In geometry, anequilateral triangleis a triangle that has all its sides equal in length.Since the three sides are equal therefore the three angles, opposite to the equal sides, are equal in measure. Therefore, it is also called anequiangular triangle, where each angle measure 60 degrees. Just like othertypes of triangles, an equilateral triangle also has its area, perimeter and height formula.Let us learn more in this article.
- Definition
- Shape
- Properties
- Comparison
- Theorem
- Formula
- Area
- Perimeter
- Height
- Centroid
- Around the center
- 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. Also, the three angles of the equilateral triangle are congruent and equal to 60 degrees.The sum of all three angles of an equilateral triangle is equal to 180 degrees.60°+ 60°+ 60°= 180°.Thus, it obeys theangle sum property of 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.e., “Equi” meaning equal and “Lateral” meaning sides. An equilateral triangle is also called aregular polygonor 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 AC are the sides of the equilateral triangle.
And
∠A = ∠B = ∠C = 60°
Based on sides there 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 three sides.
- The perpendicular drawn from vertex of the equilateral triangle to the opposite side bisects it into equal halves. Also, the angle of the vertex from where the perpendicular is drawn is divided into two equal angles, i.e. 30 degrees each.
- The ortho-centre and centroid are at the same point.
- In an equilateral triangle, median, angle bisector, and altitude for all sides are all the same.
- The area of an equilateral triangle is √3a2/ 4
- 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 two 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 measure |
Equilateral Triangle Theorem
If ABC is an equilateral triangle and P is a point on the arc BC of the circumcircle of the triangle ABC, then;
PA = PB + PC
Proof:For a cyclic quadrilateral ABPC, we have;
PA⋅BC=PB⋅AC+PC⋅AB
Since we know, for an equilateral triangle ABC,
AB = BC = AC
Therefore,
PA.AB = PB.AB+PC.AB
Taking AB as a common;
PA.AB=AB(PB+PC)
PA = PB + PC
Equilateral Triangle Formulas
We have already understood an equilateral triangle has all three sides equal in length and all three angles equal in measure. Now based on these properties the formulas for equilateral triangles are defined. The most common formulas that we consider for a triangle are:
- Area of equilateral triangle
- Perimeter of equilateral triangle
- Height of equilateral triangle
In the next section, we will be discussing all these formulas.
Area of Equilateral Triangle
Thearea of an equilateral triangleis the region occupied by it in a two-dimensional plane. The formula for the area of an equiangular triangle is given by:
A = √3a2/4
Let us derive the formula here:
If we see the above figure, the area of a triangle is given by;
Area = ½ x base x height
Here Base = a and height = h
Therefore,
Area = ½ x a x h ………(1)
Now, from the above figure, the altitude h bisects the base into equal halves, such as a/2 and a/2. It also forms two equivalent right-angled triangles.
So, for a right triangle, using Pythagoras theorem, we can write:
a2= h2+ (a/2)2
or
h2= (a)2– (a/2)2
= 3a2/4
h = √3a/2
By putting this value in equation 1, we get;
Area = ½ x a x √3a/2
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 three 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. As we know, an equilateral triangle has all equal sides. Now, if we drop an altitude from the apex of the triangle to the base, it divides the triangle into two equal right triangles.
Thus, from the above figure, we can find the height (h) of the equilateral triangle, as:
h = √3a/2
Where a is the side of the triangle.
Thus, to summarise the formulas related to equilateral triangle are:
| Area | √3a2/4 |
| Perimeter | 3a |
| Height | √3a/2 |
Centroid of Equilateral Triangle
The centroid of the equilateral triangle lies at the center of the triangle. Since all its sides are equal in length, hence it is easy to find the centroid for it.
To find the centroid, we need to draw perpendiculars from each vertex of the triangle to the opposite sides. These perpendiculars are all equal in length and intersect each other at a single point, which is known as centroid. See the figure below:
Note:The centroid of a regular triangle is at equidistant from all the sides and vertices.
Around the center
The circumcenter of equilateral triangle is the point of intersection perpendicular bisectors of the sides. Here, the circumcircle passes through all the three vertices of the triangle.
If any of the incenter, orthocenter or centroid coincide with the circumcenter of a triangle, then it is called an equilateral triangle.
- Number of Sides = 3
- Number of angles = 3
- Each interior angle = 60
- Each exterior angle = 120
- Perimeter = 3 times of side-length
- Area =√3/ 4 x (side)2
- Height =√3 (side)/2
Solved Examples on Equilateral Triangle
Q.1: Find the area of the equilateral triangle ABC, where AB=AC=BC = 4cm.
