Plane Geometry

CHAPTER V

REGULAR POLYGONS.

MEASUREMENT OF CIRCLES.

457. A regular polygon is a polygon that is both equilateral and equiangular.

458. Problem. To inscribe a regular hexagon in a circle. Construction. Draw the radius OA. With A as a center

and a radius equal to OA, describe an arc intersecting the circle at B. With B as a center and the same radius determine the point C, etc. Draw the chords AB, BC, etc. Then ABCDEF is the required regular hexagon.

Proof. AAOB is equilateral.

ZAOB=60° and AB=60°.

Why?

Hence the

Then the circle is divided into six equal arcs. chords are equal and the polygon ABCDEF is equilateral. The polygon is also equiangular, for each angle of the polygon equals in degrees one-half of four of the equal arcs into which the circle is divided.

.. Polygon ABCDEF is a regular hexagon.

Why?

459. Theorem. Any equilateral polygon inscribed in a circle is a regular polygon.

This is proved by the method that was used to prove that the equilateral hexagon is equiangular.

460. Problem. To inscribe a square in a circle.

§ 458

Draw two perpendicular diameters, complete the construction, and prove.

461. Theorem. If a circle is divided into any number of equal parts, the chords joining the successive points of division form a regular inscribed polygon; and the tangents drawn at the successive points of division form a regular circumscribed polygon. Given the circle whose center is O divided into equal parts, the chords joining A the successive points of division, and the N tangents at these points.

To prove: (1) The inscribed polygon is regular.

(2) The circumscribed polygon is regular.

Suggestion. (1) Polygon ACEGKM is equilateral by § 275, and therefore regular by § 459.

(2)

Lx=Lz= Lr= Zs, etc.

Why? Show that AABC=^CDE=AEFG, etc., and that they are isosceles.

Then show that BD=DF = FH, etc.

.. Polygon BDFHLN is regular.

Why?

462. Theorem. Tangents drawn to a circle at the vertices of a regular inscribed polygon form a regular circumscribed polygon of the same number of sides.

463. Theorem. Chords drawn from the middle points of the arcs subtended by the sides of a regular inscribed polygon to the adjacent vertices of the polygon, form a regular inscribed polygon of double the number of sides.

464. Theorem. The perimeter of a regular inscribed polygon is less than the perimeter of a regular inscribed polygon of double the number of sides, and less than the circumscribed circle.

465. Theorem. The area of a regular inscribed polygon is less than the area of a regular inscribed polygon of double the number of sides.

466. Theorem. The area of a regular polygon is less than the area within the circumscribed circle.

467. Remark. From the considerations already made, it is evident that the general problem of constructing a regular polygon depends upon the division of a circle into as many equal parts as the polygon has sides. It is to be noted that this division of the circle cannot be made by methods of elementary geometry in all cases.

As early as the time of Euclid it was known that it is possible to construct regular polygons of 2", 3.2", 5.2", and 15.2" sides. It was long supposed that these were all that could be constructed.

Gauss, a famous German mathematician who lived from 1777 to 1855, however, proved that a circle can be divided into 17 equal parts by means of the instruments of geometry. He also solved the general problem as to what polygons can be inscribed in a circle by methods of elementary geometry. The results showed that the number of sides must be: (1) a prime number of the form 22′′ +1 (as 3, 5, 17, 257); or (2) a product of such prime numbers, not repeated, (as 15, 51, 85, 255); or (3) such prime numbers times 2"; or (4) products of such prime numbers by 2".

EXERCISES

1. Show by § 460 and § 463 how to construct regular polygons of 8, 16, 32, 64.

...

sides.

2. Show how to construct regular polygons of 12, 24, 48

3. Show that an equilateral triangle may be formed by joining the alternate vertices of a regular hexagon.

4. Circumscribe a regular octagon about a

circle.

5. On a given line as a side construct a regular octagon.

Let AB be the side. The figure suggests a construction.

6. If the diagonals joining the alternate vertices of a regular hexagon are drawn, they form a second regular hexagon whose area is one-third that of the original hexagon.

7. An equilateral polygon circumscribed about a circle is regular if the number of sides is odd.

8. An equiangular polygon circumscribed about a circle is regular. 9. Where are the vertices of all the regular polygons of the same number of sides which can be circumscribed about a given circle?

461. Theorem. If a circle is divided into any number of equal parts, the chords joining the successive points of division form a regular inscribed polygon; and the tangents drawn at the successive points of division form a regular circumscribed polygon. Given the circle whose center is 0 divided into equal parts, the chords joining the successive points of division, and the N tangents at these points.

To prove: (1) The inscribed polygon is regular.

lar.

(2) The circumscribed polygon is regu

Suggestion. (1) Polygon ACEGKM is equilateral by § 275, and therefore regular by § 459.

(2)

Lx=Zz=Zr=Zs, etc.

Why? Show that AABC=ACDE=AEFG, etc., and that they are isosceles.

Then show that BD=DF=FH, etc.

... Polygon BDFHLN is regular.

Why?

462. Theorem. Tangents drawn to a circle at the vertices of a regular inscribed polygon form a regular circumscribed polygon of the same number of sides.

464. Theorem. The perimeter of a regular inscribed polygon is less than the perimeter of a regular inscribed polygon of double the number of sides, and less than the circumscribed circle.

465. Theorem. The area of a regular inscribed polygon is less than the area of a regular inscribed polygon of double the number of sides.

466. Theorem. The area of a regular polygon is less than the area within the circumscribed circle.

As early as the time of Euclid it was known that it is possible to construct regular polygons of 2", 3.2", 5.2", and 15.2" sides. It was long supposed that these were all that could be constructed.

Gauss, a famous German mathematician who lived from 1777 to 1855, however, proved that a circle can be divided into 17 equal parts by means of the instruments of geometry. He also solved the general problem as to what polygons can be inscribed in a circle by methods of elementary geometry. The results showed that the number of sides must be: (1) a prime number of the form 22"+1 (as 3, 5, 17, 257); or (2) a product of such prime numbers, not repeated, (as 15, 51, 85, 255); or (3) such prime numbers times 2"; or (4) products of such prime numbers by 2".

EXERCISES

1. Show by § 460 and § 463 how to construct regular polygons of 8, 16, 32, 64• • *, sides.

2. Show how to construct regular polygons of 12, 24, 48

sides.

3. Show that an equilateral triangle may be formed by joining the

alternate vertices of a regular hexagon.

4. Circumscribe a regular octagon about a

circle.

5. On a given line as a side construct a regular octagon.

Let AB be the side. The figure suggests a construction.

6. If the diagonals joining the alternate vertices of a regular hexagon are drawn, they form a second regular hexagon whose area is one-third that of the original hexagon.

7. An equilateral polygon circumscribed about a circle is regular if the number of sides is odd.

8. An equiangular polygon circumscribed about a circle is regular.

9. Where are the vertices of all the regular polygons of the same number of sides which can be circumscribed about a given circle?

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