Equilateral Triangle

An equilateral triangle is one of the most fundamental shapes in eogmetry. It is a triangle in which all three sides are equal and all three angles measure 60 degrees. Due to its perfect balance and symmetry, the equilateral triangle is one of the most studied figures in mathematics. It is widely utilized in real-life applications, such as architecture, engineering, and design.

In this guide, we will learn the definition of an equilateral triangle, its properties, formulas for area, perimeter, and height, and solved examples. We will also explore real-life applications, practice questions, and make the concept clear and easy to understand.

 

Table of Contents

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

 

Equilateral Triangle Definition

An equilateral triangle is a special type of triangle in which all three sides are equal in length and all three angles are equal in measure, each being 60 degrees. Because of this equality, it is one of the most symmetrical triangles in geometry.

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.

 

Equilateral Triangle Formulas

The formulas of an equilateral triangle help us to calculate its area, perimeter, and height quickly. Based on these properties, the formulas for an equilateral triangle are defined.

The most common formulas are:

Area of an Equilateral Triangle

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

The formula for the area of an equilateral triangle is:

Area=34×a2

Where ‘a is the length of a side.

Let us derive the formula step by step:

The area of any triangle is given by:

Area=12×base×height

Here, the base = a and the height = h.

Area=12×a×h …… (1)

Now, in an equilateral triangle, the altitude h bisects the base into two halves a/2 and a/2. It also forms two congruent right-angled triangles.

Using the Pythagoras theorem for one of these right triangles:

a2=h2+(a2)2

h2=a2−(a2)2

h2=a2−a24=3a24

h=32a

Substitute this value of h in equation (1):

Area=12×a×32a

Area=34a2

Hence, the area of an equilateral triangle is:

Area Of Equilateral Triangle =34a2

 

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

 

Height of an Equilateral Triangle

Formula for Height:

h=32a

Where a = length of a side of the equilateral triangle.

Derivation of the Formula

In an equilateral triangle, when we draw the height, it splits the triangle into two right-angled triangles.

The hypotenuse is the side of the equilateral triangle (a), the base is half the side (a/2), and the perpendicular is the height (h).

Using the Pythagoras theorem:

a2=h2+(a2)2

h2=a2−(a2)2

h2=a2−a24=3a24

h=32a

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

 

Perimeter of an Equilateral Triangle

The perimeter of a triangle is the total distance around it. Since all three sides of an equilateral triangle are equal, its perimeter is very easy to calculate.

Formula for Perimeter:

Perimeter=3×a

Where a = length of one side of the equilateral triangle.

Explanation

• A triangle has three sides.
• In an equilateral triangle, all three sides are equal in length.
• So, perimeter = side + side + side = 3 × a.

 

Applications of the 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=(34)×a2

$Area = \left( \frac{\sqrt{3}}{4} \right)\times 12^{2}$

 Area=(34)×144

Area=363

$ ≈ 36 × 1.732$

≈62.35cm2

 

Example 2:Find the perimeter of an equilateral triangle with each side measuring 9cm.

Solution:

Perimeter = 3 × side

= 3 × 9 = 27 cm

Thus, the perimeter of an equilateral triangle is 27 cm.

 

Example 3: Find the height of an equilateral triangle with side of 10 cm.

Solution:

Height=(34)×a

Height=(34)×10

Height=53≈8.66cm

Example 4: If the area of an equilateral triangle is1003cm2, find the length of the side.

Solution:

Use the area of an equilateral triangle formula:

Area=(34)×a2

1003=(34)×a2

Multiply both sides by 4:

4003=3×a2

Divide both sides by √3:

a2=400

a=400=20cm

 

Practice Questions

1. Find the area of an equilateral triangle with a side length of 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 the side = 5 cm.

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

6. A triangle has an area of 253cm2. 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. 

 

Frequently Asked Questions on Equilateral Triangle

1. Are equilateral triangles always 60°?

Answer: 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 an equilateral?

Answer: 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?

Answer: 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?

Answer: 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

 

Understand core math concepts like the equilateral triangle with clear explanations and solved problems at Orchids The International School.