✅ Equilateral Triangle Formula ⭐️⭐️⭐️⭐️⭐

What is an Equilateral Triangle?

• As the name suggests, ‘equi’ means equal, and an equilateral triangle is the one in which all sides are equal.
• The internal angles of any given equilateral triangle are of the same measure, that is, equal to 60 degrees.

What is the Area of Equilateral Triangle Formula?

The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle can be defined as a triangle in which all the sides are equal and the measure of all the internal angles is 60°. So, an equilateral triangle’s area can be calculated if the length of any one side of the triangle is known.

The Perimeter of Equilateral Triangle Formula

The perimeter of a triangle is equal to the sum of the length of its three sides, whether they are equal or not.

Here is a list of the area of the equilateral triangle formula, the altitude of the equilateral triangle formula, the perimeter of the equilateral triangle formula, and the semi-perimeter of an equilateral triangle.

Formulas and Calculations for an Equilateral Triangle:

• Perimeter of Equilateral Triangle: P = 3a
• Semiperimeter of Equilateral Triangle Formula: s = 3a/2
• Area of Equilateral Triangle Formula: K = (1/4) * √3 * a²
• The altitude of Equilateral Triangle Formula: h = (1/2) * √3 * a
• Angles of Equilateral Triangle: A = B = C = 60 degrees
• Sides of Equilateral Triangle: a equals b equals c.

1. Given the side of the triangle find the perimeter, semiperimeter, area, and altitude.

• a is known here; find P, s, K, h.
• P equals 3a
• s = 3a/2
• K = (1/4) * √3 * a²
• h = (1/2) * √3 * a

2. Given the perimeter of triangle find the side, semiperimeter, area, altitude.

• Perimeter(P) is known; find a, s, K, and h.
• a = P/3
• s = 3a/2
• K = (1/4) * √3 * a²
• h = (1/2) * √3 * a

3. Given the semi perimeter of a triangle find the side, perimeter, area, and altitude.

• Semiperimeter (s) is known; find a, P, K, and h.
• a = 2s/3
• P = 3a
• K = (1/4) * √3 * a²
• h = (1/2) * √3 * a

4. Given the area of the triangle find the side, perimeter, semiperimeter, and altitude.

• K is known; find a, P, s and h.
• a = √(4/√3)∗K(4/√3)∗K equals 2 * √K/√3K/√3
• P = 3a
• s = 3a / 2
• h = (1/2) * √3 * a

5. Given the altitude/height find the side, perimeter, semiperimeter, and area

• Altitude (h) is known; find a, P, s, and K.
• a = (2/√3) * h
• P = 3a
• s = 3a/2
• K = (1/4) * √3 * a²

Solved Examples

Question 1) Suppose you have an equilateral triangle with a side of 5 cm. What will be the perimeter of the given equilateral triangle?

Solution) We know that the formula of the perimeter of an equilateral triangle is 3a.

Here, a = 5 cm

Therefore, Perimeter = 3 * 5 cm = 15 cm.

FAQs (Frequently Asked Questions)

Question 1. What is the Equilateral Triangle?

Answer: An equilateral triangle is a triangle with all three sides of equal length, corresponding to what could also be known as a “regular” triangle. An equilateral triangle is known as a special case of an isosceles triangle having not just two, but all three sides equal. An equilateral triangle also has three equal.

Question 2. Is an Equilateral Triangle Unique?

Answer: A triangle has all three of its sides equal in length. An equilateral triangle is one in which all 3 sides are congruent (same length). Because it has the property that all three interior angles are equal.

Question 3. What is the Height of the Triangle or What is the Altitude of a Triangle?

Answer: Every side of the triangle can be a base, and from every vertex, you can draw the line perpendicular to a line containing the base – that’s the height of the triangle. Every triangle has three heights, which are also known as altitudes.

Area Of Equilateral Triangle

The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle is a triangle in which all the sides are equal and the measure of all the internal angles is 60°. So, an equilateral triangle’s area can be calculated if the length of its side is known.

