triangle.
Solution:

In the above figure, we can say, ∠ACD = ∠ABC + ∠BAC (Exterior angle Property)

120° = 60° + ∠BAC
∠BAC = 60°
∠A + ∠B + ∠C = 180°
∠C OR ∠ACB = 60°

Since all the three angles are 60°, the triangle is an Equilateral Triangle.

Example 2: The triangles with sides of 5 cm, 5 cm, and 6 cm are given. Find the area and perimeter of the Triangle.
Solution: 

Given, the sides of a triangle are 5 cm, 5 cm, and 6 cm
Perimeter of the triangle = (5 + 5 + 6) = 16 cm
Semi Perimeter = 16 / 2 = 8 cm
Area of Triangle = √s(s - a)(s - b)(s - c) (Using Heron's Formula)
= √8(8 - 5)(8 - 5)(8 - 6)
= √144 = 12 cm²

Example 3: In the Right-angled triangle, ∠ACB = 60°, and the length of the base is given as 4cm. Find the area of the Triangle.
Solution:

Using trigonometric formula of tan60°,
tan60° = AB / BC = AB  /4
AB = 4√3cm
Area of Triangle ABC = 1/2 
= 1/2 × 4 × 4√3 
= 8√3 cm²

Example 4: In ΔABC if ∠A + ∠B = 55°. ∠B + ∠C = 150°, Find angle B separately.
Solution:

Angle Sum Property of a Triangle says ∠A + ∠B + ∠C= 180°
Given: 
∠A + ∠B = 55°
∠B + ∠C = 150°

Adding the above 2 equations,

∠A + ∠B + ∠B + ∠C= 205°
180° + ∠B= 205°
∠B = 25°

Articles related to Triangles:

• Congurency of triangles
• Similar Triangles
• Angle Sum Property 

A star has 10 triangles that are,

∆ABH, ∆BIC, ∆CJD, ∆DFE, ∆EGA, ∆DGI, ∆GJB, ∆FAI, ∆JGB and ∆EHJ