What is an Equilateral Triangle?
As the name suggests, ‘equi’ means equal, 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.
Triangles are classified into three sorts based on the length of their sides:
Scalene triangle: The sides and the angles of the scalene triangle are not equal.
Isosceles triangle: An isosceles triangle has two equal sides and two equal angles.
Equilateral triangle: All sides and angles of the equilateral triangle are equal.
Area of Equilateral Triangle
The region enclosed by the three sides of an equilateral triangle is defined as the area of the equilateral triangle. It is expressed in square units. The common units used to express the area of an equilateral triangle are in2, m2, cm2 and yd2
Below 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 are discussed.
Area of the 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 area occupied between the sides of an equilateral triangle in a plane is calculated using the equilateral triangle area formula.
The formula for calculating the area of a triangle with a known base and height is:
Area = 1/2 × base × height
The following formula can be used to compute the area of an equilateral triangle:
Area = √3/4 × (side)2 square units
Perimeter of the 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.
An equilateral triangle's perimeter is the sum of its three sides.
P = 3a is the basic formula for calculating the perimeter of an equilateral triangle, where 'a' denotes one of the triangle's sides. The sum becomes a + a + a = 3a since all three sides of an equilateral triangle are equal.
Height = √3a/ 2
Semi perimeter = (a + a + a)/2 = 3a/2
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 the 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/√3
K/√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
1. Apply the equilateral triangle area formula and find the area of an equilateral triangle whose each side is 12 in.
Solution:
Side = 12 in
Applying the equilateral triangle area formula,
Area = √3/4 × (Side)2
= √3/4 × (12)2
= 36√3 in2
Answer: Area of an equilateral triangle area 36√3 in2
2. Calculate the perimeter and semi perimeter of an equilateral triangle with a side measurement of 12 units.
Solution:
The perimeter = 3a
Semi-perimeter = 3a/ 2
Given, side a = 12 units
Now, the perimeter of an equilateral triangle is equal to:
3 × 12 = 36 units
And, Semi-perimeter of an equilateral triangle is equal to:
36/2 = 18 units.
3. 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 on Equilateral Triangle Formula
1. How do you define Napoleon’s Theorem?
If equilateral triangles are formed on the sides of any triangle, either all outward or all inward, the lines joining the centers of those equilateral triangles produce an equilateral triangle, according to Napoleon's theorem.
The result of this technique is the inner or outer Napoleon triangle. The size of the original triangle is equal to the area difference between the outer and inner Napoleon triangles.
2. Is the Pythagorean theorem applicable to equilateral triangles?
Equilateral triangles have three equal sides and angles that are all 60 degrees. Two right triangles are formed when a perpendicular bisector line is drawn through the vertex of an equilateral. To find the missing side lengths of an equilateral triangle, utilize the Pythagorean theorem and the height of right triangles within the equilateral triangle. The area of an equilateral triangle can therefore be calculated using the formula A = √3/4 (a²). Knowing how to calculate the height and area of a triangle with equal sides makes learning other trigonometric formulas much easier.
3. Is it possible for an equilateral triangle to be obtuse?
An equilateral triangle can’t be obtuse, because an equilateral triangle has equal sides and angles, each angle is sharp and measures 60 degrees. As a result, an equilateral angle cannot be obtuse.
A triangle can’t be both right-angled and obtuse-angled at the same time. As a right-angled triangle has only one right angle, the other two angles are acute. To summarise, an obtuse-angled triangle can never form a right angle and vice versa. The side opposite the obtuse angle of a triangle is the longest.
4. An equilateral triangle is always congruent. Explain.
For this, Consider two triangles ABC and DEF with sides AB and DE equal.
AB=DE
As they are equilateral, the other two pairs of sides are also equal.
BC=EF
AC=DF
∴△ABC≅△DEF by SSS congruence criteria.
Hence, two equilateral triangles with equal sides are always congruent.