Arch Calculator

Use this arch calculator to help you determine the focus points of an ellipse. With these, you'll be able to easily draw the perimeter of the rounded part of an elliptical arch.

Focus distance from centre17 7/8 inalong the horizontal axis, √(a² − b²)

Semi-major axis (a)24 inhalf the span

Semi-minor axis (b)16 inthe rise

String length for the two-pin method48 inpin at each focus, loop this long, and trace with a pencil

Eccentricity0.745360 is a circle, approaching 1 is very flat

Arch curve length63 7/16 inhalf the full ellipse perimeter

Span48 in

Rise16 in

Rise as a fraction of span0.333330.5 gives a semicircle, less gives a flatter arch

Area under the arch603.19 sq inhalf the ellipse: π a b ÷ 2

Radius if it were a segmental arch instead26 inthe circular arc through the same three points

The formula

semi-axes a and b; foci at c = √(a² − b²) from the centre

Two pins and a loop of string

An ellipse is the set of points whose distances to two foci add to a constant. That gives the classic trammel: drive a pin at each focus, make a loop of string equal to twice the semi-major axis, and run a pencil round inside it.

When the span is exactly twice the rise the foci coincide at the centre and the ellipse is a circle — which is why a semicircular arch needs no focus construction at all. Flatten it and the foci move apart along the span.

The perimeter has no elementary closed form, so the arc length above uses Ramanujan’s approximation, which is accurate to better than one part in ten million for ordinary arch proportions.