diameter is 1, we may find the side of a trigon drawn in any other circle, by multiplying its diameter by the number which expresses the side of the trigon inscribed in the circle whose diameter is unity; or, if we divide the side of a trigon by said number, the quotient will be the diameter of the circumscribing circle.

The diameter of a circle being given, to find the side of an equal square:

Multiply the diameter by the square root of .7854, viz. by .886227. Or, the circumference being given, to find the side of an equal square-Multiply the circumference by .282094.

To find the side of an octagon drawn in a circle, its diameter being given:

Multiply the diameter by 0.3826, and the product will be the side of the inscribed octagon : or, if we divide the side of an octagon by .3826, the quotient will be the diameter of the circumscribing circle.

To find the diameter of a circle in which you may inscribe a given square, or a given rectangle :

Extract the square root of the sum of the squares of the length and breadth.

EXAMPLES.

1. What is the diameter of a eircle, in which you may inscribe a rectangle 25 inches long and 15 broad?

Ans. 29.15 inches.

2. What must be the diameter of a round log, from which you may hew a stick of timber 12 inches by 20?

Ans. 23.324 inches.

3. What is the side of a square whose area is equal to that of a circle whose diameter is 20 rods?

Ans. 17.72454 rods.

4. What is the diameter of a circle, in which you can inscribe a trigon 30 inches on each side? Ans. 34.64105 inches.

5. What is the diameter of a round log, from which you may form an octagonal prism, or column of eight equal sides, each side being 10 inches? Ans. 26.1367 inches.

23. CIRCULAR ANNULUS.

The circular annulus is the

space
included between the cir-
cumferences of two concentric
circles. The difference between
the areas of the two circles
will manifestly be the area of
the circular ring. Therefore,
to find the area of a circular
annulus:-

Multiply the sum of their diameters by their difference, and

this product by .7854, and the result will be the area.

Or,

Multiply the sum of their circumferences by their difference,

and this product by .0796, and the result will be the area.

EXAMPLES.

1. What is the area of a gravelled walk 8.25 rods wide, extending round a circular fish-pond 10 rods in diameter ? Ans. 16.4934 square rods.

2. The circumferences of two concentric circles are 62.832 and 37.6992; required the area of the annulus contained between them. Ans. 201.0624.

3. If the internal diameter of one of Saturn's rings is 30,000 miles, and the external diameter is 50,000 miles; what is the area of one side of the ring, supposing it to be a flat surface?

Ans. 1,256,640,000 square miles.

T24. CIRCULAR SECTOR.

α

b

A sector is a part of a circle bounded by two radii and an arc. Its area is equal to that of a triangle whose base is the length of the arc, and whose perpendicular altitude is equal to the radius of the circle of which the sector is a part. Therefore, to find the area of a sector:

Multiply the length of the arc by half the radius of the circle of which it is a part. Or, when the number of degrees in the arc is given, to find the area, say :—As 360 degrees are to the number of degrees in the arc, so is the area of the whole circle to the area of the sector.

When the number of degrees in the arc of the sector is given, to find the length of the arc, say :-As 360 degrees are to the number of degrees in the arc, so is the circumference of the circle to the length of the required arc. Or, multiply .0174533 by the number of degrees, and that product by the radius of the circle. Or,

From the sum of the chords of half the arc, (viz. ac and ab,) subtract the chord of the whole arc, (viz. cb,) and add one-third of the remainder to the sum of the chords of half the arc, and the result will be the length of the required arc, nearly.

The height of an arc (as ae in the above figure) is called the versed sine; and that part of the radius between the centre of the circle and the chord, (as de in the figure,) is called the apothegm.

When the chord and versed sine of an arc are given, to find the diameter of the circle:

Divide the square of half the chord by the versed sine, and to the quotient add the versed sine. Or, Divide the square of the chord by four times the versed sine, and to the quotient add the versed sine.

The apothegm and chord being given, to find the radius of the circle:

To the square of the apothegm add the square of half the chord, and extract the square root of the sum of said squares.

The versed sine and chord of an arc being given, to find the chord of the arc :—

To the square of the versed sine add the square of half the chord, and extract the square root of the sum.

EXAMPLES.

1. What is the area of a sector, whose arc is 144.666, and the diameter of the circle, of which it is a part, 144?

Ans. 5207.8.

2. If the chord of half the arc is 126, and the chord of the whole arc 216, what is the length of the arc line?

Ans. 264, nearly.

3. What is the length of an arc of 3 degrees, in a circle whose radius is 50? Ans. 2.618, nearly.

4. What is the area of a sector whose arc is 120 degrees, and the diameter of the circle 226 rods?

Ans. 13371.66 square rods.

5. If the chord of an arc is 173.2, and the versed sine 50, what is the length of the arc? Ans. 208.93, nearly.

6. If the height of an arc be 5.6, and the apothegm 8.4, what is the radius of the circle?

Ans. 14.

7. If the chord of an arc is 12, and the apothegm 10, what is the radius of the circle?

Ans. 11.6619.

8. The versed sine of an arc is 10, and the chord of the arc 24; what is the radius of the circle? Ans. 12.2.

9. What is the diameter, when the versed sine is 1 and the cord 12? Ans. 37.

10. What is the radius of an arc, whose versed sine is 6, and the chord of half the arc 15? Ans. 18.75.

11. If the chord of an arc is 1200, and its versed sine 40, what is the diameter of the circle ? Ans. 9040.

12. If the chord of an arc is 40, and the diameter of the circle 120, how many degrees does it contain?

Ans. 38° 57'.

13. What is the number of degrees in an arc, whose versed sine is 12, and radius 56 ? Ans. 760 25'. 14. What is the length of an arc of 45°, the diameter being 12? Ans. 4.712.

15. The length of a circular arc is 24, and the diameter of the circle 30; what is the area of the sector?

Ans. 180.

A circular segment is a part of a circle bounded by an arc

and a chord.