REGULAR POLYGONS AND CIRCLES

Excerpts

• REGULAR POLYGONS AND CIRCLES

  A **regular polygon** is a polygon which is both equilateral and equiangular. The equilateral triangle and the square are examples.

• REGULAR POLYGONS AND CIRCLES

  The ratio of the circumference of a circle to its diameter is constant.

• REGULAR POLYGONS AND CIRCLES

  The constant ratio of the circumference of a circle to its diameter is represented by the Greek letter $\pi$.

• REGULAR POLYGONS AND CIRCLES

  If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit.

• REGULAR POLYGONS AND CIRCLES

  The area of a regular polygon is equal to half the product of its apothem by its perimeter.

• REGULAR POLYGONS AND CIRCLES

  Therefore, to inscribe a regular decagon, divide the radius internally in extreme and mean ratio, and apply the greater segment ten times as a chord.

• REGULAR POLYGONS AND CIRCLES

  $\pi$ is incommensurable.

• REGULAR POLYGONS AND CIRCLES

  Among geometrical magnitudes which satisfy given conditions, the *greatest* is called the **maximum**; and the *smallest* is called the **minimum**.

• REGULAR POLYGONS AND CIRCLES

  Of all triangles having two given sides, that in which these sides include a right angle is the maximum.

• REGULAR POLYGONS AND CIRCLES

  Thus, the diameter of a circle is the maximum among all chords; and the perpendicular is the minimum among all lines drawn to a given line from a given external point.

• REGULAR POLYGONS AND CIRCLES

  Hence, *every* vertex lies on the circumference; that is, the maximum polygon can be inscribed in a semicircle having the undetermined side for a diameter.

• REGULAR POLYGONS AND CIRCLES

  **Isoperimetric** polygons are polygons which have equal perimeters.

Equations

• REGULAR POLYGONS AND CIRCLES

  \dfrac{(n-2) 2 \text{ rt.\ } \angle_s}{n}

  Each interior angle of a regular polygon of n sides equals the sum of its interior angles, (n-2) two right angles, divided by n.

• REGULAR POLYGONS AND CIRCLES

  \overline{OA}^2 - \overline{OP}^2 = \overline{AP}^2

  In the right triangle OAP, the square on the radius OA minus the square on the apothem OP equals the square on AP, half a side.

• REGULAR POLYGONS AND CIRCLES

  \overline{AD}^2 = DH × DC

  In the right triangle DAH, the square on the side AD equals the product of the hypotenuse DH and its segment DC.

• REGULAR POLYGONS AND CIRCLES

  AD = \sqrt{2-\sqrt{4-a^2}}

  For a circle of unit radius, the side AD of the polygon with double the sides is found from the side a of the given inscribed polygon.

• REGULAR POLYGONS AND CIRCLES

  \sqrt{R(2R - \sqrt{4R^2 - a^2})}

  For a circle of radius R, the side AD of the regular inscribed polygon with double the sides is given in terms of R and the side a of the given polygon.

• REGULAR POLYGONS AND CIRCLES

  S = \frac{1}{2}R × P

  The area of a regular polygon equals half the product of its apothem and its perimeter.

• REGULAR POLYGONS AND CIRCLES

  S' = \frac{1}{2} R × P

  The area of a regular circumscribed polygon equals half the product of the radius of the circle (its apothem) and the polygon's perimeter, for any number of sides.

• REGULAR POLYGONS AND CIRCLES

  S = \frac{1}{2}R× C

  The area of a circle equals half the product of its radius and its circumference.

• REGULAR POLYGONS AND CIRCLES

  \odot = \frac{1}{2} R × C = \frac{1}{2} R × 2\pi R = \pi R^2

  The area of a circle of radius R is pi times the square of its radius.

• REGULAR POLYGONS AND CIRCLES

  S:S' = \pi R^2:\pi R'^2 = R^2:R'^2

  The areas of two circles are to each other as the squares of their radii.

• REGULAR POLYGONS AND CIRCLES

  \pi = \dfrac{C}{2R}

  pi is the constant ratio of the circumference of a circle to its diameter.

• REGULAR POLYGONS AND CIRCLES

  C=2\pi R

  The circumference of a circle equals 2 pi times its radius.

• REGULAR POLYGONS AND CIRCLES

  2\pi R = C

  Twice pi times the radius of a circle equals its circumference.

• REGULAR POLYGONS AND CIRCLES

  \pi = \frac{1}{2}C

  When the radius is unity, pi equals half the circumference.

• REGULAR POLYGONS AND CIRCLES

  C = 6.28317

  The circumference of the unit circle, found by the inscribed-polygon computation, is approximately 6.28317.

• REGULAR POLYGONS AND CIRCLES

  \pi = 3.14159

  From the computed circumference, pi is nearly 3.14159.

• REGULAR POLYGONS AND CIRCLES

  \pi = 3.1416

  For general use, pi is taken as 3.1416.

• REGULAR POLYGONS AND CIRCLES

  \frac{1}{\pi} = 0.31831

  For general use, the reciprocal of pi is taken as 0.31831.

• REGULAR POLYGONS AND CIRCLES

  P:P' = OA:O'A' = OM:O'M'

  The perimeters of two similar regular polygons with the same number of sides are to each other as their circumscribed radii and as their inscribed radii (apothems).

• REGULAR POLYGONS AND CIRCLES

  AB:A'B' = BC:B'C'

  Two regular polygons with the same number of sides have their homologous sides proportional.

• REGULAR POLYGONS AND CIRCLES

  \triangle ACB > \triangle ADB

  Of two isoperimetric triangles with the same base AB, the isosceles triangle (AC equal to CB) has the greater area.

• REGULAR POLYGONS AND CIRCLES

  ABCDE > A'B'C'D'E'

  A polygon inscribed in a circle has greater area than an equilateral polygon with the same sides that cannot be inscribed in a circle.

• REGULAR POLYGONS AND CIRCLES

  AB = BC

  In the maximum of isoperimetric polygons with a given number of sides, the adjacent sides AB and BC are equal, because the triangle ABC on AC must be isosceles.

Problems