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Explain The Step By Step Procedure To Find The Area Of An Equilateral Triangle

Referencing Styles : APA | Pages : 1

Equilateral triangle is the triangle that are having all sides equal. The length of sides of the equilateral triangle will be all equal in length. This kind of properties of a triangle is known as congruency. One of the main characteristics of the congruency is that the angles will also be of equal value. The interior angles of an equilateral triangle are all equal. Another important characteristics of the equilateral angle is that all the interior angles of the equilateral triangle is same.

 

Figure 1: Equilateral triangle

(Source: Created by author)

In the given triangle, A, B, C are three vertex of the triangle. The three angles that can be formed using three vertices are ∠ABC, ∠CAB and ∠ACB are all same. The sum of all the three angles are equal to 180 degree. This means the value of each angle is 60 degree. Now the area of the equilateral triangle based on the congruent sides is . In the given formula S is the length of the congruent sides. Even s can be designed as length of one side of the triangle. is constant and approximately the value is around 0.433.

Apart from this process of area formation, there are others method of area calculation that can be used for various level of triangles. In order to prove the formula of area calculation let us consider the following diagram.

 

Figure 2: Equilateral triangle

In the figure above, the sides of an equilateral triangle are equal to “a” units.

We know that the area of Triangle is given by;

A = 12×base×height

To find the height, consider Triangle ABC,
Applying Pythagoras Theorem we know,

AB2=AD2+BD2

a2=h2+(a2)2

h2=a2–a24

h2=3a24

h=3√a2

Thus, we can calculate area by the basic equation,
A = 12×b×h=12×a×3√a2 Therefore, A = =3√a24unit2

Now if the perimeter of the triangle is given, then the formula of area calculation is done by a different formula. The formula of area calculation when the perimeter of the triangle is as follows.

In the given equation the perimeter p is given. The most straightforward way to identify an equilateral triangle is by comparing the side lengths. If the three side lengths are equal, the structure of the triangle is determined (a consequence of SSS congruence). However, this is not always possible.

Another useful criterion is that the three angles of an equilateral triangle are equal as well, and are thus each. Since the angles opposite equal sides are themselves equal, this means discovering two equal sides and any angle is sufficient to conclude the triangle is equilateral, as is discovering two equal angles of.

Notably, the equilateral triangle is the unique polygon for which the knowledge of only one side length allows one to determine the full structure of the polygon. For example, there are infinitely many quadrilaterals with equal side lengths (rhombus) so you need to know at least one more property to determine its full structure. In this way, the equilateral triangle is in company with the circle and the sphere whose full structures are determined by supplying only the radius. The properties of the equilateral triangle can be cited as following. The area is A=\frac{\sqrt{3}}{4} a^2. Perimeter of the triangle is P=3a. The radius of the circumscribed circle isR = \frac{a}{\sqrt{3}}. Radius of the inscribed circle is r=R/2. Geometric centre of the triangle is the center of the circumscribed and inscribed circles. And the altitude (height) from any side is h=\frac{\sqrt{3}}{2} a.

Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that: The area of the triangle is {\displaystyle \mathrm {A} ={\frac {3{\sqrt {3}}}{4}}R^{2}}. Many of these quantities have simple relationships to the altitude (“h”) of each vertex from the opposite side: The area is A=\frac{h^2}{\sqrt{3}}. In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors and the medians to each side coincide.

Finding the area of the triangle that is inscribed within a circle.

 

Figure 3: Equilateral triangle inscribed in circle

(Source: Created by author)

Applying law of sine to the triangle OBC, we get asin60=rsin30⇒a=r⋅sin60sin30⇒a=√3⋅r. Now the area of the inscribed triangle is A=12⋅AM⋅MC. Now AM=AO+OM=r+r⋅sin30=32⋅r and MC=a2=√3⋅r2 and Finally, A=12⋅(32⋅r)⋅(√32⋅r)=18⋅3⋅√3⋅r2

Using the Java programme, the area of equilateral triangle can be done in the sense that using special programmes. import java.util.Scanner;

 

public class Main {

    /**

     * Utility functions

     */

    Static void printing(String string) {

        System.out. println(string);

    }

    public static void main(String[] args) {

        Scanner sc = new Scanner(System.in); //1

        println("Enter length of a side : ");

        double side = sc.nextDouble(); //2

        double area = (Math.sqrt(3) / 4) * side * side; //3

        System.out.printf("Area of the triangle is %.2f",area); //4

    }

}

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