NovaNET - Assignment #G2A1
Geometry Semester 2
Homework on Basic Solids Name: _______________________________
Problems 1-4: Refer to the object on the right, which
is made of 5 cubes. Each cube
measures one cubic inch.
1. What is the volume of this object?
2. What is the surface area of this object?
3. Draw sketch of the right view. Be sure to show any hidden edges with dashed lines.
4. Draw sketch of the top view. Be sure to show any hidden edges with dashed lines.
5. Sketch a solid with volume 6 cubic units and the top view below:
6. Which of the following figures are prisms? (Circle the letters of all that are.)
a. b. c.
d. e.
6. Find the value of x in each of the following right rectangular prisms. Give the answers in simplified radical form:
a. b.
7. A box has a square base measuring 12 inches on a side. A rod that is 22 inches long will just barely fit in this box. How tall is the box?
8. A box is 20 inches long, 14 inches wide
and 12 inches tall. It is to be packed with
smaller boxes that are 3 inches by 2 inches
by 2 inches. What is the most number of
smaller boxes that can be packed into the
larger box?
9. A storage room is L-shaped, and the floor area is 170 square feet. A layer of boxes, each being 1 foot long, 1 foot wide and 1 foot tall, is placed on the floor.
a. At most how many boxes are on the floor?
b. If the room is 8 feet tall, at most how many boxes total can be stored in it?
10. Each cube in this “pyramid” (it isn’t really a pyramid in the mathematical sense) measures one cubic foot, and there is no empty space inside.
a. What is the volume of this “pyramid”?
b. What is the surface area of this “pyramid”?
Below is a net for the right rectangular pyramid with a square base pictured on the right:
11. What is the slant height of this pyramid?
12. What is the area of one of the faces of this pyramid?
13. What is the lateral surface area of this pyramid?
14. What is the total surface area of this pyramid?
15. The diameter of the base of a right circular cone is 8 cm and its altitude is 10 cm. What is the lateral surface area of this cone?
16. What is the lateral surface area of a right cylinder with diameter 10 cm and altitude 14 cm?
17. a. How many square inches are in a square foot?
b. How many cubic inches are in a cubic foot?
18. What is the volume of a cube which has a surface area of 384 in2?
19. When filled with water, Elaine's water-bed mattress is 7 feet long, 59 inches wide, and 7 inches high. Water weighs 62.43 pounds per cubic foot. About how much does Elaine’s water-bed mattress weigh when full?
20. Albert is at a carnival where there is a large jar of jelly-beans. If he can guess how many are in the jar he will win a prize. The jar is 10 inches in diameter and 14 inches tall. Albert happens to have a smaller jar with him. He buys enough jelly-beans to fill his own jar. It takes 162 jelly-beans, and measures 4 inches in diameter and is 6 inches tall. How many jelly-beans should he guess are in the carnival jar?
21. An artist made an aluminum scale model of a sculpture. The scale model is 18 inches tall, and the actual sculpture, also made out of aluminum, is 6 feet tall.
a. The scale model weighs 20 lbs. What is the weight of the actual sculpture?
b. The area of a face of the cube in the scale model is 6 in2. What is the area of the corresponding face in the actual sculpture?
c. What is the volume of the cube in the actual sculpture?
22. A frustum of a cone is the part of the cone that is left when a plane cuts off the part containing its apex. A repair man needs to make a frustum of a cone to bridge two circular ducts which are 5 inches apart and have diameters 6 inches and 9 inches:
He has a large sheet of thin metal from which he will cut a net that will fold into this frustum, so he needs to know three important dimensions of that net: the inner radius, r, the outer radius, R, and the angle of the sector, θ.
Find R and r to the nearest hundredth of an inch and find θ to the nearest tenth of a degree.
A Platonic Solid is a regular polyhedron. “Regular” means that all the edges are of equal length, all the angles of equal measure, and all faces are congruent shapes. It can be shown that each of the following is true of a Platonic solid:
1. All the vertices lie on a sphere.
2. All dihedral angles are equal.
3. Every face is a regular polygon.
4. All vertices are surrounded by the same number of faces.
It was known to Plato and company in ancient Greece that there are only 5 of them, and these Greeks thought they were the basic building blocks of the universe. They are named (from Greek) according to the number of faces (the Greek word for “face” is εδρον “hedron”):
4 faces: 6 faces: 8 faces: 12 faces: 20 faces :
tetrahedron hexahedron octahedron dodecahedron icosohedron
(aka “cube”)
23. Fill in the blanks to prove there are only 5 Platonic Solids:
At each vertex of a Platonic Solid, at least 3 faces must come together, because if only two came together then you wouldn’t get a solid (they would fold flat against each other). Also, the sum of the interior angles of those faces must be less than 360o, because _______________________________
________________________________________________________________________________.
Suppose each face could be an equilateral triangle:
Each interior angle of an equilateral triangle measures _______, so you could have at least _____ of them and at most _______ of them at a vertex (since the sum of the angles at the vertex must be less than 360o).
Using equilateral triangles at each vertex gives _____ Platonic Solids, the ___________________,
the _____________________, and the ___________________________.
Now suppose each face is a square:
Each interior angle of a square measures ___________, so you would have exactly ________ of them at a vertex since the sum of the angles cannot be 360o. Using squares at each vertex gives us the
________________________.
Now suppose each face is a regular pentagon:
Each interior angle of a regular pentagon measures ___________, so you could fit together only _______ of them at a vertex, and this gives the ____________________.
If you try to use a regular polygon with more than 5 sides, each interior angle would measure at least __________, and since you need at least 3 of them at a vertex, the sum of the angles there would be at least ___________, which cannot be. Thus there can be only 5 Platonic Solids.
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