Circumference Formula and Area Of Equilateral Triangle Formula

Circumference Formula

Our earth is like a sphere, and if we sliced it in half, we’d end up with a circle. Similarly, in our day to day life, we may find many objects in similar shapes. Circle and sphere both represent the 2-dimensional and 3- dimensional shape of the round objects. The circumference is the distance that surrounds a circle. We can measure the circumference of the earth by measuring the distance that we will have to walk all the way around the world. To understand circumference, this article will help a lot. Here we will discuss the circumference formula with examples. We also have to understand the meaning of diameter and radius. Let us begin!

Circumference Formula

What is Circumference?

Circumference of the circle is like the perimeter of the circle. So, it is the measurement of the boundary across any two-dimensional circular shape like a circle. To measure the circumference we need the radius or diameter of the circle.

As we know that the diameter is the distance between two points on the edge of the circle across the center of the circle. And, the radius is the distance from the center to the edge of a circle. It is the most important quantity of the circle through it we can compute the area and circumference of the circle.

The double value of the radius of a circle is called the diameter of the circle. Also, we can say that the diameter cuts the circle into two equal parts, which is called as a semi-circle.

Therefore the circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle always has a lot of importance in geometry as well as trigonometry.

Formula to Find Circumference

The circumference formula of a circle is obtained by multiplying the diameter with a constant π.  The Circumference Formula will require either the radius or the diameter of the circle for its calculation.

The mathematicians have evolved a relationship between circumference and diameter of the circle. This ratio is a constant known as π.

i.e.  Cd = π

Where C indicates circumference and d indicates diameter.

Circumference of a Circle,

C= 2 πr=πd

Where,

C	Circumference of the circle.
D	The diameter of the circle.
R	The radius of the circle.

Solved Examples

Q.1: What is the circumference of the circle with area4πsquare cm?

Solution: First we have to apply formula for area of a circle to get value of radius of the circle. Since area of circle,

A= πr²

i.e r = Aπ−−√

i.e. r = 4ππ−−√

i.e. r = 4–√

i.e. r = 2 cm

Therefore, Circumference of the Circle,

C= 2πr

i.e. C= 2π2

Thus  C  = 12.56 cm.

Thus circumference length is 12.56 cm.

Q.2: Find the radius of the circle having circumference C =  50 cm.

Solution:

It is given that circumference C is 50 cm.

As per the formula,

C= 2π r

i.e r=c2π

i.e. r=502π

i.e. r = 253.14

This implies, that

r =  7.96 cm

Therefore, the radius of the circle is 7.96 cm.

Area of Equilateral Triangle Formula

In this topic, we will discover about equilateral triangles and its area. The student will also learn the area of  equilateral triangle formula. An equilateral triangle is a triangle whose all three sides are having the same length. This is the only regular polygon with three sides. It appears in a variety of contexts, in both basic geometries as well as in many advanced topics such as complex number geometry and geometric inequalities. Let us start learning!

Area of Equilateral Triangle Formula

What is the equilateral Triangle?

We can split the word equilateral into two words as equi meaning equivalent and lateral meaning side. Therefore, an equilateral triangle is simply a triangle whose three sides are all equal. It is obvious that along with this triangle’s sides, all three angles are also equal. As we know that the sum of a triangle’s angles is always 180 degrees. Thus each angle in an equilateral triangle will be 60 degrees.

Hence, we can see that the equilateral triangle is the unique polygon for which by knowing only one side length one can determine the full structure of the polygon. In other words, the equilateral triangle is in company with the circle and the sphere whose full structures are known only by knowing the radius.

Area of Equilateral Triangle Formula:

The area is the size of a two-dimensional surface. The area of a plane surface is a measure of the amount of space covered by it. Calculating areas is a very important skill used by many people in their daily work. This are computation is highly dependent on the shape and size of the object. For the triangle, we have the formula to find out its area. For the general triangle, it is a little bit of complex calculation. But, finding the area of an equilateral triangle is comparatively easy.

Area of an equilateral triangle can be computed by the formula:

A= 3√a24

Where

A	Area of Equilateral triangle
a	Side length

Derivation of the formula:

Let one side length of the equilateral triangle is “a” units.

As we know that the area of Triangle is given by;

A = base×height2

Also, drawing a perpendicular from vertex to the base, will divide the triangle into two equal right-angled triangle. This triangle will have base length a/2 and hypotenuse length as a. thus base = a

Also length of perpendicular i.e. h will be,

h2=a2−(a2)2,    using Pythagorean theorem.

i.e. h2=3a24

i.e. h= 3√a2

Thus height = 3√a2

As we know that the area of Triangle is given by;

A = base×height2

i.e. A = a×3√a22

i.e. A= 3√a24

Hence Proved.

Solved Examples

Q.1: Find the area of an equilateral triangle with a side of length 7 cm?

Solution:

Given,

Side of the equilateral triangle i.e.

a = 7 cm

Also, area of an equilateral triangle,

A= 3√a24

i.e.  A= 3√724

i.e. A= 49×3√4 square cm.

i.e. A  = 21.21762 square cm

Thus area =  21.21762 square cm.

Q.2: Find the height of an equilateral triangle whose side is 28 cm?

Solution:

Given,

Side of the equilateral triangle,

i.e. a = 28 cm

We know, height of an equilateral triangle,

i.e. h= 3√a2

h= 3√×282

= 143–√cm

Thus height = 143–√cm.