Equilateral Triangle

An equilateral triangle is a special kind of triangle in geometry with all three sides the same length and all three angles the same, each with a measure of 60 degrees. The definition of an equilateral triangle is simply a triangle that has three equal sides and three equal angles. It is one of the most symmetrical figures in geometry and is applied extensively in mathematics, architecture, and design.

 

Table of Contents

• Equilateral Triangle Definition
• Properties of an Equilateral Triangle
• Area of Equilateral Triangle
• Area of Equilateral Triangle Formula
• Perimeter of Equilateral Triangle
• Height of Equilateral Triangle
• Applications of Equilateral Triangle in Real Life
• Solved Examples
• Practice Questions
• Conclusion
• FAQs On Equilateral Triangle

 

Equilateral Triangle Definition

The equilateral triangle definition in easy words is:

 A triangle with three equal sides and three equal interior angles that are 60 degrees.

Since all angles and all sides of an equilateral triangle are equal, it has numerous special features. The characteristics of an equilateral triangle distinguish it from any other triangle.

 

Properties of an Equilateral Triangle

The characteristics of an equilateral triangle are as follows:

• All sides are equal in measure

• All interior angles measure 60°

• It possesses three lines of symmetry

• The centroid, circumcenter, incenter, and orthocenter coincide at the same point

• Each median is also an altitude and a bisector of an angle

• The triangle is always convex

These characteristics of an equilateral triangle make solving different geometry problems easy and accurate.

 

Area of Equilateral Triangle

Equilateral triangle area is the total space within its three sides that are equal. The formula for the area of equilateral triangle is established with the help of basic trigonometry and the Pythagorean theorem.

The formula for the area of equilateral triangle is:

 Area = (√3 / 4) × a²

Where ‘a’ is the length of a side.

This formula is very important for calculating the area of an equilateral triangle in mathematical problems and real-life applications.

 

Area of Equilateral Triangle Formula

 Let’s say the side of an equilateral triangle is 6 cm.

 Using the area of equilateral triangle formula:

 Area = (√3 / 4) × 6² = (√3 / 4) × 36 = 9√3 ≈ 15.59 cm²

Therefore, the region of equilateral triangle is about 15.59 square cm.

 

Height of Equilateral Triangle

Height of an equilateral triangle is the perpendicular line from a vertex to the base. It may be calculated by the formula:

 Height = (√3 / 2) × a

 where a is side of an equilateral triangle.

This equilateral triangle height is important to learn about triangle symmetry, area calculation, and in all design uses.

 

Example:

If all sides of the equilateral triangle are 10 cm, then

Height = (√3 / 2) × 10 = 5√3 ≈ 8.66 cm

 

Perimeter of Equilateral Triangle

The perimeter of an equilateral triangle is the total distance around the triangle. As all three sides are equal, the perimeter of equilateral triangle is expressed by:

Perimeter = 3 × a

Where a is the side length.

 

Example:

If each side of the equilateral triangle is 7 cm:

Perimeter = 3 × 7 = 21 cm

The perimeter of equilateral triangle is 21 cm.

 

Applications of Equilateral Triangle in Real Life

• Employed in engineering constructions for strength and stability

• Frequent in design and architecture for visual symmetry

• Present in nature, e.g., honeycomb structures and crystals

• Utilized in logo types and artwork for its balanced appearance

 

Solved Examples:

Example 1:

Find the area of an equilateral triangle whose side is 12 cm.

Solution:

Given: Side (a) = 12 cm

We use the area of an equilateral triangle formula:

 Area = (√3 / 4) × a²

 = (√3 / 4) × 12²

 = (√3 / 4) × 144

 = 36√3

 ≈ 36 × 1.732

 ≈ 62.35 cm²

 

Example 2:

 Find the perimeter of an equilateral triangle with each side measuring 9cm.

 Solution:

 Perimeter = 3 × side

 = 3 × 9 = 27 cm

Thus, the perimeterofequilateral triangle is 27 cm.

 

Example 3:

Find the height of an equilateral triangle with side 10 cm.

Solution:

Height = (√3 / 2) × a

= (√3 / 2) × 10

= 5√3 ≈ 8.66 cm

Example 4:

If the area of an equilateral triangle is 100√3 cm², find the length of the side.

Solution:

Use the area of equilateral triangle formula:

Area = (√3 / 4) × a²

100√3 = (√3 / 4) × a²

Multiply both sides by 4:

400√3 = √3 × a²

Divide both sides by √3:

a² = 400

a = √400 = 20 cm

 

Practice Questions

1. Find the area of an equilateral triangle with side length 14 cm.

2. If the height of an equilateral triangle is 7.79 cm, determine its side length.

3. An equilateral triangle has a perimeter of 45 cm. How long is each side?

4. Use the area of equilateral triangle formula to calculate the area when side = 5 cm.

5. Calculate the height of equilateral triangle when its side is 16 cm.

6. A triangle has an area of 25√3 cm². Use the formula for the area of equilateral triangle to determine its side length.

 

Conclusion

The equilateral triangle is a basic shape in geometry with special properties. Knowing the definition of an equilateral triangle, studying the equilateral triangle properties, and using the formula for the area of an equilateral triangle are crucial to understanding geometric principles. From determining the height of equilateral triangle to determining the perimeter of equilateral triangle, this shape has numerous real-life uses as well as academic applications. Mastering the equilateral triangle serves as a foundation towards having a great grasp of geometry.

 

Related Links

Triangles - Learn about different types of triangles, their properties, angles, and classifications, with clear diagrams and practical examples.

Similar Triangles - Understand the concept of similar triangles, their properties, and the criteria for similarity with easy-to-follow examples and visual explanations.

Frequently Asked Questions on Equilateral Triangle

1. Are equilateral triangles always 60°?

Ans: Yes, in an equilateral triangle, all three interior angles are always 60 degrees.
         This is because the total angle sum of any triangle is 180°, and in an equilateral triangle, the angles are equal: 180° ÷ 3 = 60°.

 

2. What is isosceles and equilateral?

Ans:
Isosceles Triangle: A triangle with two equal sides and two equal angles.
Equilateral Triangle: A triangle with three equal sides and three equal angles (each 60°).
So, every equilateral triangle is also an isosceles triangle, but not every isosceles triangle is equilateral.

 

3. What are the 7 types of triangles?

Ans: The 7 types of triangles (based on sides and angles) are:

1. Equilateral Triangle

2. Isosceles Triangle

3. Scalene Triangle

4. Acute Triangle

5. Right Triangle

6. Obtuse Triangle

7. Isosceles Right Triangle

These types are classified by their sides and internal angles.

 

4. What are the 12 types of triangles with examples?

Ans: These are combinations of side-based and angle-based classifications:

• Equilateral Triangle - All sides 6 cm, all angles 60°

• Isosceles Triangle - Two sides 5 cm, base 3 cm

• Scalene Triangle - Sides 3 cm, 4 cm, 5 cm

• Acute Equilateral Triangle - All angles 60°

• Acute Isosceles Triangle - Two 70°, one 40°

• Acute Scalene Triangle - Angles 50°, 60°, 70°

• Right Isosceles Triangle - Two sides 7 cm, one 90° angle

• Right Scalene Triangle - Angles 90°, 60°, 30°

• Obtuse Isosceles Triangle - Angles 100°, 40°, 40°

• Obtuse Scalene Triangle - Angles 110°, 40°, 30°

• Equiangular Triangle - All 60° angles

• Triangle with Mixed Type - For example, obtuse isosceles

 

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