EXERCISES.

ON CIRCLES AND SQUARES.

(Inscriptions and Circumscriptions.)

1. Draw a circle of radius 1.5", and find a construction for inscribing a square in it.

Calculate the length of the side to the nearest hundredth of an inch, and verify by measurement.

Find the area of the inscribed square.

2.

Circumscribe a square about a circle of radius 1.5′′, shewing all lines of construction.

Prove that the area of the square circumscribed about a circle is double that of the inscribed square.

3. Draw a square on a side of 7.5 cm., and state a construction for inscribing a circle in it.

Justify your construction by considerations of symmetry.

4. Circumscribe a circle about a square whose side is 6 cm.

Measure the diameter to the nearest millimetre, and test your drawing by calculation.

5. In a circle of radius 1.8" inscribe a rectangle of which one side measures 30". Find the approximate length of the other side.

Of all rectangles inscribed in the circle shew that the square has the greatest area.

6. A square and an equilateral triangle are inscribed in a circle. If a and b denote the lengths of their sides, shew that

3a2=262.

7. ABCD is a square inscribed in a circle, and P is any point on the arc AD shew that the side AD subtends at P an angle three times as great as that subtended at P by any one of the other sides.

8.

:

(Problems. State your construction, and give a theoretical proof.)

Circumscribe a rhombus about a given circle.

9. Inscribe a square in a given square ABCD, so that one of its angular points shall be at a given point X in AB.

10. In a given square inscribe the square of minimum area. 11.

Describe (i) a circle, (ii) a square about a given rectangle. 12. Inscribe (i) a circle, (ii) a square in a given quadrant.

ON CIRCLES AND REGULAR POLYGONS.

PROBLEM 30.

To draw a regular polygon (i) in (ii) about a given circle.

Let AB, BC, CD, ... be consecutive sides of a regular polygon inscribed in a circle whose centre is O.

Then AOB, BOC, COD,

are con

OA

gruent isosceles triangles.

And if

n

the polygon has n sides, each of the

360°

LAOB, BOC, COD,...

=

n

B

(i) Thus to inscribe a polygon of n sides in a given circle, draw an angle AOB at the centre equal to

360°

n

This gives

the length of a side AB; and chords equal to AB may now be set off round the circumference. The resulting figure will clearly be equilateral and equiangular.

...

(ii) To circumscribe a polygon of n sides about the circle, the points A, B, C, D, must be determined as before, and tangents drawn to the circle at these points. The resulting figure may readily be proved equilateral and equiangular.

NOTE. This method gives a strict geometrical construction only when 360° the angle can be drawn with ruler and compasses.

n

EXERCISES.

1. Give strict constructions for inscribing in a circle (radius 4 cm.) (i) a regular hexagon; (ii) a regular octagon; (iii) a regular dodecagon. 2. About a circle of radius 1·5′′ circumscribe

(i) a regular hexagon; (ii) a regular octagon.

Test the constructions by measurement, and justify them by proof. 3. An equilateral triangle and a regular hexagon are inscribed in a given circle, and a and b denote the lengths of their sides: prove that (i) area of triangle = (area of hexagon); (ii) a2=3h2.

4. By means of your protractor inscribe a regular heptagon in a circle of radius 2". Calculate and measure one of its angles; and measure the length of a side.

PROBLEM 31.

To draw a circle (i) in (ii) about a regular polygon.

Outline of Proof. Join OD; and from the congruent ▲ OCB, OCD, shew that OD bisects the CDE. Hence we conclude that

All the bisectors of the angles of the polygon meet at O. (i) Prove that OB = OC=OD=

...

; from Theorem 6.

Hence O is the circum-centre.

(ii) Draw OP, OQ, OR,

...

perp. to AB, BC, CD, ...

Prove that OP=OQ OR = from the congruent ▲OBP,

OBQ,....

...

;

Hence O is the in-centre.

EXERCISES.

1. Draw a regular hexagon on a side of 20′′.

Draw the inscribed and circumscribed circles. Calculate and measure their diameters to the nearest hundredth of an inch.

2. Shew that the area of a regular hexagon inscribed in a circle is three-fourths of that of the circumscribed hexagon.

Find the area of a hexagon inscribed in a circle of radius 10 cm. to the nearest tenth of a sq. cm.

3. If ABC is an isosceles triangle inscribed in a circle, having each of the angles B and C double of the angle A; shew that BC is a side of a regular pentagon inscribed in the circle.

4. On a side of 4 cm. construct (without protractor)

(i) a regular hexagon; (ii) a regular octagon. In each case find the approximate area of the figure.

THE CIRCUMFERENCE OF A CIRCLE.

By experiment and measurement it is found that the length of the circumference of a circle is roughly 34 times the length of its diameter: that is to say

and it can be proved that this is the same for all circles.

A more correct value of this ratio is found by theory to be 3-1416; while correct to 7 places of decimals it is 3.1415926. Thus the value 34 (or 3·1428) is too great, and correct to 2 places only.

The ratio which the circumference of any circle bears to its diameter is denoted by the Greek letter ; so that

=

circumference diameter × π.

Or, if r denotes the radius of the circle,

circumference 2r x = 2πr;

=

where to we are to give one of the values 34, 31416, or 3.1415926, according to the degree of accuracy required in the final result.

NOTE. The theoretical methods by which is evaluated to any required degree of accuracy cannot be explained at this stage, but its value may be easily verified by experiment to two decimal places.

For example: round a cylinder wrap a strip of paper so that the ends overlap. At any point in the overlapping area prick a pin through both folds. Unwrap and straighten the strip, then measure the distance between the pin holes: this gives the length of the circumference. Measure the diameter, and divide the first result by the second.

Ex. 2. A fine thread is wound evenly round a cylinder, and it is found that the length required for 20 complete turns is 75.4". The diameter of the cylinder is 12′′: find roughly the value of π.

Ex. 3. A bicycle wheel, 28" in diameter, makes 400 revolutions in travelling over 977 yards. From this result estimate the value of #.

THE AREA OF CIRCLE.

Let AB be a side of a polygon of n sides circumscribed about a circle whose centre is O and radius r. Then we have

Area of polygon

=η. ΔΑΟΒ

=n. AB × OD

= .nAB x r

= (perimeter of polygon) × r ;

AD B

and this is true however many sides the polygon may have. Now if the number of sides is increased without limit, the perimeter and area of the polygon may be made to differ from the circumference and area of the circle by quantities smaller than any that can be named; hence ultimately

Area of circle. circumference x r

Suppose the circle divided into any even number of sectors having equal central angles : denote the number of sectors by n.

Let the sectors be placed side by side as represented in the diagram; then the area of the circle=the area of the fig. ABCD ;

and this is true however great n may be.

Now as the number of sectors is increased, each arc is decreased; so that (i) the outlines AB, CD tend to become straight, and (ii) the angles at D and B tend to become rt. angles.