Circular sectors, portions, and doubling a circle
Turning circle segments into rectilinear figures and constructing a circle of double area
This geometry sheet works on transforming circular sectors and portions and on doubling a circle. In the first note a circular portion f n is cut away and refitted into the concavity f m so that the pieces form a rectilinear triangle, and a marginal instruction asks that a portion a be made equal to a sector n. A longer construction shows how to draw a circle of double the area of another, using an indefinite line d n, a perpendicular c e, and a compass transfer of the diameter, with an alternative method built on a square. The diagrams show superimposed portions of a circle, a sector, and a circle carrying an inscribed semicircle.
On this page
Refitting a circular portion into a rectilinear triangle
a b and d c are equal; taking b from both leaves d c equal to a. If the portion f n is taken and set back into the concavity f m, the pieces make a rectilinear triangle, with a part of a portion left over at m; the lower triangle is then drawn out from the upper.
Making a portion equal to a sector
A marginal instruction directs that the portion a be made equal to the sector n, part of the sheet's study of exchanging sectors and their portions.
Varying the sector beneath its portion
Labelled b a - d c, this small figure shows a way of varying the sector that lies beneath its portion.
Constructing a circle of double area
To make one circle double another, draw the indefinite straight line d n, raise the perpendicular c e, set the semidiameter a e for the desired circle, then transfer the whole diameter c e with the compass onto e d; the line c d gives a point from which e b is the semidiameter of the doubled circle. An alternative uses a square built on the smaller circle's semidiameter.
