I. THE diagram will need paper 10 cm. X 9 cm., or 4 in. x 3 in. ABC is an equilateral triangle, each edge being 1 dcm. long. The middle points of the edges, D, E, and F, are joined, and the triangle is thus divided into four smaller equilateral triangles having edges 5 cm. long.

In English measurements, the edges of ABC may be 4 inches long, thus making the edges of the smaller triangles 2 inches long.

2. 1. How many faces has this figure? What is their shape?

2. How many edges? What is their length?

3. How many corners?

4. How many line angles? What is their size?

5. How many diedral angles? What is their size? 6. This figure is called a pyramid: why?

7. It is also called a triangular pyramid: why?

8. Has it more than one face which can be called its base? 9. Has a quadrangular pyramid more than one such face? 10. Can you explain the difference between these two cases?

3. Solid Angles. We have seen that when two edges meet, or would meet if prolonged, they form a line angle; and when two faces meet, or would meet if extended, they form a diedral angle. Now when three or more faces meet at a point and enclose all the space around the point, they make what is called a solid angle.

A Solid Angle

If you observe figures carefully you will see that it takes at least three faces to form a solid angle; for two faces would leave an open space. But there may be as many faces as you please more than three; though if you try to make a solid angle by joining pieces of paper you will find that the sum of the angles formed by the edges must not be so great as 360° or 4 right angles. If the sum were equal to 360°, the pieces of paper would lie flat and form a plane.

Notice that a solid angle has an open space in front of the point. If this space were closed by a plane cutting the other faces, the resulting figure would be a pyramid.

A Solid Angle

If a solid angle is formed by three faces, it is called a triedral (tri-e'-dral) angle, which means "having three faces." If it is formed by four or more faces, it is called a polyedral (pol-y-e'-dral) angle, which means "having many faces."

11. What is the difference between a triedral angle and a triangular pyramid ? 12. How many solid angles has a triangular pyramid?

13. How many has a cube? What is the sum of the line angles which form each solid angle?

14. How many solid angles has a quadrangular pyramid?

15. Is there a solid angle at each corner of a figure which is entirely enclosed by planes ?

16. In the triangular pyramid is the number of solid angles equal to the number

of faces ?

17. Is this true of the cube?

18. Of the triangular prism?
19. Of the quadrangular pyramid?

1. The diagram will need paper 15 X 15 cm., or 6 X 6 inches. Draw AB, 3 cm. long.

At A draw AE, 3 cm. long, and making the angle BAE 108°.

At B draw BC, 3 cm. long, and making the angle ABC 108°.

At E draw ED, 3 cm. long, and making the angle AED 108°.
Draw the line DC completing the interior part of the diagram.

Prolong the lines AB, BC, etc., in both directions until they form the fivepointed star FGHIJ.

With English measurements, I inch will be a convenient length for AB.

2. This figure is called a pentagonal (pen-tagʻ-o-nal) pyramid. Its base is a pentagon (pen'-ta-gon), which means a face having five corners. It also has five edges; for every face has as many edges as it has corners.

3. Examine the completed model, make measurements, and write out your answers to the following:

1. The number of faces.

2. The number of edges.

3. The number of corners.

4. The shapes of the faces and the number of each shape.
5. The lengths of the edges and the number of each length.
6. The number of face angles.

7. The sizes of the face angles and the number of each size.
8. The number of diedral angles.

9. The sizes of the diedral angles and the number of each size 10. The number of solid angles.

11. The number of faces which form each solid angle.