Area of an Equilateral Triangle Formula

The formula for the area of an equilateral triangle is given as:

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

Where a = length of sides

Derivation for Area of Equilateral Triangle

There are three methods to derive the formula for the area of equilateral triangles. They are:

• Using basic triangle formula
• Using rectangle construction
• Using trigonometry

Deriving Area of Equilateral Triangle Using Basic Triangle Formula

Take an equilateral triangle of the side “a” units. Then draw a perpendicular bisector to the base of height “h”.

Now,

Area of Triangle = ½ × base × height

Here, base = a, and height = h

Now, apply Pythagoras Theorem in the triangle.

a² = h² + (a/2)²

⇒ h² = a² – (a²/4)

⇒ h² = (3a²)/4

Or, h = ½(√3a)

Now, put the value of “h” in the area of the triangle equation.

Area of Triangle = ½ × base × height

⇒ A = ½ × a × ½(√3a)

Or, Area of Equilateral Triangle = ¼(√3a²)

Deriving Area of Equilateral Triangle Using Rectangle Construction

Consider an equilateral triangle having sides equal to “a”.

Now, draw a straight line from the top vertex of the triangle to the midpoint of the base of the triangle, thus, dividing the base into two equal halves.

Now cut along the straight line and move the other half of the triangle to form the rectangle.

Here, the length of the equilateral triangle is considered to be ‘a’ and the height as ‘h’

So the area of an equilateral triangle = Area of a rectangle = ½×a×h …………. (i)

Half of the rectangle is a right-angled triangle as it can be seen from the figure above.

Thereby, applying the Pythagoras Theorem:

⇒ a² = h² +  (a/2)²

⇒ h² = (3/4)a²

⇒ h = (√3/2)a ……………(ii)

Substituting the value of (ii) in (i), we have:

Area of an Equilateral Triangle

=(½)×a×(√3/2)a

=(√3/4)a²

Deriving Area of Equilateral Triangle Using Trigonometry

If two sides of a triangle are given, then the height can be calculated using trigonometric functions. Now, the height of a triangle ABC will be-

h = b. Sin C = c. Sin A = a. Sin B

Now, area of ABC = ½ × a × (b . sin C) = ½ × b × (c . sin A) = ½ × c (a . sin B)

Now, since it is an equilateral triangle, A = B = C = 60°

And a = b = c

Area = ½ × a × (a . Sin 60°) = ½ × a² × Sin 60° = ½ × a² × √3/2

So, Area of Equilateral Triangle = (√3/4)a²

Below is a brief recall about equilateral triangles:

What is an Equilateral Triangle?

There are mainly three types of triangles which are scalene triangles, equilateral triangles and isosceles triangles. An equilateral triangle has all the three sides equal and all angles equal to 60°. All the angles in an equilateral triangle are congruent.

Properties of Equilateral Triangle

An equilateral triangle is the one in which all three sides are equal. It is a special case of the isosceles triangle where the third side is also equal. In an equilateral triangle ABC, AB = BC = CA.

Some important properties of an equilateral triangle are:

• An equilateral triangle is a triangle in which all three sides are equal.
• Equilateral triangles also called equiangular. That means, all three internal angles are equal to each other and the only value possible is 60° each.
• It is a regular polygon with 3 sides.
• A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.
• A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.
• The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle.
• In an equilateral triangle, the median, angle bisector and perpendicular are all the same and can be simply termed as the perpendicular bisector due to congruence conditions.
• The ortho-centre and centroid of the triangle is the same point.
• In an equilateral triangle, median, angle bisector and altitude for all sides are all the same and are the lines of symmetry of the equilateral triangle.
• The area of an equilateral triangle is √3 a²/ 4
• The perimeter of an equilateral triangle is 3a.

Example Questions Using the Equilateral Triangle Area Formula

Question 1: Find the area of an equilateral triangle whose perimeter is 12 cm.

