Geometry and layout calculator
Circle and Arc Calculator
Calculate a full circle, solve an arc from its angle or chord, or work backward from a chord and segment rise. I will keep the arc, chord, sector, and segment from exchanging name tags while you are not looking.
Sage’s guided lesson
A chord cuts straight across; an arc takes the scenic route
Circle problems become much easier when every piece keeps its proper name. The radius reaches from center to edge, the chord joins two edge points, the arc follows the curve, and the sagitta measures how far that curve rises away from the chord.
Give each line the correct job
Circle vocabulary prevents formula mix ups
The radius runs from the center to the circumference. The diameter crosses the center and equals two radii. The circumference is the full distance around the circle.
A chord is any straight line joining two points on the circumference. The diameter is the longest possible chord. The arc is the curved portion between those same endpoints.
A sector is the wedge bounded by two radii and an arc. A circular segment is the cap bounded by a chord and an arc. The segment height, called the sagitta, is the perpendicular rise from the chord midpoint to the arc.
When measuring an existing part, write down whether the tape followed the curve or stretched straight between endpoints. Arc and chord can be close on a shallow curve, but they are never the same except at zero length.
Pi handles the full circle; the angle chooses the portion
How to calculate circle and arc values yourself
Circumference = 2πr = πd
Circle area = πr²
Arc length = 2πr × angle ÷ 360°
Chord = 2r × sin(angle ÷ 2)
Sagitta = r × [1 − cos(angle ÷ 2)]
Worked example, a 90 degree arc with a 10 inch radius
Full circumference = 2 × π × 10 = 62.8319 in
Arc length = 62.8319 × 90 ÷ 360 = 15.7080 in
Chord = 2 × 10 × sin(45°) = 14.1421 in
Sagitta = 10 × [1 − cos(45°)] = 2.9289 in
The sector area is the same fraction of the full circle area as the angle is of 360 degrees. The circular segment area is the sector area minus the triangle formed by the two radii and the chord.
I know this is off topic, but this reminds me of a joke I heard when teaching this to one of my interns, What do you call a sheep with no legs? A cloud. Yeah, I don’t see the connection either, but I think it is cute!
The little rise can reveal a very large radius
Chord and sagitta are often the easiest measurements on a real arc
Large radii are difficult to measure directly because the center may be far outside the part. A chord and sagitta let you work backward without locating that center.
Radius = chord² ÷ (8 × sagitta) + sagitta ÷ 2
Worked example, 48 inch chord with 3 inch rise
Radius = 48² ÷ (8 × 3) + 3 ÷ 2
Radius = 2304 ÷ 24 + 1.5 = 97.5 in
This method is useful for arches, bowed rails, curved walls, road crowns, tank sections, templates, windows, and checking whether a bent strip matches a drawing.
If the sagitta is less than or equal to half the chord, the measured cap is the minor segment. A larger rise describes the major segment, the long way around the circle. The calculator identifies that automatically in the chord and rise mode.
When the rise is very small, a tiny sagitta error can change the calculated radius substantially. Use the longest practical chord and measure the rise carefully at its true midpoint.
A unit designed by the circle itself
Radians make the arc length formula almost suspiciously simple
One radian is the central angle that cuts an arc equal in length to the radius. Since one complete circumference contains 2π radii, a full circle contains 2π radians.
Arc length = radius × angle in radians
Radians = degrees × π ÷ 180
One radian is about 57.2958 degrees. A 180 degree semicircle is π radians, and a 90 degree quarter circle is π ÷ 2 radians.
Degrees are convenient for dividing a revolution into familiar pieces. Radians are natural for formulas because the radius and arc length speak the same unit directly.
Lay one radius length along the circumference and the angle at the center is exactly one radian. The circle is measuring itself.
The center may be across the room
Large arcs can be drawn with a trammel, a template, or three points
A beam compass or trammel uses a long bar with a pivot at one end and a pencil or cutter at the other. It is the large scale version of an ordinary compass.
