520. Theorem. Of all regular polygons having the same area, the one which has the greatest number of sides has the least perimeter.

Given P and R, regular polygons with the same area, P having more sides than R.

To prove perimeter of P<perimeter of R.

Proof. Construct the regular polygon S, having the same perimeter as P and the same number of sides as R.

But

Then

But

Cons.

Why?

perimeter of S= perimeter of P. .. perimeter of P<perimeter of R.

521. When the number of sides of the polygon P in § 520 has been increased indefinitely, it must follow that the circumference of a circle is less than the perimeter of any regular polygon having the same area.

EXERCISES

1. A square and a circle each have an area of 36 sq. in. Which has the greater perimeter? § 521

Find the perimeter of the square and the circumference of the circle. 2. Compare the area and the perimeter of a lot 60 ft. square with the area and perimeter of a lot 90 ft. by 40 ft.

3. Compare the area and the perimeter of a lot 60 ft. square with the area and the perimeter of a lot 50 ft. by 70 ft.

4. The perimeter of the bases of two cans is the same.

The base of one

is a square and of the other a circle. Which one has the greater area of

base?

8.521

SYMMETRY

522. Two points are symmetrical with respect to a straight line as an axis if the straight line joining the two points is bisected at right angles by the axis.

The points C and D are symmetrical with respect to AB as an axis, since AB is the perpendicular bisector of CD. 523. A figure is symmetrical with respect to an axis if every point on one side of the axis has a corresponding symmetrical point on the other side of the axis.

When this is true, the two parts of the figure will coincide if one part is folded over the axis. The outline of the human form is symmetrical with respect to an axis.

B

524. Axis of Symmetry. A line in a figure is an axis of symmetry if one part of the figure when folded over it coincides with the other part.

EXERCISES

1. How many axes of symmetry has an isosceles triangle? An equilateral triangle?

2. How many axes of symmetry has a square? A rectangle? A rhombus? An isosceles trapezoid?

3. How many axes of symmetry has a regular pentagon? A regular hexagon?

4. Point out axes of symmetry in the outline figures given.

5. How many axes of symmetry has a circle?

525. Two figures are symmetrical with respect to an axis if every point in one figure has a corresponding symmetrical point in the other.

526. Center of Symmetry. If a point within a polygon bisects all lines which pass through it and are terminated by the perimeter, the point is a center of symmetry. Thus O is the center of symmetry of the square.

EXERCISE

Has an equilateral triangle a center of symmetry? A regular hexagon? A regular pentagon? A rectangle? A circle?

527. Theorem. If a figure has two axes of symmetry perpendicular to each other, it is symmetrical with respect to their point of intersection as a center of symmetry.

The polygon of the figure has two axes, AB and CD, perpendicular to each other, which intersect at 0.

To prove that O is the center of symmetry of the polygon.

A-

Proof. From any point G on the perimeter, draw GF LCD and GE LAB. Draw HK, EO, and OF.

C

-B

In like manner, it can be shown that HK || OF and HK=OF.

Then EOF is a straight line and O is its middle point. Since G is any point on the perimeter, O bisects all straight lines passing through O and terminated by the perimeter.

.. O is the center of symmetry of the polygon.

§ 526

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FORMULAS FOR REFERENCE

Rectangle, parallelogram. A=bh. b=A÷h. h=A÷b.
Triangle. Abh. b=2A÷h. h=2A÷b.

Trapezoid. A=}h(b+b′).

Right triangle. c=√a2+b2. a=√c2-b2. b=√c2-a2.
Diagonal of a square=side X √2._d=s√2.

=

Side of a square diagonalX √2, 8=d√2.

Altitude of equilateral triangle = side× √3. h=s√/3.
Area of square = (side)2= (diagonal)2. A=s2= 1⁄2d2.
Area of equilateral triangle = (side)2√3. A=1s2√3.
(Any triangle. A=√s(s—a)(s—b) (s—c), where s = {(a+b+c).
Inscribed equilateral triangle. Side=r√3
Circumscribed equilateral triangle. Side=2r√3.

Inscribed square. Side=r√2.

Circumscribed square. Side = 2r.

Inscribed regular pentagon. Sider √ 10–2√√5=1.1756r.
Inscribed regular hexagon. Side = r.

Circumscribed regular hexagon. Side-r√3.

Inscribed regular octagon. Sider √2-√2=0.7653r.
Circumscribed regular octagon. Side=2r(√2-1)=0.8284r.
Inscribed regular decagon. Side=r(√5−1) = 0.618r.
Area of regular polygon. Sap.

Circumference. c=πd=2πг.
Diameter. d=c÷π= √A÷π.
Radius. r=c÷2π = √Ã÷π.

Length of arc. s=

и

360

· πα.

§ 478

§ 496

Area of ring. A=πR2—πr2=π(R+r)(R−r). (Ex. 8, p. 249).

Area of segment of circle. G=S-2 sin u,

§ 508

§ 509