Solution:

Given: Perimeter of an equilateral triangle = 12 cm

As per formula: Perimeter of the equilateral triangle = 3a, where “a” is the side of the equilateral triangle.

Step 1: Find the side of an equilateral triangle using perimeter.

3a = 12

a = 4

Thus, the length of side is 4 cm.

Step 2: Find the area of an equilateral triangle using formula.

Area, A = √3 a²/ 4 sq units

= √3 (4)²/ 4 cm²

= 4√3 cm²

Therefore, the area of the given equilateral triangle is 4√3 cm²

Question 2: What is the area of an equilateral triangle whose side is 8 cm?

Solution:

The area of the equilateral triangle = √3 a²/ 4

= √3 × (8²)/ 4 cm²

= √3 × 16 cm²

= 16 √3 cm²

Question 3: Find the area of an equilateral triangle whose side is 7 cm.

Solution:

Given,

Side of the equilateral triangle = a = 7 cm

Area of an equilateral triangle = √3 a²/ 4

= (√3/4) × 7² cm²

= (√3/4) × 49 cm²

= 21.21762 cm²

Question 4: Find the area of an equilateral triangle whose side is 28 cm.

Solution:

Given,

Side of the equilateral triangle (a) = 28 cm

We know,

Area of an equilateral triangle = √3 a²/ 4

= (√3/4) × 28² cm²

= (√3/4) × 784 cm²

= 339.48196 cm²

Frequently Asked Questions

What is an Equilateral Triangle?

An equilateral triangle can be defined as a special type of triangle whose all the sides and internal angles are equal. In an equilateral triangle, the measure of internal angles is 60 degrees.

What does the Area of an Equilateral Triangle Mean?

The area of an equilateral triangle is defined as the amount of space occupied by the equilateral triangle in the two-dimensional area.

What is the Formula for Area of Equilateral Triangle?

To calculate the area of an equilateral triangle, the following formula is used:

A = ¼(√3a²)

What is the Formula for Perimeter of Equilateral Triangle?

The formula to calculate the perimeter of an equilateral triangle is:

P = 3a

Area of Equilateral Triangle

The area of an equilateral triangle is the amount of space that an equilateral triangle covers in a 2-dimensional plane. An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º. The area of any shape is the number of unit squares that can fit into it. Here, “unit” refers to one (1) and a unit square is a square with a side of 1 unit. Alternatively, the area of an equilateral triangle is the total amount of space it encloses in a 2-dimensional plane.

What is the Area of an Equilateral Triangle?

The area of an equilateral triangle is defined as the region covered within the three sides of an equilateral side. It is expressed in square units. Some important units used to express the area of an equilateral triangle are in², m², cm², yd², etc. Let us understand the formula to calculate the area of an equilateral triangle and its derivation in the following sections.

Area of an Equilateral Triangle Formula

The equilateral triangle area formula is used to calculate the space occupied between the sides of the equilateral triangle in a plane. Calculating areas of any plane is a very important skill used by many people in their work. For the equilateral triangle, we have the formula for its area. In a general triangle, finding the area of a triangle might be a little bit complicated for certain cases. But, finding the area of an equilateral triangle is quite an easy calculation.

The general formula for the area of a triangle whose base and height are known is given as:

Area = 1/2 × base × height

While the formula to calculate the area of an equilateral triangle is given as,

Area = √3/4 × (side)² square units

In the given triangle ABC,

Area of ΔABC = (√3/4) × (side)² square units

where,
AB = BC = CA = a (the length of equal sides of the triangle)

Thus, the formula for the area of the above equilateral triangle can be written as:

Area of equilateral triangle ΔABC = (√3/4) × a² square units

Example: How to find the area of an equilateral triangle with one side of 4 units?

Solution:

Using the area of equilateral triangle formula: (√3/4) × a² square units,

we will substitute the values of the side length.

Therefore, the area of the equilateral triangle (√3/4) × 4² = 4√3 square units.

