A circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.
What makes a circle a circle: every point on it is exactly one radius from the center. All the other facts follow from that.
Formal definition and notation
Given a point O in a plane and a positive numberr, the circle with center O and radius r is the set of all points P in the plane with OP = r. On the coordinate plane, the circle with center (h, k) and radius r is the graph of (x − h)2 + (y − k)2 = r2. Strictly, the circle is only the curve; the region inside is a disk.
Symbol
Read it as
Meaning
⊙O
circle O
the circle whose center is point O
r, d
r, d
radius and diameter, with d = 2r
π ≈ 3.14
pi
the ratio of circumference to diameter, the same for every circle
Key facts
Every radius of a circle has the same length, and every diameter has the same length.
A circle has no sides and no corners (vertices). It is not a polygon.
A circle has infinitely many lines of symmetry: every line through the center.
A chord joins two points on the circle; the diameter is the longest chord.
A full turn around the center is 360°, which is why angles are measured in a circle of degrees.
All circles are similar: any circle is a scaled copy of any other.
Circle formulas
Circle formulas
d = 2r, C = 2π r = π d, A = π r2
r
radius: center to the circle
d
diameter: across the circle through the center
C
circumference: distance around (a length)
A
area: space inside (square units)
Circumference is measured in units such as cm; area is in square units such as cm2.
Try it: change the radius
Interactive
Drag the radius or use the slider. Watch the diameter, circumference and area update, and notice that area grows much faster than circumference.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Circle examples, step by step
Example 1: Radius from diameter
Warm-up
A round table top has a diameter of 54 inches. What is its radius?
The diameter is twice the radius: d = 2r.
r = 54 ÷ 2 = 27.
Answer: The radius is 27 inches.
Example 2: Circumference and area
Core
A circle has radius 9 cm. Find its circumference and area in terms of π, then to the nearest tenth.
C = 2π r = 2π(9) = 18π.
18π ≈ 56.5 cm.
A = π r2 = π(9)2 = 81π.
81π ≈ 254.5cm2.
Answer:C = 18π ≈ 56.5 cm and A = 81π ≈ 254.5square cm.
Example 3: Point on the circle?
Test-style
A circle on the coordinate plane has center (0, 0) and passes through (6, 8). Does it pass through (−10, 0)?
The radius is the distance from the center to (6, 8): √62 + 82 = √100 = 10.
The distance from (0, 0) to (−10, 0) is 10.
Equal distance from the center means the point is on the circle.
Answer: Yes. Both points are 10 units from the center, so (−10, 0) is on the circle.
Common mistakes
✗ MistakeUsing the diameter as r: a circle with diameter 12 has area π(12)2 = 144π.
✓ Insteadr = 6, so A = π(6)2 = 36π.
The area and circumference formulas with r need the radius. Halve the diameter first.
✗ MistakeMixing up circumference and area: C = π r2.
✓ InsteadC = 2π r (a length) and A = π r2 (square units).
The squared formula gives square units, so it must be the area. Circumference is a distance.
✗ MistakeCalling an oval a circle.
✓ InsteadAn oval (ellipse) is wider one way than the other. In a circle, every diameter is the same length.
Being round is not enough: the distance from the center must be the same in every direction.
Not to be confused with
Circlevs.Sphere
A sphere is the 3-D version: all points in space at one distance from a center, like a ball. A circle is flat, like the outline of a coin.
Circlevs.Disk
Strictly, the circle is only the round curve; the disk is the curve plus the inside. Everyday speech and most tests use circle for both, as in area of a circle.
Quick check: 3 questions
Question 1 of 3
A circle has a radius of 7 m. What is its diameter?
Show answer and explanation
Answer: B.d = 2r = 14 m. 3.5 m halves instead of doubling, 49 m squares the radius, and 21.98 m is π × 7, half the circumference.
Question 2 of 3
What is the area of a circle with diameter 10 ft? Use π ≈ 3.14.
Show answer and explanation
Answer: C.r = 5, so A = 3.14 × 25 = 78.5ft2. 314 uses the diameter as the radius, 31.4 is the circumference, and 15.7 is half the circumference.
Question 3 of 3
Which statement is true for every circle?
Show answer and explanation
Answer: B. Equal radii is the definition of a circle. A circle has infinitely many lines of symmetry, d = 2r (not r2), and for r = 1 the area π is less than the circumference 2π.
Circle: frequently asked questions
Is a circle a polygon?
No. A polygon is a closed figure made of straight line segments, with a finite number of sides and corners. A circle is a smooth curve with no straight sides and no corners. You can approximate a circle with polygons that have more and more sides, which is how Archimedes estimated π, but no polygon is ever exactly a circle.
What is the difference between a circle and a sphere?
A circle is two-dimensional: a flat curve in a plane, such as the rim of a plate. A sphere is three-dimensional: the surface of a ball, every point the same distance from the center in space. A circle has circumference and area; a sphere has surface area and volume. Slicing a sphere through its center gives a circle.
How many degrees are in a circle?
A full circle is 360°. Turning all the way around the center returns you to where you started, so central angles around one point add to 360°. Half a circle is 180° and a quarter circle is 90°. In higher math, the same full turn is also measured as 2π radians.
RadiusThe radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.Geometry · Grade 6, Grade 7, Grade 8DiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7, Grade 8CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5SymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometryChordA chord is a line segment whose two endpoints both lie on a circle. The diameter is the longest chord, because it passes through the center. A chord at distance d from the center of a circle with radius r has length 2√r2 − d2, so chords closer to the center are longer.Geometry · Grade 7, GeometryDegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3SphereA sphere is a perfectly round solid shape, like a ball, in which every point on the surface is the same distance from the center. That distance is the radius. A sphere has no faces, edges or vertices. Its volume is V = 43π r3 and its surface area is 4π r2.Geometry · Kindergarten, Grade 8, GeometryArea of a CircleThe area of a circle is the amount of flat space inside the circle, measured in square units. It equals pi times the radius squared: A = π r2. A circle with radius 5 cm has area 25π, about 78.5 square centimeters. If you know the diameter, halve it first to get the radius.Geometry · Grade 7, Grade 8, GeometryPointA point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.Geometry · Grade 4, Grade 5, Grade 6PiPi (π) is the number you get when you divide any circle’s circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 227.Geometry · Grade 7, Grade 8, GeometryCentral AngleA central angle is an angle whose vertex is the center of a circle and whose sides are two radii. It cuts off an arc, and the arc’s degree measure equals the central angle. A 90° central angle marks off a quarter of the circle, and the central angles around the center add to 360 degrees.Geometry · Grade 7, GeometryArc LengthArc length is the distance along a curved part of a circle, measured in units of length. An arc with central angle θ in a circle of radius r is θ360 of the circumference: s = θ360 × 2π r when θ is in degrees, or simply s = rθ when θ is in radians.Geometry · Geometry, PrecalculusEquation of a CircleThe equation of a circle with center (h, k) and radius r is (x − h)2 + (y − k)2 = r2. It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x2 + y2 = r2.Geometry · Geometry, Algebra 2, Precalculus