When the center is inaccessible, a full size template or a flexible batten can define the curve from measured offsets. A circular arc is uniquely determined by three noncollinear points, although finding the center from those points requires constructing the perpendicular bisectors of two chords.
For repeated work, store the curve as a physical template, coordinate table, CNC program, or a series of offsets from one datum. Recreating a shallow arc from a short chord and a thick pencil line invites the radius to become a matter of opinion.
In the same circle, two equal chord lengths subtend equal angles and equal minor arcs. This gives you a practical way to divide a circumference without reading angles directly.
Circles helped invent better mathematics
Pi, chord tables, and astronomy turned circle geometry into trigonometry
Pi is the constant ratio of a circle’s circumference to its diameter. It is irrational, so its decimal digits continue without ending or repeating. For practical work, the number of digits needed depends on the size of the circle and the required tolerance, not on how dramatic the calculator display can look.
Archimedes bounded pi by comparing polygons drawn inside and outside a circle. The method replaced one difficult curve with many straight sides and tightened the estimate as the number of sides increased.
Long before modern sine buttons, ancient astronomers used tables of chords. Hipparchus is credited with an early chord table in the second century BCE, and Ptolemy later developed extensive chord calculations for astronomy. The modern chord formula, 2r sin(θ ÷ 2), still reveals that ancestry.
The symbol π came much later than the geometry. The circle was useful for surveying, wheels, gears, astronomy, navigation, architecture, and timekeeping for centuries before anyone could type pi into a phone.
Using 3.14159265 instead of 3.14 matters only when the circle size and tolerance make the difference meaningful. Precision should answer the job, not decorate the answer.
Perfect geometry meets imperfect material
Practical fabrication limits and checks
- Confirm whether the drawing dimension is radius or diameter.
- Confirm whether the measured distance is an arc or a chord.
- Measure sagitta perpendicular to the chord at its midpoint.
- Use a longer chord when checking a shallow arc, provided the endpoints remain on the intended circle.
- Keep enough precision through trigonometric steps, then round the final layout value.
- Account for cutter radius, kerf, material thickness, and the side of the line being cut.
- Expect bent material to spring back after the force or form is removed.
- Use a full size template or independent radius check before producing repeated parts.
Sheet metal bending may depend on the neutral axis rather than the inside or outside surface. Laminated wood, plastic, tubing, and spring materials also respond differently to bending. This calculator solves the intended circle; it does not predict the material’s decision about participating.
Use the calculated radius to predict both the chord and sagitta. If both match the physical arc, the geometry has passed a stronger test than either measurement alone.
The curve has identified itself
The chord stayed straight, the arc took the long route, and the sagitta finally received proper credit.
You now know how full circles, sectors, segments, radians, chords, and rise measurements connect.
That earns cheese fries. I tried arranging them as a perfect circle, but one became a chord and was immediately eaten for violating the layout.
Quick answers
Circle and arc questions
What is the difference between an arc and a chord?
An arc follows the curved circumference between two points. A chord is the straight line joining those points.
What is sagitta or segment height?
It is the maximum perpendicular distance from a chord to its arc. It may also be called the segment rise or crown.
How do I find arc length from radius and angle?
Multiply the radius by the angle in radians, or multiply the full circumference by the angle divided by 360 degrees.
How do I find radius from chord and sagitta?
Use radius = chord² ÷ (8 × sagitta) + sagitta ÷ 2.
Can one chord describe two different arcs?
Yes. The same chord belongs to a minor arc and a major arc, so the intended side of the circle must be identified.
What is a radian?
It is the angle that cuts an arc equal to the radius. One radian is about 57.2958 degrees.
Does the calculator account for material springback?
No. It solves ideal geometry. Material behavior, thickness, neutral axis, cutter radius, and fabrication tolerance require separate consideration.