Calculate the dimensions and area of a circular segment (bow segment) for arches, windows, and curved design elements.
Last updated: March 2026
Note: This calculator provides estimates based on circular geometry. Actual segment properties may vary based on precision requirements, measurement accuracy, and material properties. Always verify critical dimensions before fabrication or installation.
A circular segment (or bow segment) is the region of a circle bounded by an arc and its chord—essentially the "cap" of a circle. It's defined by the radius of the circle and the arc length (the curved distance along the segment's edge).
This calculator helps you find the chord length (the straight line across the segment), the height or sagitta (the distance from the chord to the arc's highest point), the central angle subtended by the arc, and the area of the segment. These calculations are essential for designing arched windows, doorways, bridges, decorative moldings, and any curved architectural element.
All angle calculations use radians internally. The angle is first calculated from the arc length and radius, then used to determine the other properties. Remember to use consistent units throughout.
The sagitta (also called height or rise) is the vertical distance from the center of the chord to the highest point of the arc. If you only have the chord length and sagitta, you can find the radius using: R = (H ÷ 2) + (C² ÷ 8H), where H is height and C is chord length.
Calculate dimensions for an arch with 100-inch radius and 50-inch arc length:
The sagitta is the vertical distance from the center of the chord to the highest point of the arc. It's also called the height or rise of the segment.
A sector is the 'pie slice' of a circle (including the center point), while a segment is just the 'cap' (excluding the center point and bounded only by the arc and chord).
Yes, this is perfect for calculating the dimensions of arched windows, doorways, bridges, and any other curved architectural element based on circular geometry.
You can find the radius using the formula: R = (H ÷ 2) + (C² ÷ 8H), where H is height (sagitta) and C is chord length. Then use that radius to calculate other properties.
The central angle shows how much of the full circle the segment represents. A 90° angle means the segment is 1/4 of the circle, 180° would be a semicircle.
Use the arc length to determine the curved material needed (like molding or trim). Use the area to calculate surface coverage (like panels or glass).
Yes. Angle (radians) = Arc Length ÷ Radius. Multiply by (180/π) to get degrees. This calculator performs that conversion internally.
Use consistent units for both inputs (inches vs. cm). Mixing units will produce incorrect results—switch the unit selector accordingly.
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