233. Area of a circle.

By Art. 225 the area of a regular polygon of n sides, which is inscribed in a circle of radius R, is

Let now the number of sides of this polygon be indefinitely increased, the polygon always remaining regular.

It is clear that the perimeter of the polygon must more. and more approximate to the circumference of the circle. Hence, when the number of sides of the polygon is infinitely great, the area of the circle must be the same as that of the polygon.

When n is made infinitely great the value of becomes

sin

infinitely small and then, by Art. 228, is unity.

The area of the circle therefore

square of its radius.

234. Area of the sector of a circle.

Ꮎ

TR2

=π times the

Let O be the centre of a circle, AB the bounding arc

of the sector, and let ZAOB=a radians.

By Euc. VI. 33, since sectors are to one another as the arcs on which they stand, we have

1. Find the area of a circle whose circumference is 74 feet.

2. The diameter of a circle is 10 feet; find the area of a sector whose arc is 2210.

3. The area of a certain sector of a circle is 10 square feet; if the radius of the circle be 3 feet, find the angle of the sector.

4. The perimeter of a certain sector of a circle is 10 feet; if the radius of the circle be 3 feet, find the area of the sector.

5. A strip of paper two miles long and ⚫003 of an inch thick is rolled up into a solid cylinder; find approximately the radius of the circular ends of the cylinder.

6. A strip of paper, one mile long, is rolled tightly up into a solid cylinder, the diameter of whose circular ends is 6 inches; find the thickness of the paper.

7. Given two concentric circles of radii and 2r; two parallel tangents to the inner circle cut off an arc from the outer circle; find its length.

8. The circumference of a semicircle is divided into two arcs such that the chord of one is double that of the other. Prove that the sum of the areas of the two segments cut off by these chords is to the area of the semicircle as 27 is to 55.

9. If each of 3 circles, of radius a, touch the other two, prove that

4

the area included between them is nearly equal to

a2.

25

10. Six equal circles, each of radius a, are placed so that each touches two others, their centres being all on the circumference of another circle; prove that the area which they enclose is

2a2 (3/3-π).

11. From the vertex A of a triangle a straight line AD is drawn making an angle with the base and meeting it at D. Prove that the area common to the circumscribing circles of the triangles ABD and ACD is

where ẞ and y are the number of radians in the angles B and C respectively.

235. Dip of the Horizon.

Let O be a point at a distance h above the earth's surface. Draw tangents, such as OT

and OT", to the surface of the earth. The ends of all these tangents all clearly lie on a circle. This circle is called the Offing or Visible Horizon. The angle that each of these tangents OT makes with a horizontal plane POQ is called the Dip of the Horizon.

Let r be the radius of the earth

Q

T

and let B be the other end of the diameter through A. We then have, by Euc. III. 36,

so that

OTOA. OB = h (2r+ h),

OT = √h (2r + h).

This gives an accurate value for OT.

In all practical cases, however, h is very small compared with r.

[r=4000 miles nearly and h is never greater, and generally is very considerably less, than 5 miles.]

Hence h2 is very small compared with hr.
As a close approximation we have then

so that, very approximately, we have the angle

236. Ex.

Taking the radius of the earth as 4000 miles, find the dip at the top of a lighthouse which is 264 feet above the sea and the distance of the offing.

1. Find in degrees, minutes, and seconds the dip of the horizon from the top of a mountain 4400 feet high, the earth's radius being 21 × 106 feet.

2. The lamp of a lighthouse is 196 feet high; how far off can it be seen?

3. If the radius of the earth be 4000 miles, find the height of a balloon when the dip is 1o.

Find also the dip when the balloon is 2 miles high.

4. Prove that, if the height of the place of observation be n feet, the

distance that the observer can see is

3n
miles nearly.
2

5. There are 10 million metres in a quadrant of the earth's circumference. Find approximately the distance at which the top of the Eiffel tower should be visible, its height being 300 metres.

6. Three vertical posts are placed at intervals of a mile along a straight canal each rising to the same height above the surface of the water. The visual line joining the tops of the two extreme posts cuts the middle post at a point 8 inches below its top. Find the radius of the earth to the nearest mile.