In this mini-lesson, we will explore the world of equilateral by understanding the meaning of equilateral, exploring some of the equilateral shapes, how to construct equilateral shapes, and equilateral properties, and how to apply them while solving problems. We will also discover interesting facts around them.

We encounter so many different shapes when we deal with figures in geometry.

Some of these shapes have equal side lengths while some of the shapes have equal angles.

Some of the shapes are only made up of straight lines while some of the shapes only have curves.

In this session, we are going to explore more about the geometrical shapes or polygons that are made up of equal side lengths.

**Lesson Plan**

**What Is Meant by Equilateral?**

The word equilateral is made up of two words, "equi" and "lateral."

The equilateral definition talks of polygons with equal side lengths.

**Which Geometric Shapes Are Equilateral?**

We know that polygons are shapes made up of straight-line segments, and a triangle is a polygon with the minimum number of sides.

Now, what if the side lengths of the polygons are equal? We call these equilateral shapes.

Let's explore these equilateral shapes which follow equilateral properties in more detail.

**Equilateral Triangle**

An equilateral triangle is a regular polygon with 3 equal sides.

Triangle \(\text{PQR}\) is an equilateral triangle.

\[\text{PQ} = \text{QR} = \text{RP}\]

\[{\angle}\text{PQR} = {\angle}\text{QRP} = {\angle}\text{RPQ} = 60^{\circ}\]

**Equilateral Quadrilateral**

An equilateral quadrilateral is a polygon with 4 equal sides.

An equilateral quadrilateral is called a rhombus.

The rhombus \(\text{PQRS}\) is an equilateral quadrilateral.

We cannot say that all the angles of an equilateral quadrilateral are equal.

A square is a special case of equilateral quadrilateral in which all the four angles are equal.

Square \(\text{ABCD}\) is an equilateral quadrilateral with all the four angles equal to \(90^{\circ}\).

**Equilateral Pentagon**

An equilateral pentagon is a polygon with 5 equal sides.

Pentagon \(\text{ABCDE}\) is an equilateral pentagon.

The interior angles of an equilateral pentagon can be equal or not.

The special case of an equilateral pentagon in which all the five angles are equal is called a regular pentagon.

Let's see a table of these equilateral polygons and the names of special equilateral polygons.

Number of Sides |
Name of Equilateral Shape |
Special Case |

3 | Equilateral triangle | Equilateral triangle |

4 | Rhombus | Square |

5 | Equilateral pentagon | Regular pentagon |

6 | Equilateral hexagon | Regular hexagon |

7 | Equilateral heptagon | Regular heptagon |

- What if an equilateral pentagon has all its angles equal? What is the value of each angle?

What is the value for each interior angle in the case of an equilateral and equiangular hexagon? Think about it. - If all the angles in a polygon are equal, is it always equilateral? Think about it.

**How to Construct Equilateral Shapes?**

Let's explore the simulation below to understand the construction of an equilateral polygon.

Enter the number of sides of a polygon. Make sure the number of sides for any polygon does not go beyond 6.

- All the angles of an equilateral triangle are equal to \(60^{\circ}\).
- Equiangular shapes are shapes in which all the interior angles are equal.
- A square is a special case of an equilateral quadrilateral which is also an equiangular shape.

**Solved Examples on Equilateral**

Example 1 |

Tim draws an equilateral triangle and measures one of the sides of the triangle to be 10 units. If the other two sides are of lengths \(2x - 6\) and \(3y - 2\), find the values of \(x\) and \(y\).

**Solution**

\({\triangle}\text{ABC}\) is an equilateral triangle, hence

\[\text{AB} = \text{BC} = \text{CA} \\[0.2cm]

2x - 6 = 10 = 3y - 2\]

Let's solve each of these equations separately

\[\begin{align} 2x - 6 &= 10 \\[0.2cm]

2x &= 10 + 6 \\[0.2cm]

2x &= 16 \\[0.2cm]

x &= 8 \end{align}\]

\[\begin{align} 3y - 2 &= 10 \\[0.2cm]

3y &= 10 + 2 \\[0.2cm]

3y &= 12 \\[0.2cm]

y &= 4 \end{align}\]

\(\therefore\) \(x = 8 \text{ and } y = 4\) |

Example 2 |

Paul is fencing his backyard with an iron fence. The total length of the iron fence required is 100 yards. If the backyard is in the shape of an equilateral quadrilateral, find the length of one side of the backyard.

**Solution**

Since the backyard is in the shape of an equilateral quadrilateral, all sides of the backyard are equal.

Take one of the sides of the backyard to be \(x\).

The perimeter of the backyard = \(4 \times x = 4x\)

It is given that the total length of the iron fence is 100 yards.

Hence, \(4x = 100\)

\(x = 25\)

\(\therefore\) The length of one side of the backyard is 25 yards. |

Example 3 |

Find the area of an equilateral triangle with one side equal to 4 units.

**Solution**

It is given that the side of the equilateral triangle is 4 units.

Area of an equilateral triangle with side \(a\) is \({\dfrac{\sqrt{3}}{4}}{a^2}\)

Putting the value of \(a = 4\) in the equation of the area of an equilateral triangle,

\[\begin{align} \text{Area} &= \dfrac{\sqrt{3}}{4} {a^2} \\[0.2cm]

\text{Area} &= \dfrac{\sqrt{3}}{4} {4^2} \\[0.2cm]

\text{Area} &= \sqrt{3} \times {4} \\[0.2cm]

\text{Area} &= 4\sqrt{3} {\text{ unit}}^2 \end{align}\]

\(\therefore\) \(\text{Area of the triangle is } {4}{\sqrt{3}} {\text{ unit}}^2\) |

**Interactive Questions on Equilateral **

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

This mini-lesson targeted the fascinating concept of the equilateral. The math journey around equilateral starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp but will also stay with them forever. Here lies the magic of Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

### 1. What is an equilateral parallelogram called?

An equilateral parallelogram is known as a rhombus.

A square is a special case of equilateral rhombus in which all the angles are equal to \(90^{\circ}\).

### 2. Is an equilateral triangle also a rhombus?

No, an equilateral triangle is not a rhombus. An equilateral triangle is a 3-sided polygon while a rhombus is a 4-sided polygon.

### 3.** **Are all regular polygons equilateral?

Yes, all regular polygons are equilateral.

A regular triangle is called an equilateral triangle, and a regular quadrilateral is called a square.

### 4. Is a square an equilateral?

Yes, a square is an equilateral quadrilateral with all four sides equal and all four angles equal.

### 5. What's the difference between equilateral and equiangular?

Equilateral figures are the figures with equal side lengths.

Equiangular figures are the figures with equal angles.

### 6. Do equilateral triangles have equal angles?

Yes, an equilateral triangle has equal angles. Each angle of an equilateral triangle is \(60^{\circ}\).

### 7. Are right triangles equilateral?

No, right triangles are not equilateral. In a right-angled triangle, the length of the hypotenuse is always greater than the other two sides whereas in an equilateral triangle all the sides are equal. So, a right-angled triangle does not follow the rules for an equilateral triangle.