Area of a Circle Calculator

Calculating the area of a circle is straightforward with the right formula. Knowing the radius, diameter, or circumference allows for quick and accurate results.

Whether for school, work, or everyday use, this simple method ensures precise measurements every time.

How to Find The Area of a Circle?

To find the area of a circle, you must know either the radius ("r") of a circle or the diameter ("d") of a circle. The following area of a circle calculators allow you to calculate area of a circle knowing at least one of these variables:

Note: results are rounded to the 4th decimal.

Area of a Circle Formula

A circle is a two-dimensional shape where all points are equidistant from a fixed point called the center. Here are the key parts of a circle:

Radius (r)

• The radius is the distance from the center of the circle to any point on its boundary.
• It is a fundamental measure used to calculate other properties of the circle.

Formula example: If the diameter is known, the radius is:

r = d / 2

Diameter (d)

• The diameter is the longest straight line that passes through the center of the circle, touching two points on its boundary.

Diameter is twice the radius:

d = 2 * r

Circumference (C)

• The circumference is the total distance around the circle.

It is calculated using the formula:

C = 2 * r * π = 2 * d

Area (A)

• The area is the amount of space enclosed by the circle.

It is given by the formula:

A = r² * π

Chord

• A chord is a straight line connecting two points on the circumference.
• The longest chord in a circle is the diameter.

Arc

• An arc is a portion of the circumference.
• A minor arc is smaller than half the circumference, while a major arc is larger than half.

Sector

• A sector is a region enclosed by two radii and the arc between them (like a slice of pizza).

Its area can be calculated as:

A_sector = (θ/360°)  * π * r²

where θ is the central angle in degrees.

Segment

• A segment is the area between a chord and the corresponding arc.
• The major segment is the larger part, and the minor segment is the smaller part.

Tangent

• A tangent is a straight line that touches the circle at exactly one point.
• It is always perpendicular to the radius at the point of contact.

Secant

• A secant is a straight line that intersects the circle at two points.

These parts form the basis of many geometric and real-world applications involving circles, such as in engineering, physics, and design.

A Short History of Calculating the Area of a Circle

The problem of calculating the area of a circle has fascinated mathematicians for thousands of years. Ancient civilizations developed various approximations and methods, gradually refining their understanding over time.

Ancient Egypt and Babylon (c. 2000 BCE - 1500 BCE)

The Egyptians and Babylonians had early approximations for the area of a circle. The Rhind Mathematical Papyrus (c. 1650 BCE) suggests that Egyptians used a formula similar to:

A ≈ [( 8 / 9) * d]²

where d is the diameter. This gives an approximation of π ≈ 3.16, which is quite close to the actual value.

The Babylonians, around the same period, approximated π as 3, using it to estimate circular areas.

Ancient Greece (c. 500 BCE - 200 CE)

Greek mathematicians made significant advancements in understanding circles:

• Anaxagoras (5th century BCE) explored the concept of squaring the circle, trying to construct a square with the same area as a given circle using only a compass and straightedge.
• Hippocrates of Chios (c. 440 BCE) showed that certain curved areas could be compared to squares, an early step in integral calculus.
• Archimedes (c. 250 BCE) provided the most accurate approximation of π for his time. He inscribed and circumscribed polygons around a circle, narrowing π between 3.1408 and 3.1428. He also derived the area formula:

A = π * r²

though he expressed it geometrically rather than algebraically.

India and China (c. 400 CE - 1200 CE)

• Indian mathematicians, such as Aryabhata (5th century CE), used approximations of π and methods resembling calculus.
• Brahmagupta (7th century CE) contributed to understanding circular segments.
• Chinese mathematicians, including Zu Chongzhi (5th century CE), improved the value of π to 355/113 (accurate to six decimal places).

Islamic Golden Age (c. 800 CE - 1400 CE)

Mathematicians like Al-Khwarizmi and Al-Tusi built on Greek and Indian knowledge, further refining the concept of π and circle calculations.

Renaissance to Modern Mathematics (1500 CE - Present)

• The development of calculus in the 17th century by Newton and Leibniz allowed for precise area calculations.
• The symbol π was introduced by William Jones in 1706 and popularized by Euler in the 18th century.
• Today, the formula A = πr² is a fundamental part of mathematics and engineering, with π known to trillions of decimal places.

From ancient approximations to modern precision, the journey of calculating a circle’s area reflects humanity’s progress in mathematics and science.