PROBLEMS FOR COMPUTATION

523. (1.) Find the number of degrees in an angle of each of the following regular polygons: (a) triangle, (b) pentagon, (c) hexagon, (d) octagon, and (e) decagon.

(2.) What is the area of a regular pentagon inscribed in a circle whose radius is 12 cm.?

(3.) If the side of a regular hexagon is 10 m., find the number of square feet in its area.

(4.) Find the area of a regular octagon inscribed in a circle whose radius is 12 cm.

(5.) If the radius of a circle is R, find the side and the apothem of a regular inscribed (a) triangle, (b) square, (c) hexagon.

(6.) If, in the above example, R= 15.762, find the numer ical value of the side and apothem for each of the three polygons.

(7.) Prove that the side of a regular octagon, inscribed in a circle whose radius is R, is equal to RV2-V2.

(8.) Find the apothem of a regular octagon inscribed in a circle whose radius is R.

(9.) If the radius of a circle is R, find the side of a regular inscribed decagon.

x

R/R

(10.) What is the apothem of the above decagon? (11.) Find the side of a regular hexagon circumscribed about a circle whose radius is R.

(12.) If the radius of a circle is R, prove that the area of a regular inscribed dodecagon is 3R.

(13.) There are three regular hexagons; the side of the first is 20 in., that of the second is I m., that of the third 5 ft. Find in meters the side of a fourth regular hexagon whose area is equal to the sum of the areas of the first three.

(14.) A wheel, having a radius of 1.5 ft., made 3360 revolutions in going over the road from one town to another. How many miles apart are the towns?

(15.) If the circumference of a circle is 50 in., find the radius.

(16.) If a wheel has 35 cogs, and the distance between the middle' points of the cogs is 12 in., find the radius of the wheel.

(17.) Find the width of a ring of metal the outer circumference of which is 88 m. in length, and the inner circumference 66 m.

(18.) If the radius of a circle is 16 cm., how many degrees, minutes, and seconds are there in an arc 10 cm. long?

(19.) Find the number of feet in an arc of 20° if the radius of the circle is 12 m.

(20.) How many degrees are there in an arc whose length is equal to the radius of the circle?

(21.) If an arc of 30° 12.5664 in., find the radius of the circle.

(22.) If the radius of a circle is 15 cm., find the length of the arc subtended by a chord 15 cm. in length.

(23.) If the circumference of a circle is c, find its radius. and diameter.

(24.) Find the area of a circle whose radius is (a) 11 in.; (b) 17.146 m.; (c) 35 ft.

(25.) Find the ratio of the areas of two circles if the radius of one is the diameter of the other.

(26.) If the circumference of a circle is 60 ft., find the

area.

(27.) The radius of a circle is 13 in. Find the side of a square whose area is equal to that of the circle.

(28.) The side of an inscribed square is 23 m. What is the area of the circle?

(29.) What is the area of a circle inscribed in a square whose surface contains 211 ares?

(30.) Find the side of the largest square that can be cut from the cross-section of a tree 14 ft. in circumference.

(31.) If the diameter of a given circle is 5 cm., find the diameter of a circle one-fourth as large.

(32.) A rectangle and a circle have equal perimeters. Find the difference in their areas if the radius of the circle is 9 in. and the width of the rectangle is three-fourths its length.

(33.) If the radius of a circle is 25 m., what is the radius of a concentric circle which divides it into two equivalent parts?

(34.) The radii of two concentric circles are respectively 9 and 6 in. Find the area of the ring bounded by their circumferences.

(35.) The chord of a segment of a circle is 34 in. in length, and the height of the segment is 8 in. Find the radius. (36.) In a circle whose radius is 18 in., find the height of a segment whose chord is 28 in. in length.

(37.) If the radius of a circle is 16 cm., what is the area of a sector having an angle of 24°?

(38.) The radius of a circle is 9 in. Find the area of a segment whose arc is 60°.

Hint.-Area of segment AEBD area of sector AEBC minus area of triangle ABC.

(39.) If the radius of a circle is R, find the area of the segment subtended by the side of a regular hexagon.

(40.) If the radius of a circle is R, find the area of a segment subtended by the side of (a) an inscribed equilateral triangle, (b) an inscribed regular octagon, (c) an inscribed regular decagon.

GEOMETRY OF SPACE

BOOK VI

STRAIGHT LINES AND PLANES

524. Def.-A plane has already been defined as "a surface such that, if any two points in it are taken, the straight line passing through them lies wholly in the sur face."

$8

A plane is regarded as indefinite in extent, but is usually represented to the eye by a parallelogram lying in it.

525. Def.-A plane is determined by given conditions,

if it is the only plane fulfilling these conditions.

A