Derivation of Area of an Equilateral Triangle Formula

In an equilateral triangle, all the sides are equal and all the internal angles are 60°. So, an equilateral triangle’s area can be calculated if the length of one side is known. The formula to calculate the area of an equilateral triangle is given as,
Area of an equilateral triangle = (√3/4) × a² square units
where,
a = Length of each side of an equilateral triangle

The above formula to find the area of an equilateral triangle can be derived in the following ways:

• Using the general area of a triangle formula
• Using Heron’s formula
• Using trigonometry

Deriving Equilateral Triangle’s Area Using Area of Triangle Formula

The formula used to calculate the area of an equilateral triangle can be derived using the general area of the triangle formula. To do so, we require the length of each side and the height of the equilateral triangle. We will thus calculate the height of an equilateral triangle in terms of the side length.

The formula for the area of an equilateral triangle comes out from the general formula of an equilateral triangle which is the area of the triangle is equal to ½ × base × height. Derivation for the formula of an equilateral triangle is given below.

Area of triangle = ½ × base × height

For finding the height of an equilateral triangle we use the Pythagoras theorem (hypotenuse² = base² + height²).

Here, base = a, and height = h

Now, apply the Pythagoras theorem in the triangle.

a² = h² + (a/2)²

⇒ h² = a² – (a²/4)

⇒ h² = (3a²)/4

Or, h = ½(√3a)

Now, put the value of “h” in the area of the triangle equation.

Area of Triangle = ½ × base × height

⇒ A = ½ × a × ½(√3a)

Or, area of equilateral triangle = ¼(√3a²)

So, the formula of height comes as ½ × (√3 × side) and further the area of the equilateral triangle becomes ¼(√3 × side²).

Deriving Area of Equilateral Triangle Using Heron’s Formula

Heron’s formula is used to find the area of a triangle when the length of the 3 sides of the triangle is known. In mathematics, Heron’s formula named after Hero of Alexandria. Who gives the area of any triangle when the length of all three sides is known. We do not use angles or other distances for finding the area of a triangle using Heron’s formula.

Steps are given below for finding the area of a triangle:

Consider the triangle ABC with sides a, b and c.

Heron’s formula to find the area of the triangle is:

Deriving Area of Equilateral Triangle With 2 Sides and Included Angle (SAS)

For finding the area of a triangle side angle side (SAS) formula is a way to use the sine trigonometric function to calculate the height of a triangle and use that value to find the area of the triangle. There are three variations to the same formula based on which sides and included angle are given.

Consider a,b, and c are the different sides of a triangle.

• When sides ‘b’ and ‘c’ and included angle A is known, the area of the triangle is: 1/2 × bc × sin(A)
• When sides ‘b’ and ‘a’ and included angle B is known, the area of the triangle is: 1/2 × ab × sin(C)
• When sides ‘a’ and ‘c’ and included angle C is known, the area of the triangle is: 1/2 × ac × sin(B)

In an equilateral triangle, A = B = C = 60°

sin A = sin B = sin C

Now, area of ABC = 1/2 × b × c × sin(A) = 1/2 × a × b × sin(C) = 1/2 × a × c × sin(B)

For equilateral triangle, a = b = c

Area = 1/2 × a × a × sin(C) = 1/2 × a² × sin(60°) = 1/2 × a² × √3/2

So, area of equilateral triangle = (√3/4)a²

How to Find the Area of Equilateral Triangle?

The following steps can be followed to find the area of an equilateral triangle using the side length,

• Step 1: Note the measure of the side length of the equilateral triangle.
• Step 2: Apply the formula to calculate the equilateral triangle’s area given as, A = (√3/4)a², where, a is the measure of the side length of the equilateral triangle.
• Step 3: Express the answer with the desired unit.

Now, that we have learned the formula and method to calculate the area of the equilateral triangle, let us see few solved examples for better understanding.

Examples on Area of Equilateral Triangle

What Is the Formula of Equilateral Triangle Area?

