Circle Calculator: Finding c, d, a, r

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If you want to find out the radius, area, diameter, or circumference of a circle, then you are going to need a circle calculator. If math is not your strong suit, a circle calculator can be a valuable asset every time you need to work out one of these values.

Below, we will help you understand and identify circle formulas, the definition of a circle, and even how to find the center of one. In essence, you can find everything you need to know about calculating a circle below – including the circle calculator itself! 

C = Circumference
D = Diameter
R = Radius
A = Area

What is the Definition of a Circle?

Circle is a Greek word which means a ring or hoop, and it refers to a closed curve with several points from which you can measure the distance. The circle tends to relate more to the line creating the shape as opposed to the area inside it. If you referred to both the boundary line and included area within it, it would be a disc. The disc, in most terminologies, is the curve with a plane surface.

The Circle’s Lines

To be able to work out what the circle’s circumference is, you need to understand every term involved in the process. These are circumference, diameter, chord, and radius.

Circumference: The distance around the circle.
Radius: A line that connects from the center of the circle to the outer edge.
Diameter: A line that goes from an endpoint and passes through the center to the other side.
Chord: A line that goes from one outer edge of the circle to another side.

There are also other terms you may like to learn for other circle calculations, such as arc or tangent.

A Circle’s Properties

A circle is a simple shape, but it’s quite complicated too. A circle is: 
	A shape with the largest area regarding its perimeter
	Symmetric – the line will reflect a mirror image 
	Unique – you can construct it from any three points on the plane 

All circles are very similar and have a dozen or more other unique properties you may want to research if you want to find out more

Formulas for a Circle

When it comes to working out solutions to any circle-related problems, you will find they all follow simple formulas. 

These include:
Circumference - c = 2πr
Area - a = πr²
Diameter – d = 2r

There are also several more complex formulas available. However, if you are looking for a circle formula, you can find it in an equation of a circle calculator.

Working with the Unit Circle

You can describe a unit circle as a regular circle whose radius is one. It will have its center at the origin, also known as (0,0) if you are using a Cartesian coordinate system.

You will need a graphic like this.

Locate x and y on the circle which equates to the lengths of the triangle sides. The radius ends up being the hypotenuse of the first length (x). You may want to use the Pythagorean Theorem to work out the following: 

x² + y² = 1

Because x² = (−x)², you can rightly assume that you are able to use the same formula for every quadrant instead of only the initial one. 

When it comes to sine and cosine of the right triangle, however, the following formula is what you will need.

cos(α) = x / 1 = x
sin(α) = y / 1 = y

Followed by 
sin²(α) + cos²(α) = 1

This equation is a Pythagorean trigonometric identity.

How to Find the Circumference of a Circle Using a Circle Calculator: Find C

You can define the circumference of a circle as the length of a line you would craft with using a compass. The method of finding the circumference of a circle can also depend on whether you have the radius or diameter, or you don’t have any information at all.

If you have the radius or diameter, follow this formula:
c = 2πr = πd

If you don’t, follow this one:
c = 2√(πa)

How to Find the Diameter of a Circle using a Circle Calculator: Find D

Firstly, you have to understand what the diameter is, by definition. The diameter is the length of a line that goes through the center point of your circle and joins two points. 

There are three possible formulas for finding a circle’s diameter, depending on whether you know the circle’s radius, area, or circumference. 

If you know the circle’s radius:
d = 2r

If you don’t know the radius or area:
d = c / π

If you don’t know the radius or circumference:
d = 2√ ( a / π )

How to Find the Area of a Circle using a Circle Calculator: Find A

The ‘a’ in any circle formula stands for the area of the circle. The area is also written in squared units. You can calculate the area of a circle in two different ways. 

If you know the radius or diameter of the circle:
a = πr² = π * (d / 2)²

If you don’t know the diameter or radius or the circle:
a = c² / 4π

How to Find the Radius of a Circle using a Circle Calculator: Find R

If you see an R in a formula pertaining to circles, it stands for the radius. Radius is the length of a line that goes from the center of your circle to the edge. There are three potential formulas, depending on how many other measurements you have. 

If you know the circle’s diameter:
r = d / 2

If you don’t know the circle’s diameter or area:
r = c / 2π

If you don’t know the circle’s circumference or diameter:
r = √(a / π)

How to Find the Center of a Circle?

There isn’t one single way to help you to find the center of the circle. Rather, there are several. However, some are more straightforward than others. Here are two of the best methods to find the center. 

First method:
The first method offers more of an estimate than an exact center. If you don’t need it to be 100 percent accurate, this is an effective method for you. 

