Angles Of An Equilateral Triangle

In geometry, an equilateral triangle is a triangle that has all its sides equal in length. Since the three sides are equal therefore the 3 angles, contrary to the equal sides, are equal in measure. Therefore, it is likewise called an equiangular triangle, where each angle measure out 60 degrees. Just like other types of triangles, an equilateral triangle likewise has its expanse, perimeter and summit formula. Let us learn more in this article.

Tabular array of contents:

Definition
Shape
Properties
Comparison
Theorem
Formula
- Surface area
- Perimeter
- Top
Centroid
Circumcenter
Examples
FAQs

What is an Equilateral Triangle?

As we have already discussed in the introduction, an equilateral triangle is a triangle that has all its sides equal in length. Too, the 3 angles of the equilateral triangle are congruent and equal to threescore degrees.The sum of all three angles of an equilateral triangle is equal to 180 degrees. 60 ° + threescore ° + lx ° = 180 °. Thus, it obeys the bending sum property of triangle .
Equilateral Triangle

Shape of Equilateral Triangle

The shape of an equilateral triangle is regular. The word 'Equilateral' is formed by the combination of two words, i.due east., "Equi" pregnant equal and "Lateral" meaning sides. An equilateral triangle is besides chosen a regular polygon or regular triangle since all its sides are equal.

Suppose, ABC is an equilateral triangle, then, as per the definition;

AB = BC = AC, where AB, BC and Air-conditioning are the sides of the equilateral triangle.

And

∠A = ∠B = ∠C = sixty°

Based on sides at that place are other two types of triangles:

Scalene Triangle
Isosceles Triangle

Properties of Equilateral Triangle

All three sides are equal.
All three angles are congruent and are equal to 60 degrees.
It is a regular polygon with 3 sides.
The perpendicular drawn from vertex of the equilateral triangle to the opposite side bisects information technology into equal halves. Also, the angle of the vertex from where the perpendicular is drawn is divided into two equal angles, i.e. thirty degrees each.
The ortho-eye and centroid are at the same point.
In an equilateral triangle, median, bending bisector, and altitude for all sides are all the same.
The area of an equilateral triangle is √3a²/ iv
The perimeter of an equilateral triangle is 3a.

Comparison: Scalene, Isosceles and Equilateral Triangles

Equilateral triangle	Isosceles triangle	Scalene triangle
All three sides are equal	Any ii sides are equal	All three sides are unequal
All the three interior angles are equal to 60 degrees	Angles opposite to equal sides are equal	All the angles are unequal in mensurate

Equilateral Triangle Theorem

If ABC is an equilateral triangle and P is a signal on the arc BC of the circumcircle of the triangle ABC, so;

PA = Lead + PC

Proof: For a circadian quadrilateral ABPC, we have;

PA⋅BC=Lead⋅AC+PC⋅AB

Since we know, for an equilateral triangle ABC,

AB = BC = AC

Therefore,

PA.AB = Atomic number 82.AB+PC.AB

Taking AB as a mutual;

PA.AB=AB(PB+PC)

PA = Pb + PC

Hence, proved.

Equilateral Triangle Formulas

We have already understood an equilateral triangle has all 3 sides equal in length and all three angles equal in measure. At present based on these properties the formulas for equilateral triangles are divers. The most common formulas that nosotros consider for a triangle are:

Surface area of equilateral triangle
Perimeter of equilateral triangle
Height of equilateral triangle

In the adjacent section, we volition exist discussing all these formulas.

Surface area of Equilateral Triangle

The surface area of an equilateral triangle is the region occupied by information technology in a ii-dimensional plane. The formula for the expanse of an equiangular triangle is given by:

A = √3aⁱⁱ/iv

Let us derive the formula here:

Equilateral triangle area

If we see the to a higher place figure, the area of a triangle is given by;

Area = ½ x base of operations x pinnacle

Hither Base = a and height = h

Therefore,

Surface area = ½ x a ten h ………(1)

Now, from the above figure, the altitude h bisects the base of operations into equal halves, such as a/2 and a/2. Information technology too forms 2 equivalent right-angled triangles.

