Equilateral Triangle

An equilateral triangle is a type of triangle in which all three sides are equal in length and all three interior angles are equal. Since the sum of the interior angles of a triangle is 180°, each angle in an equilateral triangle measures 60°.

Perimeter

The perimeter of a triangle is equal to the sum of its three side lengths.

In the figure given above, ∆ABC is an equilateral triangle with equal sides that measure "a" unit. 

So, the perimeter of an equilateral triangle (P) = (AB + BC + AC) units

(P) = a + a + a = 3a units

Area

The total region bounded by the three sides of a triangle in a two-dimensional plane is known as the area of a triangle.

Area of an Equilateral Triangle = √3/4 a²

where "a" is side length of the triangle.

Using Heron's Formula

We know that the area of a triangle can be calculated using Heron's formula if all three of its side lengths are given. In the figure given above, ∆ABC is an equilateral triangle with equal sides that measure "a" unit. 

So, AB = BC = CA = a

We know that,

Area of Triangle = \sqrt{s(s - a)(s - b)(s - c)}

• s is the semi-perimeter.
• a, b, and c are Side Lengths of a triangle.

Here, a = b = c = a

So, s = (a + a + a)/2 = 3a/2

Now, substitute the values in the formula.

A = √{3a/2(3a/2-a)(3a/2-a)(3a/2-a)}

A = √{(3a⁴)/(4)²}

A = (√3/4) a²

Hence, Area of Equilateral Triangle = √3/4 a²

Centroid

The centroid of the triangle, also called the center of the triangle, is a point that is at the center of the triangle. This point is equidistant from all three vertices of the triangle. For an equilateral triangle, as all the sides are equal in length, it is easy to find the centroid for it.

If we draw perpendiculars from all the vertices of the equilateral triangle to their opposite sides, the point where they all meet is the centroid of the equilateral triangle.

We know that the meeting point of all three perpendiculars of the triangle is called the orthocenter of the triangle. Thus, for an equilateral triangle, the centroid and orthocenter are the same points.

For any equilateral triangle ABC, its centroid is denoted using point A in the image added below.

In equilateral triangle with length “a” the distance from the centroid to the vertex is equal to √(3a/3)

Circumcenter of Equilateral Triangle

The center of the circle passing through all three vertices of the triangle is called the circumcenter of the triangle. It is calculated by taking the intersection of any two perpendicular bisectors of the triangle.

If the length of the side of the equilateral triangle is aaa, then the circumcenter is at a distance of:

Circumradius (R)=\text{Circumradius (R)} = \frac{a}{\sqrt{3}}

Where:

• a = length of the side of the equilateral triangle.
• R = circumradius, the radius of the circumcircle.

Note: In an equilateral triangle, the incenter, orthocenter, and centroid all coincide with the circumcenter of the equilateral triangle.

Properties

Some important characteristics of an equilateral triangle are,

• All three side lengths of an equilateral triangle always measure the same.
• The three interior angles of an equilateral triangle are congruent and equal to 60°.
• According to the angle sum property, the sum of the interior angles of an equilateral triangle is always equal to 180°.
• Equilateral triangles are considered regular polygons since their three side lengths are equal.
• The perpendicular drawn from any vertex of an equilateral triangle bisects the opposite side into two halves. The perpendicular also bisects the angle at the vertex from which it is drawn into 30° each.
• In an equilateral triangle, the orthocenter and centroid are at the same point.
• The median, angle bisector, and altitude for all sides of an equilateral triangle are the same.
• The area of an equilateral triangle is √3/4 a², where "a" is the side length of the triangle.
• The perimeter of an equilateral triangle is 3a, where "a" is the side length of the triangle.

Formulas

Equilateral Triangle Theorem

The equilateral triangle theorem states that,

For any equilateral triangle ABC, if P is any point on the arc BC of the circumcircle of the triangle ABC, then PA = PB + PC

Proof: 

In cyclic quadrilateral ABPC, we have,
PA⋅BC = PB⋅AC + PC⋅AB

As ABC is an equilateral triangle,
AB = BC = AC

Thus,
PA.AB = PB.AB + PC.AB

Simplifying,
PA.AB = AB(PB + PC)
PA = PB + PC

Hence, proved.

Scalene vs Isosceles vs Equilateral Triangles

Major differences between scalene triangles, isosceles triangles, and equilateral triangles are added in the table below.

Scalene Triangle	Isosceles Triangle	Equilateral Triangle
All three side lengths of a scalene triangle are always unequal.	There will be at least two equal side lengths in an isosceles triangle.	All three side lengths of an equilateral triangle always measure the same.
All three interior angles of a scalene triangle are always unequal.	The interior angles opposite the equal sides of an isosceles triangle are equal.	The three interior angles of an equilateral triangle are congruent and equal to 60°.

Solved Examples

Example 1: Determine the area of an equilateral triangle whose side length is 10 units.
Solution:

Given,

• Side length (a) = 10 units

We know that,

Area of Equilateral Triangle = √3/4 a²
A = √3/4 × (10)²

⇒ A = √3/4 × 100 
⇒ A = 25√3 square units ≈ 43.301 square units

Hence, the area of the given equilateral triangle is approximately equal to 43.301 square units.

Example 2: Determine the height of an equilateral triangle whose side length is 8 cm.
Solution:

Given,

• Side length (a) = 8 cm

We know that,

Height of Equilateral Triangle = √3a/2

⇒ H = √3/2 × 8
⇒ H = 4√3 cm
⇒ H ≈ 6.928 cm

Hence, the height of given equilateral triangle is approximately equal to 6.928 cm.

Example 3: Determine the perimeter of an equilateral triangle whose side length is 13 cm.
Solution:

Given,

• Side length (a) = 13 cm

We know that,

Perimeter of Equilateral Triangle (P) = 3a units
⇒ P = 3 × 13 = 39 cm.

Hence, the perimeter of the given equilateral triangle is 39 cm.

Example 4: What is the area of an equilateral triangle if its perimeter is 36 cm?
Solution:

Given,

Perimeter of Equilateral Triangle (P) = 36 cm

We know that,

Perimeter of Equilateral triangle (P) = 3a units

⇒ 3a = 36
⇒ a = 36/3 = 12 cm

We know that,

Area of Equilateral Triangle = √3/4 a²

⇒ A = √3/4 × (12)²
⇒ A = √3/4 × 144
⇒ A = 36√3 sq. cm

Hence, Area of the given equilateral triangle is 36√3 sq. cm.