Solution:
By the formula, we know;
Area = √3a2/4
Given a = 4cm
Hence, by putting the value we get;
Area = √3(4)2/4
A = 4√3
Q.2: Find the altitude of an equilateral triangle whose sides are equal to 10cm.
Solution:
By the formula, we know,
Height 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
Q1
What is an equilateral triangle?
A triangle that has all its sides equal in dimension and each angle measure up to 60 degrees, is called an equilateral triangle.
Q2
What is the area of equilateral triangle formula?
The formula to find area of equilateral triangle is given by:
A = (√3/4)a2, where a is the length of side of equilateral triangle.
Q3
What is the perimeter of an equilateral triangle?
The perimeter of an equilateral triangle is the sum of all its three equal sides.
Perimeter = 3a, where a is the length of its sides.
Q4
What is the height of equilateral triangle?
The formula to find the height of equilateral triangle is:
Height, h = (√3/2)a, where a is the side of the equilateral triangle.
Q5
What type of polygon is an equilateral triangle?
An equilateral triangle is a regular polygon or a regular triangle.
Q6
What is the different name of triangles based on sides?
Based on sides, there are three different kinds of triangles. Their names are:
Scalene triangle
Isosceles triangle
Equilateral triangle
Q7
How many sides does an equilateral triangle has?
All triangles have 3 sides only.
Q8
What is the perimeter of equilateral triangle with side equal to 10cm?
Perimeter = 3 x sides of equilateral triangle
Perimeter = 3 x 10 = 30 cms
FAQs
Equilateral Triangle - Definition, Properties, Formulas & Examples? ›
Formulas and Calculations for an Equilateral Triangle:
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.
Formulas and Calculations for an Equilateral Triangle:
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.
An equilateral triangle is also called 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 AC are the sides of the equilateral triangle.
What is the definition and properties of equilateral triangle? ›In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
What are equilateral triangles examples? ›An equilateral triangle is a type of triangle. It is a regular polygon and has special properties: all three sides are equal in length, and all three angles in the corners are 60º. Examples in real life can include traffic signs and tortilla chips.
What is the formula list of properties of triangle? ›- The sum of all the angles of a triangle (of all types) is equal to 180°.
- The sum of the length of the two sides of a triangle is greater than the length of the third side.
- In the same way, the difference between the two sides of a triangle is less than the length of the third side.
: a triangle in which all three sides are the same length.
Why is the formula for an equilateral triangle different? ›Unless a side is given, the area of an equilateral triangle differs from each other since area is a measure of dimension, like ft², m² and the linear dimension of the triangle is missing.
What is the formula for the volume of an equilateral triangle? ›The volume of an equilateral triangular prism can be easily found out by using the formula, Volume = (√3/4)a2 × h, where,'a' is side length and 'h' is the height of the equilateral triangular prism.
What are the 7 properties of triangle? ›- A triangle has three sides and three angles.
- The sum of the angles of a triangle is always 180 degrees.
- The exterior angles of a triangle always add up to 360 degrees.
- The sum of consecutive interior and exterior angle is supplementary.
What are the properties of an equiangular triangle? ›
Equiangular triangles have equal sides and equal angles. The sum of all the interior angles of a triangle is equal to 180°, and each angle of an equiangular triangle is equal to 60°. An equilateral triangle has a predictable shape.
What is the formula for the diagonal of an equilateral triangle? ›equilateral triangle properties
The equilateral triangle diagonal formula given as √3a2/4. The orthocenter, circumcenter, incenter, centroid are all the same point. The circumradius of an equilateral triangle is a√3/3.
Sal proves that the angles of an equilateral triangle are all congruent (and therefore they all measure 60°), and conversely, that triangles with all congruent angles are equilateral.
How do you find the sides of an equilateral triangle? ›When the perimeter is given: Clearly the perimeter of an equilateral triangle is thrice the length of its side, dividing the perimeter by 3 would yield the length of the side of such a triangle. Hence the length of side of an equilateral triangle is one- third of the perimeter of the triangle.
What are the formulas of properties? ›- Associative Property of Addition Formula.
- a + (b + c) = (a + b) + c.
- Associative Property of Multiplication Formula.
- (a × b) × c = a × (b × c)
There are two important formulas related to triangles, i.e., the Heron's formula and Pythagoras theorem. The sum of the interior angles of a triangle is 180° and is expressed as ∠1 + ∠2 + ∠3 = 180°.
What are the three formulas of triangle? ›Let ABC be a triangle such that the length of the 3 sides of the triangle is AB = c, BC = a and CA = b. Then, the area of triangle ABC = √[s × (s – a) × (s – b) × (s – c)]. Learn how to find the area of different types of triangles using Heron's formula.
