Equilateral Triangle

Triangle with all three sides equal are known as Equilateral Triangle

Examples of Equilateral Triangle
Given below are the examples of equilateral triangles.
Notice the triangle’s three sides are equal

What is an equilateral triangle

Angles of equilateral triangle

Since all the sides of equilateral triangle are equal, the angles are also equal to each other
In equilateral triangle, each angle measures 60 degrees.

Each angle of equilateral triangle measured 60 degree

Angles of equilateral triangle

ABC is an equilateral triangle with equal sides and equal angles

Let the angle of equilateral triangle be x degree

We know that sum of angle of triangle is 180 degree
x + x + x = 180
3x = 180
x = 60

Hence, the value of each angle is 60 degree

equilateral triangle example

Area of Equilateral Triangle

You can calculate the area of equilateral triangle using following formula:
Area of Equilateral Triangle = \frac{\sqrt{3}}{4} a^{2}


Area of equilateral triangle

Given the equilateral triangle with sides a cm and perpendicular bisector of height h cm

We know that:
Area=\ \frac{1}{2} \times \ base\ \times height\\\ \\ Area\ =\ \frac{1}{2} \times \ a\ \times h\ \ \ \ \ \ \ \ \ -\ -\ eq( 1)

Taking triangle AOB
Since height h is perpendicular to BC, triangle AOB is a right angled triangle

Using Pythagoras Theorem
h^{2} +\left(\frac{a}{2}\right)^{2} =a^{2}\\\ \\ h^{2} +\left(\frac{a}{2}\right)^{2} =a^{2} \ -\frac{a^{2}}{4}\\\ \\ h\ =\ \frac{\sqrt{3}}{2} \ a\ \ \ \ \ \ \ \ \ -\ -\ -eq( 2)

Putting value of h in eq(1) we get:
Area = \frac{\sqrt{3}}{4} a^{2}

Equilateral Triangle Properties

(01) In equilateral triangle all sides and angles are equal

Properties of equilateral triangle

(02) In equilateral triangle, the value of each angle is 60 degrees

(03) In equilateral triangle, the perpendicular drawn from vertex to opposite sides divides the sides and angles in equal halves

equilateral triangle definition

Given above is equilateral triangle.
Here the line AD is perpendicular to side BC.

So according to above mentioned property, the perpendicular line divides the angle and side into equal halves,
\angle 1=\angle 2
Side BD = DC

(04) Equilateral triangle has three line of symmetry

Line of symmetry divides the triangle into equal halves.
Since all the sides are equal, equilateral triangle has three symmetry lines

Given below is the image of three possible line of symmetry in equilateral triangle

Line of symmetry in equilateral triangle

(05) Circumcenter of equilateral triangle

Circumcenter is the center of the circle which passes through each of the polygon’s vertex.
It is located at the point of intersection of two perpendicular bisectors

Circumcenter of equilateral triangle

Given above is the equilateral triangle ABC circumscribed by circle with center O.
The point O is called Circumcenter of the triangle

How to locate the circumcenter?
The point of intersection of any two perpendicular bisector of triangle is the circumcenter

Finding circumcenter of equilateral triangle

In the above equilateral triangle, AM & CN are perpendicular bisectors.
These bisectors meet at point O, which is also the circumcenter of the triangle.

the radius R of circumcenter is calculated by formula:
R = \frac{a}{\sqrt{3}}

(06) Median, Angle bisector and Altitude

In equilateral triangle, the median line, angle bisector line and altitude line is the same.
Hence the centroid and angle bisector point is also the same

(07) Centroid of Equilateral Triangle

Centroid is the central point of any geometrical figure.
For example, centroid of a circle is the center point called Radius.

But for other figures like triangle, the location of center point is not that straight forward.
For geometrical figures, the location of centroid is the intersection of medians (line which divides side equally)

Centroid of equilateral triangle

The above image is of equilateral triangle ABC with medians AM, BL and CN.
Since the median meet at point O, it is the centroid of the triangle.

As we have understood earlier that in equilateral triangle, the median, altitude and angle bisector are same line. So the points like centroid, circumcenter, orthocenter are also located at the same point

Hence in equilateral triangle, the centroid, circumcenter, orthocenter are all the same.

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