An equilateral triangle, also known as a triangle with equal sides, is a fundamental shape in geometry. Each of its sides is of equal length, and its interior angles are all 60 degrees, making it a 60-degree triangle. It is also a perfectly symmetrical shape.
In the figure given below, ∆ABC is an equilateral triangle with equal sides that measures "a" unit, i.e., AB = BC = AC = a.

Height of Equilateral Triangle
The height or altitude of an equilateral triangle is equal to √3a/2, where "a" is the side length of the triangle. It can be determined by using the Pythagorean formula. In the figure given below, ∆ABC is an equilateral triangle with equal sides that measures "a" unit. We can observe that a perpendicular is drawn from the vertex A to the opposite side BC, bisecting it into two halves at point D. ABD and ADC are two equal right angles. The length of AD is the height of the given triangle ∆ABC.
Height of an Equilateral Triangle = √3a/2
where "a" is the side length of the triangle.
Perimeter of Equilateral Triangle
Perimeter of a triangle is equal to the sum of its three side lengths. We know that all three sides of an equilateral triangle are equal. So, the perimeter of an equilateral triangle is 3a, where "a" is the side length of the triangle.
In the figure given above, ∆ABC is an equilateral triangle with equal sides that measures "a" unit.
So, the perimeter of an equilateral triangle (P) = (AB + BC + AC) units
(P) = a + a + a = 3a units
Perimeter of an equilateral triangle (P) = 3a units
where "a" is the side length of the triangle.
Area of Equilateral Triangle
Total region bounded by the three sides of a triangle in a two-dimensional plane is known as the area of a triangle. The area of an equilateral triangle is √3/4 a2, where "a" is the side length of the triangle.
Area of an Equilateral Triangle = √3/4 a2
where "a" is side length of the triangle.
We know that the area of a triangle can be calculated using Heron's Formula if all its three side lengths are given. In the figure given above, ∆ABC is an equilateral triangle with equal sides that measures "a" unit.
So, AB = BC = CA = a
We know that,
Area of Triangle = √{s(s-a)(s-b)(s-c)}
where,
- s is Semi-Perimeter
- a, b, and c are Side Lengths of 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 of Equilateral Triangle
Centroid of the triangle also called the centre of the triangle is a point which is at the centre 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 perpendicular 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 orthocentre of the triangle. Thus, for an equilateral triangle, the Centroid and Orthocentre 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 centre 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.

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Properties of Equilateral Triangles
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.
- Median, Angle Bisector and Altitude for all sides of an equilateral triangle are the same.
- Area of an Equilateral Triangle is √3/4 a2, where "a" is the side length of the triangle.
- Perimeter of an Equilateral Triangle is 3a, where "a" is the side length of the triangle.
Formula for Equilateral Triangles |
---|
Each Interior Angle of Equilateral Triangle | 60° |
Each Exterior Angle of Equilateral Triangle | 120° |
Perimeter of Equilateral Triangle | 3 × Sides |
Height of Equilateral Triangle | √3/2 × (Side) |
Area of Equilateral Triangle | √3/4 × (Side)2 |
Equilateral Triangle Theorem
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.
Difference Between Scalene, Isosceles, and Equilateral Triangles
Major differences between Scalene Triangle, Isosceles Triangle and Equilateral Triangle is added in the table below,
Scalene vs Isosceles vs Equilateral Triangles |
---|
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°. |
 |  |  |
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Solved Examples on Equilateral Triangle
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,
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,
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.
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