So, for a right triangle, using Pythagoras theorem, we can write:

a^two = h² + (a/2)^two

h² = (a)² – (a/two)²

= 3a²/4

h = √3a/2

By putting this value in equation 1, we become;

Area = ½ 10 a 10 √3a/two

A = √3a²/4

Hence, the area of the equilateral triangle equals to √3a²/4.

Perimeter of Equilateral Triangle

In geometry, the perimeter of any polygon is equal to the length of its sides. In the case of the equilateral triangle, the perimeter will be the sum of all iii sides.

Suppose, ABC is an equilateral triangle, then the perimeter of ∆ABC is;

Perimeter = AB + BC + AC

P = a + a + a

P = 3a

Where a is the length of sides of the triangle.

Height of Equilateral Triangle

The height of an equilateral triangle can be determined using the Pythagoras theorem. It is also called altitude of an equilateral triangle. Every bit we know, an equilateral triangle has all equal sides. At present, if we drop an altitude from the apex of the triangle to the base, it divides the triangle into ii equal right triangles.

Thus, from the above figure, we tin find the acme (h) of the equilateral triangle, as:

h = √3a/two

Where a is the side of the triangle.

Thus, to summarise the formulas related to equilateral triangle are:

Area	√3a^two/4
Perimeter	3a
Elevation	√3a/2

Centroid of Equilateral Triangle

The centroid of the equilateral triangle lies at the middle of the triangle. Since all its sides are equal in length, hence it is easy to discover the centroid for it.

To find the centroid, we demand to draw perpendiculars from each vertex of the triangle to the reverse sides. These perpendiculars are all equal in length and intersect each other at a unmarried point, which is known as centroid. See the effigy below:

Centroid of equilateral triangle

Note: The centroid of a regular triangle is at equidistant from all the sides and vertices.

Circumcenter

The circumcenter of equilateral triangle is the point of intersection perpendicular bisectors of the sides. Here, the circumcircle passes through all the 3 vertices of the triangle.

If whatsoever of the incenter, orthocenter or centroid coincide with the circumcenter of a triangle, then it is called an equilateral triangle.

Facts of Equilateral Triangle:

Number of Sides = 3
Number of angles = 3
Each interior angle = 60
Each outside bending = 120
Perimeter = three times of side-length
Area = √3/ 4 x (side) ⁱⁱ
Tiptop = √iii (side)/two

Solved Examples on Equilateral Triangle

Q.1: Notice the area of the equilateral triangle ABC, where AB=Air conditioning=BC = 4cm.

Solution:

By the formula, we know;

Expanse = √3a²/iv

Given a = 4cm

Hence, by putting the value we go;

Area = √3(iv)²/4

A = 4√3

Q.2: Discover the altitude of an equilateral triangle whose sides are equal to 10cm.

Solution:

Past the formula, we know,

Tiptop of an equilateral triangle = √3a/2

Since, a = 10cm

Hence,

h = √3 x (10/2)

h = 5√3

Video Lesson on Types of Triangles

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Frequently Asked Questions on Equilateral Triangle

What is an equilateral triangle?

A triangle that has all its sides equal in dimension and each bending measure out up to lx degrees, is called an equilateral triangle.

What is the surface area of equilateral triangle formula?

The formula to find expanse of equilateral triangle is given by:
A = (√three/4)a2, where a is the length of side of equilateral triangle.

What is the perimeter of an equilateral triangle?

The perimeter of an equilateral triangle is the sum of all its iii equal sides.
Perimeter = 3a, where a is the length of its sides.

What is the superlative of equilateral triangle?

The formula to find the height of equilateral triangle is:
Superlative, h = (√three/2)a, where a is the side of the equilateral triangle.

What type of polygon is an equilateral triangle?

An equilateral triangle is a regular polygon or a regular triangle.

What is the dissimilar name of triangles based on sides?

Based on sides, at that place are three different kinds of triangles. Their names are:
Scalene triangle
Isosceles triangle
Equilateral triangle

How many sides does an equilateral triangle has?

All triangles accept iii sides only.

What is the perimeter of equilateral triangle with side equal to 10cm?

Perimeter = 3 ten sides of equilateral triangle
Perimeter = 3 ten ten = 30 cms