11. The areas of circles are to one another as the squares of their diameters. For they are to one another as the similar elementary triangles into which they are divided, and these are as the squares of the radii.
12. The circumferences of circles are as their diameters (Cor. 1).
13. The circumference of sectors having equal central angles are proportional to their radii. Hence if a, a′ denote the arcs of two sectors, which subtend equal angles at the centres, and if r, r′ be their radii,
=
.
14. The area of a circle is equal to half the rectangle contained by the circumference and the radius. This is evident by dividing the circle into elementary triangles, as in Ex. 8.
15. The area of a sector of a circle is equal to half the rectangle contained by the arc of the sector and the radius of the circle.
PROP. XXI.—Theorem.
Rectilineal figures (A, B), which are similar to the same figure (C), are similar to one another.