383. THEOREM. If from any point A of a circle two chords AB and AC are drawn on the same side of A, the first a side of a regular inscribed decagon, and the second a side of a regular inscribed hexagon, then the chord BC is a side of a regular inscribed polygon of fifteen sides.

SUGGESTION. What fraction of four right angles does the chord AB subtend at the centre ? AC? BC?

PROPOSITION XIII

384. Given the radius of a circle and a side of an inscribed regular polygon of n sides; to find the length of a side of an inscribed regular polygon of 2n sides.

C

B

K

Let OA (r) be a radius of the given circle, AB (=p) be a side of an inscribed regular polygon of n sides, and AC (= q) be a side of an inscribed regular polygon of 2 n sides. The problem is to find the value of q in terms of r and p.

Join OC. This line bisects AB at right angles, at K.

or

Now

whence

q= √2 p2 — r √4 p2 — p2.

Writing instead of r, this formula becomes

If now we let Q2n represent the perimeter of an inscribed regular polygon of 2n sides, while p represents the length of one side of a regular inscribed polygon of n sides, we have

Q2n = 2nq= n√2 d2 - 2 d√d2-p2.

EXERCISES

1. Does the above formula for the value of q hold true when p = 2r, in which case q is a side of an inscribed square? Compare the result with that given on page 251.

2. Apply the formula to find the side of an inscribed regular octagon. 3. In a circle of radius r, the side of an inscribed regular dodecagon equals r √2√3.

4. In a circle of radius 3 feet, what is the length of a side of an inscribed regular dodecagon ?

5. The area of a regular octagon inscribed in a circle is equal to the product of the sides of the inscribed and circumscribed squares.

6. The square of the side of an inscribed equilateral triangle is equal to the sum of the squares of the sides of the inscribed square and of the inscribed regular hexagon.

PROPOSITION XIV

385. To find approximately the value of ".

First, find the perimeter of a regular inscribed polygon of 12 sides.

Next, find the perimeter of a regular inscribed polygon of 24

When the number of sides of the inscribed polygon is indefinitely increased, Qan approaches the circumference C as its

limit.

But

C = d x π.

(Art. 362.)

Hence the multipliers of d in the above computation are approximate values of corresponding to inscribed polygons of 12, 24, 48, 96... sides, respectively.

The value = 3.1415904 is correct to five decimal places.

If the number of sides of the polygon were again doubled, a still closer approximation would be obtained.

The value of has been computed correctly to over seven hundred decimal places. In practice it is customary to use π = 3.1416.

= 22 is an approximate value correct to two decimal places, while π = 35 is correct to five decimal places.

A rough approximation to the value of ☛ can be obtained by rolling a circular piece of cardboard, of say 6 inches radius, along a straight line until it makes a complete revolution, then measuring carefully the length of the line-segment so traced, and comparing it with the diameter of the cardboard. A smaller error will probably be made if the cardboard is given two or three revolutions, and the whole distance divided by the number of revolutions.

A second method is to mark out a circle on a piece of cardboard ruled in squares, ordinary centimetre paper, then find approximately the area of the circle by counting the squares enclosed by it, reckoning each fractional square enclosed by the circle, a half square, this being an approximate average value. Setting this area equal to πr2 gives an approximate value for π.

EXERCISES

In these exercises use 22 as the value of π.

1. The diameter of a circle is 5 feet, what is its circumference ?

2. The radius of a circle is 1 foot 8 inches, what is its circumference?

3. A wheel is twelve feet in circumference, what is its diameter ?

4. A circular field is 1000 yards in circumference, what is its diameter and its area?

5. Two fields are each 1600 yards around. One is circular and the other is square. What is the difference in their areas?

6. From a circular piece of paper of 10 inches radius, a circular piece is cut which has a radius of the first for its diameter. Find the area of the remaining piece.

S

MISCELLANEOUS EXERCISES

1. Divide a given circle into two arcs such that any angle inscribed in one arc is three times an angle inscribed in the other.

SUGGESTION.

required arcs.

The side of an inscribed square is the chord of the

2. Divide a given circle into two arcs such that any angle inscribed in one arc is five times an angle inscribed in the other arc.

3. From a point without a circle two tangents are drawn which, with their chord of contact, form an equilateral triangle whose side is 18 inches. Find the diameter of the circle.

4. In a regular polygon of n sides the straight lines which join any vertex to the non-adjacent vertices divide the angle at that vertex into n2 equal parts.

5. An equilateral triangle and a regular hexagon are inscribed in a given circle; show that—

(1) The area of the triangle is half that of the hexagon;

(2) The square on a side of the triangle is three times the square on a side of the hexagon.

6. If ABCDE is a regular pentagon and AC, BE intersect at H, show that

(1) CH and EH are each equal to a side of the pentagon ;

(2) AB is a tangent to the circle circumscribed about the triangle BHC. 7. Show that the area of a regular hexagon inscribed in a circle is three-fourths of that of the corresponding circumscribed hexagon.

8. The area of a square circumscribed about a circle is double of the area of the inscribed square.

9. If ABCD is a square inscribed in a circle and P is any point on the circle, show that the sum of the squares on PA, PB, PC, PD is double the square on the diameter.

10. An equilateral triangle is inscribed in a circle, and tangents are drawn at its vertices; prove that

(1) The resulting figure is an equilateral triangle;

(2) Its area is four times that of the given triangle.

11. What is the area of a circular ring if the radii of the outer and inner circles are 132 and 120 feet, respectively?

12. A carriage wheel makes 168 revolutions in going half a mile. What is its height?