I have told you about figures which have three and four sides, and we now come to such as have five, six, seven or more sides, and of course you can understand that a figure will always have as many angles as it has sides a triangle has three angles, and three sides; a square has four angles, and four sides--so that I need not always say both sides and angles, for you will know now that if I say a figure has five sides, it will have five angles; and if I say a figure has six sides, you will know that it has six angles. Now I do not want you to learn long lessons, but you know we cannot get on with any study if we do not try to remember something about it; so I will give you just a couple of lines or so, which you must not forget.

All figures which have more than four sides are called POLYGONS, which means “many. angles."

But all polygons are not alike; and therefore, each one of them has a name by which you can tell how many sides it has; thus

A PENTAGON has FIVE sides.
A HEXAGON has SIX sides.
A HEPTAGON has SEVEN sides.
An OCTAGON has EIGHT sides.
A NONAGON has NINE sides.
A. DECAGON has TEN sides.

F

A figure having five sides is called a PENTAGON (which means five angles).

TO DRAW A PENTAGON, of which the line A B is to be one side, Fig. 44.

Draw a perpendicular at A and the QUADRANT C B.

Divide this quadrant into five equal parts 1, 2, 3, 4, 5.

Continue the quadrant beyond the point C, and mark on it one of the five divisions; so that now you will have six equal divisions set off on the arc from B to D. You must be very careful in dividing the quadrant, for I cannot give you any rule for

doing it. You must depend entirely on yourself, and try again and again, until you get all the five parts quite equal, and this practice will be very useful to you through life; for it will teach you to be accurate in your work. The whole of this figure depends on this quadrant being rightly divided; for if any of the parts are unequal, it will cause a want of exactness in all the sides of the figure.

Having then, marked off one of the divisions beyond C, thus obtaining the point D, draw the line D A, which will be a second side of the pentagon.

Now draw a perpendicular at B, and a horizontal from D, which will pass through the perpendicular in the point E.

On this horizontal mark the point F, as far from the point E, as D is from the opposite side.

Draw the line F B, and this will be the third side of the pentagon.

Draw the perpendicular G in the middle of the line A B.

And from the point F draw a line to H, in the middle of B D.

These two lines will cross each other in the point O; this is the centre of the pentagon, and all the five angles will be equally distant from it.

Now you will see that we have four points of the pentagon, and therefore only require one more ; but I think I have given you a hint by which you can find the fifth, for you know I have told you that O is the centre, and that all the points are equally distant from it.

Therefore, from O mark off on the perpendicular the length O F, which will give you the point I. Draw the lines F I and D I, which will complete the pentagon.

When all the sides of a polygon are equal, and all its angles equal, they are called REGULAR.

Our figure is, therefore, a REGULAR PENTAGON. And now I am going to ask you to remember another short rule.

Every Polygon may be divided into as many triangles as it has sides, by drawing lines from the angles to the centre.

And thus the lines drawn from A, B, D, F, and I to O will divide the pentagon into five triangles. These will be equal to each other, but they will not be equilateral, for the base will be rather longer than the two sides; they will therefore, you know, be isosceles triangles.

I will now tell you how to use this figure for

two other lessons; but I do not intend to draw the lines, but only to speak of them, so that this may be an exercise for you.

(1) To inscribe a Circle in a Regular Pentagon.

Find the centre by drawing perpendiculars from the middle of two sides which are next to each other (G and H), which will intersect in O.

Draw similar perpendiculars from the middle of each side to the centre, and also lines to the centre from each of the angles.

Mark off on these lines from the centre the length of the perpendiculars (as O G), and you will then have ten points through which to draw your circle; namely, five at the ends of the perpendiculars, and five on the lines drawn from the angles.

(2) To describe a Circle ABOUT a Regular Pentagon.

Begin just as you did in the last figure, but draw the lines out further than the sides of the pentagon, until they are as long as the lines O A or O B. Then draw your circle touching the ends of these lines and the angles of the pentagon.