C= 2 * R.
In the equality, the area of the O = } R X C,

substitute 2 . R for C;
then the area of the O= } R X 2 7 R,

=ī R.
That is, the area of a 0 = times the square on its radius.

382. Cor. 2. The area of a sector equals the product of its radius by its arc; for the sector is such part of the circle as its arc is of the circumference.

383. DEF. In different circles similar arcs, similar sectors, and similar segments, are such as correspond to equal angles at the centre.

PROPOSITION XI. THEOREM. 384. Two circles are to each other as the squares on their radii.

We are to prove

Let R and R' be the radii of the two circles Q and Q'.

@ _R2

Ō = R2
Now
Q=R2,

§ 381
(the area of a O = times the square on its radius),
and
Q=0 R2

§ 381 Then

@ R2 R2
ē = + R2 = R'2'

Q. E. D. 385. COROLLARY. Similar arcs, being like parts of their respective circumferences, are to each other as their radiï ; similar sectors, being like parts of their respective circles, are to each other as the squares on their radii.

PROPOSITION XII. THEOREM.

386. Similar segments are to each other as the squares on their radii.

§ 163

AL...-----------B

AL------------- B
PI

P Let A C and A' C' be the radii of the two similar segments A B P and A' B' P'.

1. ABP A 02
We are to prove

A' B' P A' C2
The sectors A C B and A' C' B' are similar, § 383

(having the is at the centre, C and C", equal).
In the A AC B and A'C' B'
LC=LC',

§ 383 (being corresponding 5 of similar sectors).

AC=CB,
A' C' = C'B';

§ 163 .. the A AC B and A' C' B' are similar, § 284 (having an 2 of the one equal to an Zof the other, and the including sides

proportional).
Now

sector ACB À C2
sector A' C'

B AT 02
(similar sectors are to each other as the squares on their radii);

Δ Α Β A CO2
and

$ 342 A A'C' B' AT C2? (similar $ are to each other as the squares on their homologous sides). Hence sector AC B-A ACB AC?

sector A' C' B' — A A'C' B' A' C12
or,
segment A BPĀ C2

$ 271 segment A' B'

P ATC2 (if two quantities be increased or diminished by like parts of each, the results will be in the same ratio as the quantities themselves).

Q. E. D.

§ 385

EXERCISES.

1. Show that an equilateral polygon circumscribed about a circle is regular if the number of its sides be odd.

2. Show that an equiangular polygon inscribed in a circle is regular if the number of its sides be odd.

3. Show that any equiangular polygon circumscribed about a circle is regular.

4. Show that the side of a circumscribed equilateral triangle is double the side of an inscribed equilateral triangle.

5. Show that the area of a regular inscribed hexagon is three-fourths of that of the regular circumscribed hexagon.

6. Show that the area of a regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.

7. Show that the area of a regular inscribed octagon is equal to that of a rectangle whose adjacent sides are equal to the sides of the inscribed and circumscribed squares.

8. Show that the area of a regular inscribed dodecagon is equal to three times the square on the radius.

9. Given the diameter of a circle 50; find the area of the circle. Also, find the area of a sector of 80° of this circle.

10. Three equal circles touch each other externally and thus inclose one acre of ground; find the radius in rods of each of these circles.

11. Show that in two circles of different radii, angles at the centres subtended by arcs of equal length are to each other inversely as the radii.

12. Show that the square on the side of a regular inscribed pentagon, minus the square on the side of a regular inscribed decagon, is equal to the square on the radius.

ON CONSTRUCTIONS.

PROPOSITION XIII. PROBLEM. 387. To inscribe a regular polygon of any number of sides in a given circle.

Let Q be the given circle, and n the number of sides

of the polygon.

It is required to inscribe in Q, a regular polygon having n sides.

Divide the circumference of the O into n equal arcs.

Join the extremities of these arcs.

Then we have the polygon required.

$ 181

For the polygon is equilateral,
(in the same O equal arcs are subtended by equal chords);

and the polygon is also regular,
(an equilateral polygon inscribed in a О is regular).

§ 364

Q. E. F.

PROPOSITION XIV. PROBLEM.

388. To inscribe in a given circle a regular polygon which has double the number of sides of a given inscribed regular polygon.

E

Let A B C D be the given inscribed polygon.

It is required to inscribe a regular polygon having double the number of sides of A B C D. Bisect the arcs A B, BC, etc.

Draw A E, EB, BF, etc.,
The polygon A E B FC, etc., is the polygon required.
For the chords A B, BC, etc., are equal, § 363

(being sides of a regular polygon).
.-. the arcs A B, BC, etc., are equal,

§ 182 (in the same O equal chords subtend equal arcs).

Hence the halves of these arcs are equal,
or, AE, EB, BF, FC, etc., are equal ;
.. the polygon A EB F, etc., is equilateral.
The polygon is also regular,

§ 364 (an equilateral polygon inscribed in a is regular); and has double the number of sides of the given regular polygon.

Q. E. F.