How Do You Find the Perimeter and Area of an Equilateral Triangle?

How Do You Calculate the Height Using Area of an Equilateral Triangle?

What Are the Properties of an Equilateral Triangle?

Equilateral Triangle Formulas

An equilateral triangle is a triangle for which the measure of all sides is equal and all its angles measure 60º. Equilateral triangle formulas are used to calculate the perimeter, area and height,of an equilateral triangle. 

What Are Equilateral Triangle Formulas?

An equilateral triangle is the one in which all sides of the triangle are equal and have equal angles measures. Equilateral triangle formulas are used to calculate the area, altitude, perimeter, and semi-perimeter of an equilateral triangle.

Equilateral Triangle Formulas

Equilateral triangle formulas are used to calculate different parameters of an equilateral triangle.

• Formula to calculate area of an equilateral triangle is given as:
  Area of an equilateral triangle, A = (√3/4)a², where a is the side of the equilateral triangle.
• Formula to calculate perimeter of an equilateral triangle is given as:
  Perimeter of an equilateral triangle, P = 3a, where a is the side of the equilateral triangle.
• Formula to calculate semi-perimeter of an equilateral triangle is given as:
  Semi perimeter of an equilateral triangle = 3a/2, where a is the side of the equilateral triangle.
• Formula to calculate height of an equilateral triangle is given as:
  Height of an equilateral triangle, h = (√3/2)a, where a is the side of the equilateral triangle.

Examples Using Equilateral Triangle Formulas

Example 1: If the perimeter of an equilateral triangle is 99 units, what would be the length of each side of an equilateral triangle?

Solution:   

To find: Length of each side of an equilateral triangle.

Given: Perimeter of an equilateral triangle = 99 Units

Using equilateral triangle formulas for perimeter,

The perimeter of an equilateral triangle = 3a

99 = 3a

a = 99/3

a = 33 Units

Answer: The length of each side of the given equilateral triangle is 33 Units.

Example 2: If each side of an equilateral triangular park measures 200 m, calculate its area.

Solution: 

To find: Area of an equilateral triangular park.

Length of each side of the park = 200m

Using equilateral triangle formulas for the area,

Area of an Equilateral Triangle = (√3/4)a²

=√3/4(200)²

= 17,320.508 m²

Answer: The area of an equilateral triangular park is 17,320.508 m².

Example 3: Calculate the area, altitude, perimeter, and semi-perimeter of an equilateral triangle whose side is 10 units.

Solution:

Side of an equilateral triangle = a = 10 units

Area of an equilateral triangle

= √3/4 a²

= √3/4 × 10² unit²

= √3/4 × 100 unit²

= 43.30 unit²

Altitude of an equilateral triangle

= √3/2 a

= √3/2 × 10 units

= 8.66 units

Perimeter of an equilateral triangle

= 3a

= 3 × 10 units

= 30 units

Semi-Perimeter of an equilateral triangle

= 3a/2

= (3×10)/2 units

= 15 units

Answer: Area = 43.3 sq units, Altitude= 8.66 units , Perimeter = 30 units, Semi-perimeter = 15 units

FAQs on Equilateral Triangle Formulas

Equilateral Triangle Shape

Calculator Use

An equilateral triangle is a special case of a triangle where all 3 sides have equal length and all 3 angles are equal to 60 degrees. The altitude shown h is hb or, the altitude of b. For equilateral triangles h = ha = hb = hc.

If you have any 1 known you can find the other 4 unknowns. So if you know the length of a side = a, or the perimeter = P, or the semiperimeter = s, or the area = K, or the altitude = h, you can calculate the other values. Below are the 5 different choices of calculations you can make with this equilateral triangle calculator. 

