Arc - Math Is Fun

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Math is Fun Arc

Part of the circumference of a circle. Or part of any curve.

See an arc in action (drag the points):

images/circle-prop.js?mode=arc

Arc Length

circular sector arc length

The arc length is:

L = θ × r (when θ is in radians)

L = θ × π180 × r (when θ is in degrees)

images/circle-prop.js?mode=arclen

Finding the Radius from Width and Height

Imagine we know the width and height of an arc (maybe it is on the top of a door) and we want to make that arc using some string and a pencil ... what radius do we use?

arc width and height

radius = height2 + width28 × height

Example: Sam loves this double door, and wants to make one just like it.

door arch

The door width is 1500mm, the side height is 1950mm and total height at center is 2200mm, so:

  • The arc width is 1500mm
  • The arc height is 2200 − 1950 = 250mm

Sam calculates the arc radius

radius = 2502 + 150028 × 250

radius = 125 + 1125 = 1250

And it looks like this:

arc width and height result

Now Sam can mark out and cut the wood.

Intersecting chords arc height length radius

Note: the formula comes from the Intersecting Chords Theorem, so h(2r-h) = (w/2)(w/2), can you work out the rest?

Circle Builder's Math Geometry Index Copyright © 2025 Rod Pierce

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