Why is pi irrational?

Pi cannot be expressed as a ratio of two integers. Its decimal expansion (3.14159265...) never repeats or terminates. This was proven by Johann Lambert in 1768. Pi is also transcendental (proven by Lindemann in 1882), meaning it is not the root of any polynomial with rational coefficients.

How many decimal places of pi are known?

As of 2024, pi has been calculated to over 200 trillion decimal places. In practical engineering, pi to 15 decimal places (3.14159265358979) is more than sufficient for any physical calculation. Ten digits of pi is enough to calculate Earth's circumference to within a millimeter.

What is the relationship between a circle and a sphere?

A sphere is a 3D circle. Volume = (4/3)*pi*r^3. Surface area = 4*pi*r^2. The cross-section of a sphere through its center is always a great circle with the same radius.

An annulus is the region between two concentric circles. Area = pi*(R^2 - r^2) where R is the outer radius and r is the inner radius. Applications: pipe walls, washers, ring-shaped gardens, tire cross-sections.

What is the relationship between radius and diameter?

The diameter is exactly twice the radius: d equals 2r. The radius is the defining measurement of a circle.

How is area different from circumference?

Area (pi times r squared) measures the space inside the circle in square units. Circumference (2 times pi times r) measures the distance around the circle in linear units.

What is a sector of a circle?

A sector is a pie-slice region bounded by two radii and the arc between them. Its area equals (theta divided by 360) times pi times r squared.

Circle Calculator — Calculover

Circle Parts & Formulas

Part	Definition	Formula
Radius (r)	Distance from center to edge	r = d/2 = C/(2π) = √(A/π)
Diameter (d)	Distance across circle through center	`d = 2r`
Circumference (C)	Perimeter / total boundary length	`C = 2πr = πd`
Area (A)	Space enclosed by circle	`A = πr²`
Semicircle Area	Half of full circle	`A = πr²/2`
Arc Length	Curved portion for angle θ	`Arc = rθ` (θ in radians)
Sector Area	Pie-slice for angle θ	`A = ½r²θ`
Chord Length	Straight line joining two points on circle	`chord = 2r·sin(θ/2)`
Segment Area	Area between chord and arc	`A = ½r²(θ − sin θ)`
Annulus Area	Ring between two concentric circles	`A = π(R²−r²)`

Part

Definition

Formula

Radius (r)

Distance from center to edge

r = d/2 = C/(2π) = √(A/π)

Diameter (d)

Distance across circle through center

d = 2r

Circumference (C)

Perimeter / total boundary length

C = 2πr = πd

Area (A)

Space enclosed by circle

A = πr²

Semicircle Area

Half of full circle

A = πr²/2

Arc Length

Curved portion for angle θ

Arc = rθ (θ in radians)

Sector Area

Pie-slice for angle θ

A = ½r²θ

Chord Length

Straight line joining two points on circle

chord = 2r·sin(θ/2)

Segment Area

Area between chord and arc

A = ½r²(θ − sin θ)

Annulus Area

Ring between two concentric circles

A = π(R²−r²)

How to Use This Calculator

Enter the Radius

Type the radius of the circle into the input field. You can also enter the diameter and the calculator will halve it.

View the Results

The calculator displays the area, circumference, and diameter instantly.

Review the Steps

A step-by-step breakdown shows the formulas and intermediate calculations used.

Formula & Methodology

Area of a Circle

A = πr²

Multiply pi by the square of the radius to find the enclosed area.

Circumference

C = 2πr

The circumference is the distance around the circle, equal to pi times the diameter.

Diameter

d = 2r

The diameter is twice the radius; it is the longest chord of the circle.

Key Terms

Radius (r)

The distance from the center of the circle to any point on its circumference.

Diameter (d)

The distance across the circle through its center; equal to twice the radius.

Circumference (C)

The total distance around the circle; the circle's perimeter.

Pi (π)

An irrational constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter.

Area (A)

The total space enclosed within the circle, measured in square units.

Radius	Diameter	Circumference	Area
1	2	6.28	3.14
5	10	31.42	78.54
10	20	62.83	314.16
25	50	157.08	1,963.50
100	200	628.32	31,415.93

Radius

Diameter

Circumference

Area

6.28

3.14

31.42

78.54

62.83

314.16

157.08

1,963.50

100

200

628.32

31,415.93

The Circle: Geometry's Most Perfect Shape

Pi: The Universal Constant

Pi (π) appears throughout mathematics and physics, from the geometry of circles to the oscillation of waves. It is irrational—its decimal expansion never terminates or repeats. Despite millennia of effort, pi has been computed to trillions of digits, yet for all practical engineering purposes, 3.14159 provides more than enough precision.

Circles in Engineering and Nature

Wheels, gears, pipes, and satellite orbits are all based on circular geometry. Nature favors circles because they enclose the maximum area for a given perimeter, minimizing material or energy. Soap bubbles, planet cross-sections, and tree trunks all approximate circles for this reason.

Circle Calculator

Circle Parts & Formulas

π (Pi) Reference

How to Use This Calculator

Enter the Radius

View the Results

Review the Steps

Formula & Methodology

Area of a Circle

Circumference

Diameter

Key Terms

Real-World Examples

Circle Measurements by Radius

The Circle: Geometry's Most Perfect Shape

Pi: The Universal Constant

Circles in Engineering and Nature

Circle Calculator

Circle Parts & Formulas

π (Pi) Reference

How to Use This Calculator

Enter the Radius

View the Results

Review the Steps

Formula & Methodology

Area of a Circle

Circumference

Diameter

Key Terms

Real-World Examples

Circle Measurements by Radius

The Circle: Geometry's Most Perfect Shape

Pi: The Universal Constant

Circles in Engineering and Nature

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