Circular Segment / Arch Area

Circular Segment Square Footage Calculator

Enter your measurements and get the area instantly — in square feet, yards, meters and more. Add a price to estimate material cost for flooring, paint or tile.

Open Calculator Read the Guide
Circular Segmentarea
Area = ½ r² (θ − sin θ)
Circular Segment
Area = ½ × r² × (θ − sin θ)
FormulaArea = ½ × r² × (θ − sin θ)
Need material? Add a depthMulch, gravel, topsoil, sand or concrete — get cubic yards & bags ▾
Chord Height or radius + angle Area = ½ r² (θ − sin θ)

How to Calculate the Square Footage of a Circular Segment

A circular segment is the region between a straight line (the chord) and the curved arc of a circle — picture slicing a thin curved sliver off the edge of a pie, not a full wedge from the center. It is the shape of the floor under a curved bay window, a rounded patio extension, or an arched alcove. Because the boundary is part straight and part curved, it needs more than a simple length × width, but the calculator handles it from measurements you can actually take on site.

Two ways to measure a segment

Most people have one of two pairs of measurements, so the calculator offers both input methods:

  • Radius and central angle: If you know the radius of the underlying circle and the angle the segment spans, the area is ½ × r² × (θ − sin θ), with θ in radians (the calculator converts from degrees).
  • Chord and height: More practical on site — measure the straight chord across the opening and the height (how far the arc bulges out from the chord at its deepest). The calculator derives the radius and angle from these, then applies the same formula.

Measuring a segment in the real world

The chord-and-height method is almost always easier for a room or patio. Stretch a tape straight across the mouth of the curved area — that is the chord. Then measure from the midpoint of that line to the furthest point of the curve — that is the height, sometimes called the sagitta. These two numbers are enough; you do not need to find the circle's center, which is usually off in mid-air anyway.

Worked example

Consider a curved bay window with a chord of 10 feet across the opening and the curve bulging out 2 feet at its deepest.

From these, the underlying radius works out to 7.25 feet, and the segment area comes to about 13.75 square feet of floor inside the bay. That is the extra floor area the bay adds beyond the straight wall line.

Where circular segments appear

  • Bay and bow windows: The floor area inside a curved window projection is a segment.
  • Rounded patios and decks: A patio with one curved edge adds a segment to the rectangular part.
  • Arched alcoves and niches: The floor or surface of a curved recess.
  • Pool and pond edges: Rounded extensions on otherwise straight-sided features.

Related shape calculators

If you need the full pie-slice from the center rather than the outer sliver, use the sector calculator. For a half circle use the semicircle calculator, and for the whole round area the circle calculator. A curved patio is often a rectangle plus a segment.

Frequently Asked Questions

What is a circular segment?
It is the region between a straight chord and the arc of a circle — the curved sliver at the edge of a circle, such as the floor inside a bay window or under a rounded patio edge.
Which measurements do I need for a segment?
Either the radius and the central angle, or — more practically — the chord (the straight distance across the opening) and the height (how far the curve bulges out from the chord). Choose the matching input method.
What is the segment height or sagitta?
It is the distance from the midpoint of the chord to the furthest point of the arc. It tells the calculator how deeply the curve bows out, which together with the chord determines the area.
What is the formula for a circular segment?
With radius r and central angle θ in radians, Area = ½ × r² × (θ − sin θ). When you supply chord and height, the calculator first derives r and θ, then applies the same formula.
How do I measure a bay window floor?
Use the chord-and-height method: measure straight across the window opening for the chord, then from the middle of that line to the deepest point of the curve for the height.
Do I need to find the circle's center?
No. The chord-and-height method avoids it entirely, which is helpful because the center is usually a point in empty space outside the room.
How many square feet does a 10 ft chord, 2 ft deep bay add?
About 13.75 square feet of additional floor area.
Can I add a segment to a rectangular patio total?
Yes. Calculate the rectangle and the segment separately, then add them for the total area of a patio with one curved end.