Formulas and Calculations for a equilateral triangle:

• Perimeter of Equilateral Triangle: P = 3a
• Semiperimeter of Equilateral Triangle: s = 3a / 2
• Area of Equilateral Triangle: K = (1/4) * √3 * a²
• Altitude of Equilateral Triangle h = (1/2) * √3 * a
• Angles of Equilateral Triangle: A = B = C = 60°
• Sides of Equilateral Triangle: a = b = c

1. Given the side find the perimeter, semiperimeter, area and altitude

• a is known; find P, s, K and h
• P = 3a
• s = 3a / 2
• K = (1/4) * √3 * a²
• h = (1/2) * √3 * a

2. Given the perimeter find the side, semiperimeter, area and altitude

• P is known; find a, s, K and h
• a = P/3
• s = 3a / 2
• K = (1/4) * √3 * a²
• h = (1/2) * √3 * a

3. Given the semiperimeter find the side, perimeter, area and altitude

• s is known; find a, P, K and h
• a = 2s / 3
• P = 3a
• K = (1/4) * √3 * a²
• h = (1/2) * √3 * a

4. Given the area find the side, perimeter, semiperimeter and altitude

• K is known; find a, P, s and h
• a = √ [ (4 / √3) * K ] = 2 * √ [ K / √3 ]
• P = 3a
• s = 3a / 2
• h = (1/2) * √3 * a

5. Given the altitude find the side, perimeter, semiperimeter and area

• h is known; find a, P, s and K
• a = (2 / √3) * h
• P = 3a
• s = 3a / 2
• K = (1/4) * √3 * a²

The Formula for the Area of Equilateral Triangles

Before we begin, let’s review what an equilateral triangle is — a triangle with three equal side lengths and three equal internal angles of 60° each. Now, let’s get one thing straight: The area of an equilateral triangle is not the perimeter of an equilateral triangle. It’s the total space of the triangle’s surface.

As you know, there are many different types of triangles: right triangles, scalene triangles, and isosceles triangles. Again, in an equilateral triangle, the length of the sides of an equilateral triangle are equal.

To determine the area of an equilateral triangle, you must know its side lengths. So, before diving into the equilateral triangle area formula, let’s look at how to find the side lengths.

How to Find the Side Lengths of an Equilateral Triangle

When you know the height of the triangle, you can determine the side lengths. Once you’ve found the side length, you can then determine the area of an equilateral triangle.

When you divide an equilateral triangle into two right triangles, you see the height of an equilateral triangle. This is done by slicing an equilateral triangle in half from the tip of a vertex to the midpoint of one side to form an angle bisector.

The perpendicular bisector, the straight line that forms two 90° angles, represents the height of the equilateral triangle, as marked by height h. By creating this bisector, we’ve divided this equilateral into two right triangles. To find the height, you can use the Pythagorean theorem:

Since all sides of an equilateral triangle are the same, side A=side C. And since the base of the right triangle is half the side length of the equilateral triangle, side A=side C/2. Now let’s plug in the height, base, and side length of C for the hypotenuse to isolate the value of h:

If you only know the height of an equilateral triangle’s perpendicular bisector, you can use this formula to determine the length of each equal side:

Now that we know how to use the height of an equilateral triangle to determine its missing side length, let’s learn how to solve for the area.

Formula for the Area of Equilateral Triangles

The figure above is an equilateral triangle. The following formula is used to determine the area of the triangle:

Mastering the Area of Equilateral Triangles

Equilaterals are triangles with three equal sides and angles that all measure 60°. When you create a perpendicular bisector line through the vertex of an equilateral, you form two right triangles. You can use the Pythagorean theorem and height of the right triangles within the equilateral to determine the missing side lengths of an equilateral triangle.

Then, you can use the formula A = √3/4 (a²) to determine the area of an equilateral triangle. Knowing how to find the height and area of a triangle with equal sides makes it so much easier to learn other trigonometry formulas.

Equilateral Triangle

triangle, outer Napoleon triangle, second Morley triangle, Stammler triangle, and third Morley triangle.

The smallest equilateral triangle which can be inscribed in a unit square (left figure) has side length and area

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