(c) Through E, the second mark on the diameter, draw a line from C to meet the circumference at D.

(d) Draw AD.

(e) Use AD as a radius and mark off four other equal arcs on the circumference.

(f) Join these points in succession to form a regular pentagon.

2. (a) Make another regular pentagon.

(b) Measure the at each vertex in both figures. How many degrees are in each ? How many right?

(c) Draw all the radii.

(d) How many and what kind of A are formed?

(e) How many degrees are in each

of each A?

3. What would you do to the pentagon to make a regu

lar 10-sided polygon, that is, a decagon?

4. (a) Draw an irregular pentagon.

(b) Measure the 4 at the vertices and find their sum, measured in degrees and in right 4.

(c) How does the sum of the

with the sum of the

in this one compare

of a regular pentagon?

(d) Does the sum of the interior

change with its shape or size?

5. Does the sum of the interior

with the number of sides?

of a pentagon

of a polygon change

6. Make a table showing the sum of the interior of the different polygons constructed and the size of the at each vertex, measured in rt. 4 and in degrees.

7. (a) Draw a regular pentagon and all possible diagonals.

(b) How many points are in the star?

(c) What kind of A are formed on each side of the pentagon?

(d) What kind of a polygon is left in the center?
(e) Compare its position with the original pentagon.
(f) There are now three at each vertex.

Compare them in size.

(g) How many isosceles A can you find in the figure? (h) Without measuring, can you compute the size of each of the four at one vertex of the inner

pentagon?

8. Can you find a rhombus in the figure? How many? 9. (a) Can you fit four regular pentagonal tiles together? (b) Can you fit three together?

(c) How much of an angle will be left?

(d) Is there any regular polygon that will fit in exactly? 10. Make and shade or color several original designs made from pentagons.

11. It is impossible to construct exact regular polygons of 7, 9, or 11 sides with compass and ruler. But the method given for the pentagon, which gives approximate results only, may be used for the other polygons.

To construct a heptagon or 7-sided polygon, divide the diameter into 7 equal parts, and from the vertex of the equilateral triangle draw a line through the second point of division. Proceed as in the construction of the pentagon.

CYLINDERS AND CIRCLES

A. CYLINDERS

1. We have seen that many objects in our surroundings are more or less rectangular. But there is another group of objects about us that have one general shape different from the rectangular solid.

2. Think for a moment of the shape of a telegraph pole, the trunk of a tree, the water glass for the table, the ice. cream freezer, the fruit jar, the rows of tin cans containing vegetables on the grocer's shelves, smoke stacks on steamships, the pencil with which you write, the pillars in front of some public buildings. All of these things are cylindrical in shape, or like a cylinder.

3. The tin can is probably the most convenient example of a cylinder. (a) How many surfaces has it? (b) How many of these are flat or plane?

(c) How many are curved?

(d) What is the shape of the flat
surfaces?

(e) What is their relative posi

tions?

4. (a) To test a plane surface, put the edge of your ruler on it in several different positions. If the

edge touches the surface in all positions, it is a flat or plane surface.

(b) Put the edge of your ruler along the side of a tin can or other cylindrical object, from top to