Sug. Can you show how A B F D and G H O N are related, etc.? Can you show how the polyedral ▲ at A is related to the polyedral at G? Complete proof.

Draw conclusion, and call this Prop. XXIV.

To Prove-That they are to each other as the cubes of their homologous edges.

Sug. See $517.

Write the proposition, and call it Prop. XXV.

533.

Cor. Show that any two similar polyedrons are to each other as the cubes of their homologous edges. EXERCISES.

478. In Ex. 455, compare the volume of the solid formed with the original solid.

479. In Ex. 455, compare the volume of the solid formed with the original solid.

480. A farmer wishes to build a cubical bin that will hold 100 bushels of wheat. What will be an inside edge in inches?

481. What is the edge of a cube whose entire surface is 1 square foot?

482. What is the entire surface of a common building brick?

483. What is the edge of a cube that will contain a gallon, dry measure?

484. The base of a pyramid is 12 square feet and its altitude is 6 feet. What is the area of a section parallel to the base and 2 feet from it?

485. Prove the diagonals of a rectangular parallelopiped equal.

486. The volume of any triangular prism equals one half the product of any lateral face by its distance from the opposite edge. Prove.

487. Show that the four diagonals of a parallelopiped bisect each other. (The point of intersection is called the center of the parallelopiped.)

488. Can you prove that a straight line passing through the center of a parallelopiped and terminated by two faces is bisected at the center?

489. Can you show that the middle points of the edges of a regular tetraedron are the vertices of a regular octaedron. Is the altitude of a regular tetraedron equal to the sum of the perpendiculars to the faces from any point within the figure?

490. Find the volume of a regular triangular pyramid whose basal edge is 4 feet and whose altitude is V3 feet.

491. Find the lateral edge, lateral area, and volume of a frustum of a regular triangular pyramid the sides of whose bases are 10 Vg and 2 √3 and whose altitude is 10. 492. If the homologous edges of two similar polyedrons are 2 and 3, what is the ratio of their entire surfaces and of their volumes?

493. Can you show that the volume of a regular tetra

1

edron equals the cube of an edge multiplied by V??
12

494. Can you show that the volume of a regular octa-
1
3

edron equals the cube of an edge multiplied by √2?

495. A side of a cube is the base of a pyramid whose vertex is at the center of the cube. Compare the volumes of the cube and pyramid.

496. If a pyramid is cut by a plane parallel to its base, the pyramid cut off is similar to the first, and the two pyramids are to each other as the cubes of any two homologous edges. Prove.

REGULAR POLYEDRONS.

DEFINITION.

A regular polyedron is a polyedron whose faces are equal regular polygons and whose diedral angles are all equal. [See §456 for figures.]

QUESTIONS.
534.

1. What is the fewest number of faces necessary to form a polyedron? Name the solid.

2. How many faces meet at each vertex of the tetraedon. 3. What is the fewest number of faces required to form

a convex polyedral angle? What is it called?

4. Illustrate how the sum of the face angles of any polyedral angle compares with 4 right angles.

5. What is the angle of an equilateral triangle? How many equilaterals are required to form a solid angle? What is the sum of the face angles forming the solid angle? Could

a convex polyedral angle be formed with four equivalent As? Illustrate. With five? Illustrate. With six? Illustrate. How many regular solid figures may be made using equilateral s?

6. What is the limit of the number of squares that can be used in forming regular convex polyedrons? Illustrate. How many convex polyedrons can be formed with squares? Name the solid.

7. How many regular pentagons may be used to form a solid angle? Illustrate. How many convex polyedrons can be formed with regular pentagons? Name the solid.

8. What is the limit of the number of regular polyedrons formed of regular hexagons? Why?

9. What is the limit of the number of sides of a regular polygon that can be used in forming regular polyedrons.

10. What is the greatest number of regular convex polyedrons? Name those that are bounded by triangles; by squares; by pentagons. These five regular polyedrons are called the Platonic Bodies.

535.

The point within a regular polyedron equally distant from the sides is called the center of the polyedron.

How is the center located from the vertices? Prove it. What is the locus of all points equally distant from a given point?

Do you see how a regular polyedron may be related to the surface of a sphere?

[See § 456 for diagram of the Platonic Bodies.]

Problem: With an edge equal to A B, can you construct a regular tetraedron?

(1) How many faces are required for the solid?

(2) Where is the center of a regular tetraedron? Let V - A B C be the tetraedron required.

B

(3) Can you find the center of the base A B C? How do you erect a perpendicular to a plane at a given point?

(4) In what perpendicular to the base does the vertex of the tetraedron lie? How can you find the required point?

537..

PROPOSITION XXVII.

Problem: Construct a regular hexaedron whose edge equals a given line?