   1.	Choose any point on the circle
   2.	Get a triangle measuring tool or something that has straight edges and a 90-degree angle.
   3.	Across the circle, draw two perpendicular lines that intersect with the circle.
   4.	Draw lines through the points to establish your diameter. 
   5.	Construct the bisector of the diameter OR choose another point, draw the lines, and create a second diameter. The intersection of the two will be the circle’s approximate center.

Second method:
This one is more accurate than the first. 

   1.	Use a ruler and draw two chords
   2.	Construct perpendicular bisectors for one of those chords.
   3.	Draw overlapping circles with a compass, with their centers at the chord’s endpoints. Locate where the circles intersect and draw a line through them with a ruler. 
   4.	Repeat for the second chord. Where the two bisectors cross, that’s the circle’s center.

A Circle’s Chord

The chord, part of ancient trigonometry, is the line that joins two points to any curve. Most commonly, it’s a circle, but it can also be an ellipsis or oval. If the chord passes through the circle’s center, then it’s the longest chord and becomes the diameter. 

You can calculate the chord’s length using the circle calculator formula below.
chord(α) = √[(1 - cos(α))² + sin²(α)] = √[2 - 2cos(α)] = 2sin(α/2)

What is a Concentric Circle?

Concentric circles are circles with the same center. They form a kind of ring with one smaller circle inside of another. The area in the middle of the two circles with different radii is an annulus.

If you can’t seem to picture a concentric circle in your head, then several everyday items are concentric circles. An archery board, dart board, tree growth rings, and a CD’s grooves are all examples of concentric circles.

What is Pi?

Most people have at least some idea of Pi or have at least heard of it. Pi is a mathematical constant – but it also refers to the diameter to circumference ratio of a circle. The ratio, no matter the circle’s size, will be c/d (with the formula π = circumference/diameter) and most people refer to Pi as its first few digits (3.14159265).

Pi is unique in that it’s an irrational number that has a never-ending amount of digits. No figure will ever represent pi and its value. Even with the help of computer technology which traced over 22 trillion digits, the numbers continued to go on. It just didn’t end!

Such is the appeal of Pi that there is a Guinness World Record for how many numbers someone can recite correctly in a row. A retired Japanese Engineer holds the unofficial record, quoting over 100,000, while the official record is 70,000. 

Fun Facts About Pi
	Pilish is a style of writing – word lengths match the pi digits 
	Books exist that contain full works of Pi, with as many as 10,000 digits in pilish language
	March 14 is Pi Day, a day to celebrate pi and eat round-shaped treats like pies and pizza
	You only need 38 pi digits to measure the universe’s circumference

Circle in a Square

A common problem that people used to try to solve was building a square that had the same volume as a circle. Seems easy enough, right? Except, you could only use a straight edge like a ruler, and a compass. In 1882, “squaring the circle” was proven impossible by mathematician Ferdinand von Lindemann. Ferdinand said that because π is a transcendental number, you cannot construct π√.
However, it’s entirely possible to do if you have more tools at your disposal than a ruler and compass.

Circles and 3D Shapes

As most people know, a circle is 2D. When it’s 3D, it’s a geometric shape. A cylinder has two circle bases, while a cone has one circle. Because we worked out the circle formula above, we can quickly work out the cylinder and cone’s volumes. 

Cylinder volume:

Cone volume:

A sphere, on the other hand, is a round 3D shape with an entirely circular surface – akin to a circle or disc but in three dimensions. It can also form part of a circle of spheres, circles that sit on a sphere. The circle has a plane that passes through the sphere’s center and features in cartography and geography.

What is the Roundest Country in the World?

If you are fully immersed in the measurement of circles, then why not find out what is the roundest country in the world? Gonzalo Ciruelos, a math expert and blogger, created a roundness perimeter before ranking the countries in order of most round to least round. Sierra Leone is the rounded country, and Marshall Island’s is the least round. 

You can check out the rankings here: https://gciruelos.com/what-is-the-roundest-country.html

How to Use a Circle Calculator?

If you still need help learning how to work a circle calculator, please let this perfectly round pizza be of assistance.

1. Identify the item and what you want to do with it. (Pizza, work out the size of it).
2. Type the dimensions you have into the circle calculator. A simple search will tell you that a medium pizza is 12 inches in diameter.

Your pizza’s parameters are below:
Diameter = 12 inches
Radius = 6 inches
Circumference = 37.7 inches
Area = 113.1 inches-squared

Try the circle calculator out for yourself with something circular that you might have lying around the house!

Circle Calculator: Finding c, d, a, r

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