731. EXERCISE.

The radius of a circle is 5 ft. Find the side and the

area of the inscribed square.

732. EXERCISE. Find the side and the area of a square circumscribed about a circle, having a diameter 6 in. long.

733. EXERCISE. The area of a square is 16 sq. in.

Find the radius

of the inscribed circle and also the radius of the circumscribed circle.

PROPOSITION VI. PROBLEM

734. To inscribe a regular hexagon in a circle.

D

E

B

Let O be the center of the given circle.

Required to inscribe a regular hexagon in the circle.

Draw the radius OA. Lay off the chord AB 04. Draw OB. ▲ OAB is equilateral, and angle o contains 60°.

.. the arc AB is of the circumference, and the chord AB is one side of a regular hexagon.

Complete the hexagon ABCDEF.

Q.E.F.

735. COROLLARY I. The chords joining the three alternate vertices form an inscribed equilateral triangle.

736. COROLLARY II. Tangents drawn at the vertices of the inscribed hexagon and of the triangle form a regular circumscribed hexagon and a regular circumscribed triangle.

737. COROLLARY III. If the arcs are bisected and chords and tangents are drawn according to § 710, regular polygons of 12, 24, 48, etc., sides will be inscribed in and circumscribed about the circle.

738. EXERCISE. The side of the inscribed equilateral triangle is R√3, and its area is § R2√3.

739. EXERCISE. The side of the circumscribed equilateral triangle is 2 R√3, and its area is 3 R2√3.

740. EXERCISE. area is R2√3.

741. EXERCISE. R√3, and its area

742. EXERCISE.

The side of a regular inscribed hexagon is R, and its

The side of a regular circumscribed hexagon is is 2 R2√3.

The area of a regular inscribed hexagon is double that of an equilateral triangle inscribed in the same circle. [Show this in two ways: 1st, by comparing the values of their areas as derived in §§ 738 and 740; 2d, by a geometrical demonstration using the figure of § 734.]

743. EXERCISE. What is the area of a regular hexagon inscribed in a circle, the radius of which is 4 in.?

744. EXERCISE. The area of a regular inscribed hexagon is 10 sq. in. What is the area of a regular hexagon circumscribed about the same circle ?

745. EXERCISE. The area of an equilateral triangle is 48 √3 sq. ft. Find the radii of the inscribed and of the circumscribed circles.

746. EXERCISE. The area of a regular hexagon is 54 a2 √3. Find the radii of the inscribed and of the circumscribed circles.

747. EXERCISE.

Show that the circumscribed equilateral triangle is 4 times the inscribed equilateral triangle; that the circumscribed square is 2 times the inscribed square; and that the circumscribed regular hexagon is of the inscribed regular hexagon.

748. EXERCISE. compasses only.

Divide a circumference into quadrants by the use of

[SUGGESTION. The side of an inscribed square is the altitude of an isosceles triangle whose base is 2 R and one of whose sides is R √3.]

PROPOSITION VII. PROBLEM

749. To inscribe a regular decagon in a circle.

B

A

Let ́o be the center of the given circle.

Required to inscribe a regular decagon in the circle.

Draw the radius 04. Divide it in extreme and mean ratio, OB being the greater segment.

Lay off AC =0B. Draw BC and OC.

ZA+ZACO +20=180°. (?)

220+220 +20=180°. (?) 40=36°.

.. the arc AC, the measure of 0, contains 36° of arc, and is of the circumference.

The circumference can therefore be divided into ten parts, each equal to the arc AC, and the chords joining the points of division form a regular inscribed decagon.

Q.E.F.

750. COROLLARY I. The chords joining the alternate vertices of a regular inscribed decagon form a regular inscribed pentagon.

751. COROLLARY II. Tangents drawn at the vertices of the regular inscribed pentagon and decagon form a regular circumscribed pentagon and a regular circumscribed decagon.

752. COROLLARY III. If the arcs are bisected and chords and tangents are drawn according to § 710, regular inscribed and circumscribed polygons of 20, 40, 80, etc., sides will be formed. The length of the side of a regular inscribed decagon

753. EXERCISE. is (√5-1)r.

754. EXERCISE. Find the length of a side of a regular inscribed pentagon. [In the R.A. ▲ ADC (see the figure of § 749), AC is the side of the decagon, and AD is one half the difference between the radius and the side of the decagon.]

√10 – 2√5
2

γ.

755. EXERCISE. Show that the sum of the squares described on the sides of a regular inscribed decagon and of a regular inscribed hexagon equals the square described on the side of a regular inscribed pentagon.

[Represent the sides of the pentagon, hexagon, and decagon by p, h, and d, respectively.

756. EXERCISE. What is the length of the side of a regular decagon inscribed in a circle having a diameter 4 in. long?

757. EXERCISE. If the side of a regular pentagon is 2√5 in., that the radius of the circumscribed circle is √10 + 2√5 in.

show

PROPOSITION VIII. PROBLEM

758. To inscribe a regular pentedecagon in a circle.

B

A

Let o be the center of the given circle.

Required to inscribe a regular polygon of fifteen sides in the circle.

Lay off the chord AB = side of regular inscribed hexagon, and the chord 4C side of regular inscribed decagon.

AC: =

The arc AB contains 60°, (?) and the arc AC, 36°. (?)

.. the arc BC contains 24° and is of the circumference. The circumference can therefore be divided into fifteen parts, each equal to BC; and the chords joining the points of division form a regular inscribed pentedecagon.

Q.E.F.

759. COROLLARY I. Tangents drawn at the vertices of the inscribed pentedecagon form a regular circumscribed pentedecagon.

760. COROLLARY II. If the arcs are bisected, and chords and tangents are drawn as described in § 710, regular inscribed and circumscribed polygons of 30, 60, 120, etc., sides will be formed.

761. SCHOLIUM. In Propositions V., VI., VII., and VIII. we have seen that the circumference can be divided into the following numbers of equal parts: