Answer Key: Three-Dimensional Cross Sections

Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection of a particular three-dimensional object analyze the cross-section of a three-dimensional object Activities. You may be familiar with the generic two-dimensional drawing of a three-dimensional cube created like this: In order to represent this three-dimensional object, this drawing attempts to show all three dimensions of the cube. x-axis z-axis y-axis Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

If you were to consider this as a picture in a picture frame, you would see how the x- and y-axes are in the same plane as the frame. x-axis z-axis y-axis Because the z-axis cannot be drawn perpendicular to the x- and y-axes, nor is it in the same plane as the frame, you have what is a called a vanishing point. This is a point in the distance towards which the third dimension would be drawn. A good example of a vanishing point can be seen in these example drawings of roads. Notice how the roads converge to a single point. A drawing with a single vanishing point is called a one-point perspective. The example of the cube above is in one-point perspective; note that if you extend the parallel sides not in the x-y plane, they appear to intersect. What happens if the z-axis is directly perpendicular to your vision using the picture frame example above? In other words, what happens if the z-axis were coming directly out at you perpendicular to the frame? Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

z-axis x-axis y-axis Now you cannot see the third dimension at all, and the cube looks like a square. Notice that the z-axis is represented by a single point. The cube can also be shown in two-point perspective in which you have two vanishing points. z-axis y-axis x-axis Notice how neither the z-axis nor the x-axis is perpendicular to the picture frame any longer. The z-axis vanishing point is off to the upper right of the frame. The x-axis vanishing point is off to the upper left. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

Finally, consider three-point perspective. As you might expect, this is where none of the three axes are perpendicular to our frame (or even appear to be perpendicular to each other). z-axis y-axis x-axis Remember that a figure may not be fully transparent either. So consider the two- and threepoint perspectives with no picture frame, no axes, and with opaque sides. Here are some examples. Determine what perspective the figure is in. Remember to look for a dimension perpendicular to our picture frame. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

Perspective: One-point perspective Perspective: Three-point perspective Solution: Perspective: Two-point perspective. Consider that the y-axis is still perpendicular to a picture frame. It would be the line from the top of the pentagon base to the midpoint of the pentagon base. This figure is part of the answer: Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 5

. In order to look at a two-dimensional view of a three-dimensional object, you can use a mechanism known as a projection. Remember how the previous activity used the analogy of a picture frame to help describe the various perspectives? The picture frame itself is an example of a projection. Consider this diagram: The eye represents the point from which a person would view the object. The plane represents a picture frame (or possibly a window). With a projection, you describe the object as it would be seen on the picture frame. Since a picture is two-dimensional but the object is threedimensional, essentially you are representing a three-dimensional object in two dimensions. The easiest way to think of this is a silhouette. If you were to put back lighting behind the object, only the silhouette would be visible in a window. The silhouette is an example of a projection. Here s an example. Use the picture frame as a projection medium again. If treated the above object as a silhouette, the projection would look like this: Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 6

Technically, any time you are trying to draw a three-dimensional object in a two-dimensional space (such as on this paper), you are creating a projection. All of the diagrams and pictures in this entire lesson are technically projections (shading is only used to create the illusion of depth). Now look at another example with the projection medium at a different angle: Notice how the pentagonal prism is projected on to the plane. Based on the angle, the projection is also a pentagon. Draw the projection of these objects relative to the plane of this paper. Use the silhouette approach as shown in the example. A. Answer: Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 7

. Finally, examine the cross-sections of three-dimensional objects. The cross-section is the intersection of the object with a plane. It is similar to a projection except the plane actually intersects the object. Here is the basic cube again but with an intersecting plane: Notice how the cross section is a square. Here is another example: Because the plane intersects the cube at an angle, it looks like the cross-section is a parallelogram; more accurately, it is a rectangle, but tilted at an angle. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 9

Describe the cross-section of the following figures: A. Answer: triangle B. Answer: rectangle C. Answer: circle Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 0

Geometry A Unit Answer Key: Perimeter and Volume Name Date Objectives In this lesson, you will: describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed apply this idea to solve problems Activities. Before determining the effect of changing a dimension on perimeter, first recall the various formulas used to calculate perimeter. The basic formula for perimeter is: P = a+ b+ c+ d +... + n (n = the number of sides) For rectangles and parallelograms, this can be more specific: P = a+ b For equilateral figures, this can be simplified to: P = an (n = the number of sides) For circles, the perimeter is called a circumference, and the formula for this is: C =πd If the dimensions of a particular polygon change, how does that affect the perimeter? Consider an irregular polygon first: e a b e a b d c d c Notice that the length of side a has been doubled. Look at how that affects the perimeter: P = a+ b+ c+ d + e P = a+ b+ c+ d + e P = ( a+ b+ c+ d + e) + a Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

By doubling the length of one side, the total perimeter value was increased by that one length, or P = P +a. Using the same approach, determine how much the perimeter of the second figure will increase or decrease if the length of all sides were changed. A. First square: Second square: a a Find the perimeter of the first square: P = a Find the perimeter of the second square: P = a = a How much bigger or smaller is the perimeter of the second square? P = P. The perimeter of the second square is one half times the perimeter of the first square. B. 6 6 + Find the perimeter of the first figure: P = 6+ + + = 6 Find the perimeter of the second figure: P = 6+ + 6+ = 8 Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

How much bigger or smaller is the perimeter of the second figure? P = P +. The perimeter of the second figure is units more than the perimeter of the first figure. C. Find the circumference of the first figure: C = π Find the circumference of the second figure: C = π How much bigger or smaller is the circumference of the second figure? C = C. The circumference of the second circle is three times larger than the circumference of the first circle.. Now, applying the same approach that you learned above, you will determine the effect of changing dimension on area. Here is a review of the various formulas for area. The area of a rectangle (and a square, which is a specific type of rectangle) is: A = lh (l = length, h = height) The area of a triangle is: A = bh (b = length of base, h = height) The area of a trapezoid is: A = ( a+ b) h (a and b are the parallel sides of the trapezoid, h is the height) The area of a circle is: A =πr (r = radius) Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

How does changing the dimensions affect the area? Consider this example: 6 Notice how the length of one side of the rectangle is halved. Look at how that affects the area. A = lh A = lh A = (6)() = 6 A = ()() = (6)() = A = A A. a a a a a a a a Find the area of the first square: A = ( a)( a) = a Find the area of the second square (note that the length of all sides is changed): A = = a a a How much bigger or smaller is the area of the second square? A = A. The area of the second square is one fourth the size of the first. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

B. Find the area of the first figure: A = bh = ( )( ) ( )( ) 8 = = Find the area of the second figure: A = = ( )( ) = ( )( ) 6 bh = How much bigger or smaller is the area of the second figure? A = A. The second figure s area is of the first figure s area. Notice that h = h. C. Given two rectangles, and, the area of rectangle =, and the area of rectangle = 0. Rectangle has the same length as rectangle. Determine how much larger the width of rectangle is compared to the width of rectangle. Show your work in the space provided below. A = lw and A = lw = lw and 0 = lw Solving for w and w you get: w w 0 =, and w = l l = xw 0 = 5 x 0 = x( ) = x l l w 5 = w Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 5

. Now, applying the same approach that you learned above, you will determine the effect of changing dimension on volume. Here is a review of some common volume formulas. The basic formula for a rectangular solid is: V = lwh (l = length, w = width, h = height) The basic formula for a prism is: V = Bh (B = area of the base figure, h = height) (B is the shaded area) For cylinders: V =πr h (r = radius, h = height) For pyramids: V = Bh (B = area of the base figure, h = height) (B is the shaded area) For cones: V = πr h (r = radius, h = height) For spheres: V = πr (r = radius) Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 6

If the dimensions of a particular solid are changed, how does that affect the volume? Look at these pyramids: The dimensions of the base of the second figure are the same as the dimensions of the base of the first figure, but the height of the second figure is expanded by the height of the first figure. Thus, h = h. V = Bh V = Bh V V = B h = Bh By substituting V = Bh into the formula for V, you can see that V = V. A. Given two cubes, with the first cube having a side of h, and the second cube having a side of h, what is the relation of the volume of the first cube to the volume of the second (note that all three dimensions have changed)? V = h V ( ) = h V = h V = 7h V = 7V. The second cube s volume is 7 times more than the volume of the first cube Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 7

B. 5 Compare the volume of the pyramid above and a second pyramid in which the height is doubled and the side currently equal to is tripled. Express the volume of the second pyramid in terms of the first (note that the base is a rectangle). Show your work in the space provided. V = Bh and B = lw V = Bh V = lwh V V = 6V V = lw h = lwh Check by plugging in numbers: V = ()()(5) V = ()()()()(5) V = ()(5) V = ()()(0) V = 0 V = 0 C. Given two spheres, if the volume of the first sphere is 6π, and the volume of the second sphere is 88 π, what is the relation of the second radius to the first radius? Show your work in the space provided. V = πr 6π= πr V = πr 88π = πr 7 = r 6 = r = r 6 = r r = r Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 8

Geometry A Unit Answer Key: Surface Area and Volume of a Sphere Name Date Part : Surface Area and Volume of Spheres Objective In this part of the lesson you will find the surface area and volume of a sphere, given the formulas. Activities. Review the formulas for area and circumference. To calculate the circumference of a circle, a special mathematical value is needed. This value is called pi ( π ). Pi is the ratio of the circumference to the diameter and is approximately equal to.59. The use of pi essentially deals with the fact that a circle is round and, therefore, requires a different approach than polygons. The perimeter, or circumference, will always be in terms of pi (or a decimal approximation). The formula for circumference is: Circumference = pi diameter = π d The area of a circle will also be in terms of pi. The formula for area is: Area = pi radius = πr A sphere is essentially the same shape in three dimensions that a circle is in two dimensions. Just like a circle, a sphere has a center point, and all other points on the sphere are equidistant from the center point. In real life, a good example of a sphere is a ball. Earth is often represented as a sphere as well (although, technically, it s not perfectly round). r Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

The surface area of a sphere is the amount of space covering the entire surface of the sphere. The formula for surface area is: Surface area = pi radius = πr Calculate the surface area of these spheres, using. as an approximation for π : Sphere with radius of : Surface area = 6π or 00.96 Sphere with radius of 7: Surface area = 96 π or 65. Sphere with radius of : Surface area = 6 π or 50. Sphere with radius of : Surface area = 6 π or.0. Now calculate the volume of a sphere. Volume = = pi radius πr Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

Calculate the volumes of these spheres, using. as an approximation for π : Sphere with radius of : Volume = 56 or 67.95 π Sphere with radius of 7: Volume = 7 or 6.0 π Sphere with radius of : Volume = or.9 π Sphere with radius of : Volume = 6 π or.0 Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

Part : Describing Points on Spheres Objective In this part of the lesson you will describe sets of points on the surface and the interior of spheres: chords, tangents, great circles, and small circles. Activity. There are different ways to describe geometric objects associated with a sphere. A chord is a line segment in which both endpoints lie on the surface of the sphere. r A tangent to a sphere is a line or line segment (or plane, for that matter) that touches the sphere at exactly one point. r Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

A great circle is a circle on the edge of a sphere in which the circumference of the circle is equal to the circumference of the sphere. This circle divides the sphere into two hemispheres. A good example of this would be the Earth s equator (the Earth is not perfectly round, so this is just an approximation). A great circle is the largest circle that can be drawn on a sphere. If you were to intersect a plane with a sphere, and it passed through the center of the sphere, you would have a great circle. r A small circle is a circle constructed by a plane intersecting a sphere that does not go through the center. Given these intersections with a sphere, describe the resulting set of points: r The intersection of the sphere and this plane is a small circle. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 5

r The heavy line is a chord. r The point shows that this plane is tangent to the sphere r The dark area, or intersection, is a great circle. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 6

Geometry A Unit Answer Key: Polyhedron Name Date Objectives In this lesson, you will: understand the definition of polyhedron and net (or pattern) describe the polyhedron that can be made from a given net (or pattern) describe the net for a given polyhedron Links Polyhedron Nets http://mathworld.wolfram.com/topics/polyhedronnets.html Paper Models of Polyhedra http://www.korthalsaltes.com/ The Platonic Solids http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html The Uniform Polyhedra http://www.mathconsult.ch/showroom/unipoly/unipoly.html Activities. What is a polyhedron? A polyhedron is a three dimensional shape made up of a number of polygonal faces. The faces meet in edges, and the edges meet in vertices. A polyhedron is analogous to a three-dimensional polygon. The plural of polyhedron is polyhedra. Polyhedra are described by a number of characteristics, including: Convex: a polyhedron of which any plane section is a convex polygon Vertex-uniform: all vertices are the same (i.e., each vertex is surrounded by the same faces in the same order) Edge-uniform: all edges have the same length Face-uniform: all faces are congruent Regular: the polyhedron is vertex-uniform, edge-uniform, and face-uniform However, not all polyhedra share the characteristics listed above. Polyhedra are also described by the following characteristics: number of sides on each face number of faces of the polyhedron number of edges of the polyhedron number of vertices of the polyhedron Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

There are exactly five regular and convex polyhedra. They are known as the Platonic solids. They are the only polyhedra in which the faces are all the same regular polygon. Tetrahedron: faces vertices 6 edges all faces are equilateral triangles Cube (also known as a hexahedron): 6 faces 8 vertices edges all faces are squares Octahedron: 8 faces 6 vertices edges all faces are equilateral triangles Dodecahedron: faces 0 vertices 0 edges all faces are equilateral pentagons Icosahedron: 0 faces vertices 0 edges all faces are equilateral triangles Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

Consider the following polyhedra. Describe the number of faces, vertices, and edges. A. This is called a truncated tetrahedron: Faces: 8 Vertices: Edges: 8 B. This is called a truncated octahedron: Faces: Vertices: Edges: 6 Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

C. This is called a rhombicuboctahedron: Faces: 6 Vertices: Edges: 8. All polyhedra can be constructed from a two-dimensional pattern. This pattern is also referred to as a net or a map. Consider this particular map: E A B C D F Notice that if you were to cut it out, fold it along the edges, and tape it together, it would form a cube. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning

A. Right face B. Bottom face C. Left face D. Top face E. Back face F. Front face Maps are not unique and can be generated in several possible ways. Here is another possible cube map. A B C D E F A. Right face B. Bottom face C. Left face D. Front face E. Top face F. Back face Notice that it is easy to see the number of edges and faces from a net. You can use this example, as well as the shape of the faces, to help identify the particular polyhedron created by any particular net. Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 5

In the following examples, identify the particular polyhedron. You may need to visit the websites listed at the beginning of this lesson to get the specific names of the polyhedron. A. octahedron B. icosahedron C. truncated tetrahedron Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 6

. Try the reverse. Given the polyhedron, create the map. Be careful, though. The map for the cube is not just a bunch of 6 squares put together. A B C D E F Notice that in this example, creating a cube is not possible because faces A and F will overlap. It also would leave one face empty. Try some examples where you are given the polyhedron, and you create the map. A. Answer: B. Answer: Copyright 006 PLATO Learning, Inc. All rights reserved. PLATO is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning 7