Radius is the connecting measurement
The radius r runs from the centre to the circumference. The diameter d passes through the centre from one side to the other, so d = 2r. A circle is completely defined by either one. Once the radius is known, circumference and area follow.
Circumference C = 2πr = πdArea A = πr²Circumference is a length around the edge, so it uses linear units. Area fills the interior and uses square units. A circle with radius 5 cm has diameter 10 cm, circumference 10π ≈ 31.416 cm and area 25π ≈ 78.540 cm².
Chord, arc and central angle
A chord is a straight line joining any two points on the circle. The diameter is the longest possible chord. An arc is the curved part of the circumference between two points. The angle formed by radii to those endpoints is the central angle θ.
Angles in calculator inputs are normally shown in degrees, but arc formulas use radians internally. Convert degrees by multiplying by π/180. A complete turn is 360° or 2π radians.
Arc length L = rθChord length c = 2r sin(θ/2)Worked example: a 60° arc on a radius-12 circle
Convert 60° to π/3 radians. Then:
L = 12 × π/3 = 4π ≈ 12.566c = 2(12)sin30° = 12The chord equals the radius in this special case, making an equilateral triangle with the two radii.
Sector: the slice that includes the centre
A sector is bounded by two radii and their arc. It takes the same fraction of the full circle as its central angle takes of 360°. Its area can be written using degrees or radians.
Sector area = (θ° / 360°)πr² = ½r²θ radiansFor r = 12 and θ = 60°, the sector is one-sixth of the circle, so its area is 24π ≈ 75.398 square units. The Sector & Arc Calculator connects radius, angle, arc length, chord and sector area.
Segment: the region between chord and arc
A circular segment is not the same as a sector. It is bounded by a chord and its arc, excluding the triangle between the chord and the centre. Subtract that triangle from the sector.
Segment area = ½r²(θ − sin θ)The segment’s rise from the chord to the arc is often called its sagitta or segment height h. For a minor segment, h = r[1 − cos(θ/2)]. This geometry appears in arches, part-full pipes and horizontal tanks. Explore it in the Circular Segment Calculator.
Annulus: the ring between two circles
An annulus is the region inside an outer circle of radius R but outside a concentric inner circle of radius r. Subtract the inner area from the outer area.
Annulus area = π(R² − r²)With outer radius 8 and inner radius 5, the area is π(64 − 25) = 39π ≈ 122.522 square units. This is useful for washers, gaskets, tracks and the cross-section of pipe wall material. The Annulus Calculator draws both radii in proportion.
Relationships worth checking visually
- The radius is always half the diameter.
- A chord cannot be longer than the diameter.
- Arc length is at least as long as the chord between the same endpoints.
- A minor sector is larger than its corresponding minor segment because the sector also contains the central triangle.
- As a segment height approaches the diameter, the segment approaches the full circle.
- An annulus requires an outer radius greater than its inner radius.
A diagram will not prove the arithmetic, but it makes a radius/diameter mix-up or an unexpected major segment much easier to notice.
Working backwards from area or circumference
Circle problems often give the quantity that is easiest to measure rather than the radius. Rearrange before substituting. From circumference, r = C/(2π). From area, r = √(A/π). For a measured circumference of 100 mm, the radius is 100/(2π) ≈ 15.915 mm and the diameter is approximately 31.831 mm.
If the area is 2000 mm², r = √(2000/π) ≈ 25.231 mm. Keep the calculator’s internal precision through the reverse step; rounding the radius too early changes every later result.
A major arc or segment uses the remainder of the full circle. If a minor segment subtends angle θ, the corresponding major segment covers 2π − θ radians, and its area is the full circle area minus the minor segment area. State which region is intended rather than relying on a drawing orientation alone.
Choosing the right circle calculator
Use the Circle Calculator for radius, diameter, circumference and area. Use the sector tool when the centre and angle matter, the segment tool when a chord cuts off a region, and the annulus tool for a ring. For part-full cylindrical capacity, continue to the horizontal tank guide and Tank Fill Calculator.
Approximation, rounding and unit behaviour are explained in Methodology & Accuracy.