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# Equilateral|Definition & Meaning

## Definition

An **equilateral triangle** has sides of the **same length**, making it a special kind of triangle in geometry. **Three right angles** are also equal in size since their opposite sides are also equal in length. Therefore, it is also termed an equiangular triangle, in which each angle measures **60 degrees**.

It is possible to calculate the **area**, **perimeter**, and **height** of an equilateral triangle in the same way that these measurements can be calculated for other triangles.

**What Is a Triangle With Equilateral Sides?**

Additionally, all three of the **triangle’s angles** add up to a total of **sixty degrees** and are **congruent**. The total angle equals 180 degrees when all **three angles** are added together in a triangle with equilateral sides. **60° + 60° + 60° = 180°**. As a result, it satisfies the requirements of the angle sum attribute of triangles.

**The Shape of an Equilateral Triangle**

An equilateral triangle displays the outline of a **regular shape**. It is sometimes termed a **regular polygon** or regular triangle since all its sides are equal.

Let us suppose that the **triangle ABC** is equilateral by definition;

The formula for the equilateral triangle is as follows: **AB = BC = AC**, where AB, BC, and AC are the sides of the triangle.

And

**∠A = ∠B = ∠C = 60°**

There are also **two** other kinds of triangles, distinguished by the sides they have:

**Scalene Triangle****Isosceles Triangle**

**Characteristics of Equilateral Triangles**

**Equilateral Triangles** Have the Following Characteristics:

- Every one of the
**three sides**is equivalent. - The
**sum**of the three angles, which add up to**sixty degrees**, makes them**congruent**. - The
**polygon**has three straight sides, making it a**regular shape**. - The equilateral triangle is divided into two halves by the
**perpendicular**. In addition, the angle of the**vertex**, measured from the point where the perpendicular is drawn, is cut in half to create two equal angles, each of which is equal to**30 degrees**. - The same point serves as both the
**orthocenter**and the**centroid**of the figure. - When describing an equilateral triangle, it is essential to note that each side’s median,
**angle bisector**, and**altitude values**are identical. - The area of an equilateral triangle can be computed with the formula:
**[($\sqrt{3}$/4).b²]**. - A triangle with equilateral sides has a perimeter equal to
**3b**.

**Equilateral Triangle Theorem **

If triangle **XYZ** is an equilateral triangle, and point P is located on the circumcircle of triangle XYZ, then the following statements are **factual**:

**PX = PY + PZ**

As evidence, consider the following for a circular **quadrilateral XYPZ**:

**PX.YZ = (PY.XZ) + (PZ.XY)**

Given that we are aware, in the case of an **equilateral triangle XYZ:**

**XY = YZ = ZX**

Therefore:

**PX.AY = (PY.XY) + (PZ.XY)**

Using **XY** as a common denominator:

**PX.AY = XY(PY+PZ)**

**PX = PY + PZ**

**Therefore, it is proven.**

**Area of a Triangle With Equilateral Sides**

The region of a **two-dimensional plane** occupied by an equilateral triangle is referred to as the **triangle’s area**. The following equation is used to calculate the area of an **equilateral triangle**:

**A = [($\sqrt{3}$/4).a²]**

**The Perimeter of a Triangle With Equal Sides**

In **geometry**, the length of the sides of a polygon is used to calculate its** perimeter**. The perimeter of an equilateral triangle is equal to the total of its **three sides**. Hence this value is used to describe the shape.

Let’s say that **XYZ** is an equilateral triangle. In that case, the **perimeter** of **ABC** is as follows:

**Perimeter **= XY + YZ + ZX

**P **= b + b + b

**P = 3b**

In this equation, **b** represents the length of one of the **triangle’s sides**.

**Measurement of the Axis of an Equilateral Triangle**

The **height** of an equilateral triangle, having similar sides, can be computed using the **Pythagorean theorem**, given its **base** and **height**. In some contexts, it may also be referred to as the altitude of an **equilateral triangle**. Now, if we take an **elevation** from the **triangle’s apex** and drop it down to the base, we may divide it into **two right triangles** that are **equal in size**.

As a result, we can get the **height (h)** of the equilateral triangle using the diagram that was just presented by using the formula:

**h = $\sqrt{3}b/2$**

Where b represents one of the **triangle’s sides**.

**The Apex of a Right Triangle**

At its **geometric center**, an equilateral triangle’s **centroid** can be found. Since all sides are the same length, determining its **centroid** is a breeze.

The **centroid** can be located by connecting the opposite sides of the triangle with **perpendiculars** drawn from the** three vertices**. These perpendiculars all meet at the **same point**, the centroid, and are the same length.

**Circumcenter**

The **circumcenter** of an equilateral triangle is the position of **union perpendicular bisectors** of the sides. This triangle has a circumcircle that touches all three of its corners.

In geometry, a triangle is equilateral if its in-center, **orthocenter**, or centroid lies on the same line as its **circumcenter.**

### Example 1

Find the base of the triangle, having two similar sides. Given that an **equilateral triangle** has a side of **16 cm**, which has a perimeter similar to an **isosceles triangle**, having a side of **23 cm**.

### Solution

Since the perimeter is the same for both triangles, it can be written as follows:

Perimeter of Equilateral = Perimeter of Isosceles Triangle

3 x 16 = (2 x 23) + Z

48 = 46 + Z

**Z = 2 cm**

Here, the side of the equilateral triangle is multiplied by three since all sides are equal, whereas isosceles has two sides equal; therefore, multiplied by two. And the variable Z is left as the base.

### Example 2

An **equilateral triangle** has a perimeter of **8 cm**; find the height of the triangle.

### Solution

a = (8 / 3) = 2.67

h² = [a² – (a² / 4)]

h = $(\sqrt{3} / 6)$ x 8

**h = 2.3 cm**

The height of the triangle is 2.3 cm.

*All mathematical drawings and images were created with GeoGebra.*