The circumferences of circles are to each other as their radii, and the areas are to each other as the squares of their radii.

Let C and O be the centres of two circles whose radii are CA and OB: then the circumferences are to each other as their radii, and the areas are to each other as the squares of their radii.

M

F

K

For, let similar regular polygons MNPST and EFGKL be inscribed in the circles: then the perimeters of these polygons are to each other as their apothems, and the areas are to each other as the squares of their apothems, whatever may be the number of their sides (P. IX.).

If the number of sides is made infinite (P. X., Sch.), the polygons coincide with the circles, the perimeters with the circumferences, and the apothems with the radii: hence, the circumferences of the circles are to each other as their radii, and the areas are to each other as the squares of the radii; which was to be proved.

Cor. 1. Diameters of circles are proportional to their radii: hence, the circumferences of circles are proportional to their diameters, and the areas are proportional to the squares of the diameters.

Cor. 2. Similar arcs, as AB and DE, are like parts of

parts of the circles to which they belong: hence, similar ares are to each other as their radii,

and similar sectors are to each other as the squares of their radii.

Scholium. The term infinite, used in the proposition, is to be understood in its technical sense. When it is proposed to make the number of sides of the polygons infinite, by the method indicated in the scholium of Proposition X., it is simply meant to express the condition of things, when the inscribed polygons reach their limits; in which case, the difference between the area of either circle and its inscribed polygon, is less than any appreciable quantity. We have seen (P. XII.), that when the number of sides is 16384, the areas differ by less than the millionth part of the measuring unit. By increasing the number of sides, we approximate still nearer.

The

area of a circle is equal to half the product of its circumference and radius.

Let O be the centre of a circle, OC its radius, and ACDE its circumference: then the area of the circle is equal to half the product of the circumference and radius.

For, inscribe in it a regular polygon ACDE. Then the area of this polygon is equal to half the product

F

of its perimeter and apothem, whatever may be the number of its sides (P. VIII).

If the number of sides is made infinite, the polygon coincides with the circle, the perimeter with the circumference, and the apothem with the radius: hence, the area of the circle is equal to half the product of its circumference and radius; which was to be proved.

Cor. 1. The area of a sector is equal to half the product of its arc and radius.

Cor. 2. The area of a sector is to the area of the circle, as the arc of the sector to the circumference.

To find an expression for the area of any circle in terms of its radius.

Let C be the centre of a circle, and CA its radius. Denote its area by area CA, its radius

by R, and the area of a circle whose radius is 1, by # (P. XII., S.).

Then, because the areas of circles are to each other as the squares of their radii (P. XIII.), we have,

That is, the area of any circle is 3.1416 times the square of its radius.

To find an expression for the circumference of a circle, in terms of its radius, or diameter.

Let C be the centre of a circle, and CA its radius.

GEOMETRY.

Denote its circumference by circ. CA, its radius by R, and From the last Proposition, we have,

its diameter by D.

That is, the circumference of any circle is equal to 3.1416 times its diameter.

Scholium 1. The abstract number 7, equal to 3.1416, denotes the number of times that the diameter of a circle is contained in the circumference, and also the number of times that the square constructed on the radius is contained in the area of the circle (P. XV.). Now, it has been proved by the methods of higher mathematics, that the value of is incommensurable with 1; hence, it is impossible to express, by means of numbers, the exact length of a circumference in terms of the radius, or the exact area in terms of the square described on the radius. It is not possible, therefore, to square the circle-that is, to construct a square whose area shall be exactly equal to that of the circle.

are

Scholium 2. Besides the approximate value of ", 3.1416, usually employed, the fractions and also sometimes used to express the ratio of the diameter to the circumference.

EXERCISES.

1. The side of an equilateral triangle inscribed in a circle is 6 feet; find the radius of the circle.

2. The radius of a circle is 10 feet; find the apothem of a regular inscribed hexagon.

3. Find the side of a square inscribed in a circle whose radius is 5 feet.

4. Draw a line whose length shall be √3.

5. The radius of a circle is 4 feet; find the area of an inscribed equilateral triangle.

6. Show that the sums of the alternate angles of an octagon inscribed in a circle are equal to each other.

7. The area of a regular hexagon, whose side is 20 feet, is 1039.23 square feet; find the apothem.

8. One side of a regular decagon is 20 feet, and its apothem 15.4 feet; find the perimeter and the area of a similar decagon whose apothem is 8 feet.

9. The area of a regular hexagon inscribed in a circle is 9 square feet, and the area of a similar circumscribed hexagon is 12 square feet; find the areas of regular inscribed and circumscribed polygons of 12 sides.

10. Given two diagonals of a regular pentagon that intersect; show that the greater segments will be equal to each other and to a side of the pentagon, and that the diagonals cut each other in extreme and mean ratio.

11. Show how to inscribe in a given

circle a regular polygon of 15 sides.

12. Find the side and the altitude of an equilateral triangle in terms of the radius of the inscribed circle.