What is the formula for the area of an equilateral triangle without height? ›To find the area of an equilateral triangle, you need to calculate the length of half the side length and substitute it into the Pythagorean theorem to find the height. You could also substitute it into sin60∘ , cos30∘ , tan30∘ , or tan60∘ to find the height.
Which best explains why all equilateral triangles? ›Answer and Explanation:
All equilateral triangles are similar because they all share the same features. An equilateral triangle must have three angles equal to 60 degrees and three sides that are exactly equal.
The length of the median of an equilateral triangle is always equal. As we know that the length of all sides of an equilateral triangle is equal, it follows that the length of the median of an equilateral triangle bisecting these sides is also equal.
What is unique about an equilateral triangle? ›
An equilateral triangle is a triangle whose three sides all have the same length. They are the only regular polygon with three sides, and appear in a variety of contexts, in both basic geometry and more advanced topics such as complex number geometry and geometric inequalities.
What is the formula for equilateral triangle based pyramid? ›The volume of a triangular pyramid can be found using the formula V = 1/3AH where A = area of the triangle base, and H = height of the pyramid or the distance from the pyramid's base to the apex.
What is the formula for surface area of an equilateral triangular pyramid? ›A triangular pyramid has an equilateral triangle as its base with side lengths 6 in and a height of 8 in. What is the volume and surface area of the pyramid? To find the surface area of a pyramid, we use the formula SA=B+12ps, where B is the area of the base, p is the perimeter of the base, and s is the slant height.
What is the rule of the triangle? ›The rule of the sides of a triangle is that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This rule is also known as the triangle inequality theorem. This implies that we cannot have a triangle with lengths 3, 4, 9 as 3 + 4 = 7 < 9.
What are the 5 theorems of a triangle? ›The Side–Side–Side Theorem, Side Angle Side Theorem, Angle Side Angle Theorem, Angle-Angle Side Theorem, and Right angle-Hypotenuse-Side or the Hypotenuse Leg Theorem are the five triangle congruence theorems.
What are the 4 types of triangle? ›| Based on their Sides | Based on their Angles |
|---|---|
| Scalene Triangle | Acute Triangle |
| Isosceles Triangle | Obtuse Triangle |
| Equilateral Triangle | Right Triangle |
Every equilateral triangle is isosceles, but the converse is not always true. A triangle with all three equal sides is called equilateral. If two of its sides are equal, a triangle is called isosceles.
Is every equilateral triangle isosceles? ›Yes. For a triangle to be isosceles,any two sides should be equal in length. In equilateral triangle, all the three sides are equal in length. The minimum criterion of two equal sides is hence met.So, all the equilateral triangles are isosceles triangle too.
Are all equilateral triangles equiangular? ›Every equilateral triangle is also an isosceles triangle, so any two sides that are equal have equal opposite angles. Therefore, since all three sides of an equilateral triangle are equal, all three angles are equal, too. Hence, every equilateral triangle is also equiangular.
How do you find the line of symmetry of an equilateral triangle? ›How many types of equilateral triangles are there? ›
The equilateral triangle is considered as a regular polygon or a regular triangle as angles are equal and sides are also equal. The triangles are categorized into three different types based on their sides.
What is the name of a triangle with only two equal sides? ›An isosceles triangle is a triangle with two equal sides.
What else is an equilateral triangle called? ›An equilateral triangle is a triangle with all three sides of equal length , corresponding to what could also be known as a "regular" triangle.
What triangle is a Dorito? ›Doritos, candy corn, and mountains all have the same thing in common - they are shaped like triangles, but what if I told you all triangles are not the same. In this lesson, you will learn about a special type of triangle called the equilateral triangle.
What is the formula of triangular? ›The basic formula for the area of a triangle is equal to half the product of its base and height, i.e., A = 1/2 × b × h. This formula is applicable to all types of triangles, whether it is a scalene triangle, an isosceles triangle, or an equilateral triangle.
What is the formula of area of an equilateral triangle as a function of the length s of its sides? ›In general, the height of an equilateral triangle is equal to √3 / 2 times a side of the equilateral triangle. The area of an equilateral triangle is equal to 1/2 * √3s/ 2 * s = √3s2/4.
What is the 45 45 90 triangle formula? ›The 45-45-90 triangle rule states that the three sides of the triangle are in the ratio 1:1:\(\sqrt{2}\). So, if the measure of the two congruent sides of such a triangle is x each, then the three sides will be x, x and \(\sqrt{2}x\). This rule can be proved by applying the Pythagorean theorem.
What is the formula of triangle with 3 equal sides? ›If a triangle has 3 equal sides, it is called an equilateral triangle. The area of an equilateral triangle can be calculated using the formula, Area = a2(√3/4), where 'a' is the side of the triangle.