I would like to dedicate this book to my brothers and sisters: Mary Lilja, Virginia Bellrichard, Thomas Errthum, Christine Breuer, Charles Errthum, and Joseph Errthum. A special thank you to Charles, for encouraging me to patent the original models. It is impossible for me to express my true appreciation for the guidance and support all of you have provided me over the years. You are wonderful brothers and sisters. I would like to also thank Vincent Kotnik III and his wife, Cathy, who have been like a brother and sister to me. Vincent, you have been a wonderful friend and a great colleague. Thank you for mentoring me my first three years of teaching; you guided me well. Cathy, your positive attitude is an inspiration not only for me, but for anyone who meets you. I cherish the fact that I have both of you as friends. I have great respect and admiration for each of you. You are truly very special people. Please know that you are all near and dear to my heart. Emily Errthum ETA 75545 Copyright 2005 by Classroom Products. All rights reserved. Except as permitted under the United States Copyright Act and where expressly permitted, no part of this publication may be reproduced or distributed in any form by any means, or stored in a database or retrieval system, without prior written permission from the publisher. However, the publisher grants permission to classroom teachers to reproduce activity pages for one classroom only. The reproduction of any part for an entire school or system is strictly prohibited. Send all inquiries to: ETA/Cuisenaire 500 Greenview Court Vernon Hills, IL 60061
INTRODUCTION Prior to Getting Into Solids, the solids were just that solid. The lateral faces were made of plastic that tightly enclosed each solid. It was impossible to get inside the model to identify the height, apothem, radius, central angle measure, and the right triangles needed to find missing lengths. Leeanne Branham, from Fresno Pacific University, best identified the problem with solids in the March 1998 issue of CMC ComMuniCator. In the article Getting Into Pyramids Leeanne wrote, (referring to the solid lateral face models) but one drawback that these models all share was that I couldn t get inside them. Her article was designed, with worksheets, to assist students in solving surface area and volume problems by redrawing the parts of a figure on one-dimensional paper. I identified with her frustration as well as observed the frustration level of my own students. This inspired me to design the three-dimensional open models, Getting Into Solids Pyramids and Getting Into Solids Prisms. These new models literally allow the learner to get inside the solid. The Getting Into Solids Prisms models allow students to: A. Identify the different parts of the regular prisms and right cylinders by colors. 1. Brass rod height 2. Green string lateral edge, height 3. Etched green line on the base radius 4. Etched red line on the base apothem 5. Etched blue line on the base base edge B. Attach numbers to the model for ease of visualization. 1. Write on the plexiglass base with overhead markers 2. Attach stick-on notes with unit measure on the height C. Employ all learning styles. 1. Visual 2. Auditory (when guided by the teacher or students teaching others in groups) 3. Kinesthetic/Tactical 1
Section One of this book contains reproducible worksheets for finding the surface area and volume of regular prisms and right cylinders. Each model has three worksheets. The first worksheet includes dimensions. The second worksheet serves as an answer key for the first worksheet. Finally, the third worksheet is blank so that the instructor or student can assign dimensions to any illustration specific from a classroom textbook. The worksheets in this section are intended to guide students through a frequently long and difficult process. Helping students to organize their mathematical steps provides students with immediate success and, as a result, promotes a positive experience for finding surface area and volume. Section Two of this book (beginning on page 22) also contains reproducible worksheets for finding the surface area and volume of regular prisms and right cylinders. This section is less structured than section one. It is designed for students to take on the challenge of organizing their thought process in a logical order. Students will need to show multiple mathematical steps to yield the desired result. With prior work in section one, students are easily able to make this transition. Students may find it more difficult, but many will show extreme growth in their organizational skills. Section two utilizes concepts from the book Getting Into Solids Pyramids. Students will need to know how to calculate the volume of a pyramid and the volume of a cone. Brass rods are provided to build and display each of the six prism models at one time. A shorter brass rod is included to allow the construction of a 16 cm x 16 cm x 16 cm cube, or to change the height of any of the other models. Easy to assemble instructions: (see diagram below) Green String 1. Screw the brass rod into polygon center (representing height). 2. Screw a congruent polygon on the opposite side of the brass rod. 3. Slide the end of the green string (representing lateral edge, height) into the notch of each base. The string should be perpendicular to the bases. 4. Repeat Step 3 for all green strings, alternating from side to side. 5. Stick-on notes can be attached to the green string, with unit measure (representing lateral edge, height). 6. Turning one of the prism bases 180º will create symetrical pyramid models. Use twist ties, string, or tape to retain the desired shape. Brass Rod Notches Twist to Create Pyramids Polygon Base 2
Vocabulary for Regular Prisms and Right Cylinders L.A. = lateral area P = perimeter of base r = radius S.A. = surface area B = area of base h = height V = volume π = pi Formulas for Regular Prisms and Right Cylinders Regular Prisms Right Cylinders L.A. = P h L.A. = 2 π r h S.A. = L.A. + 2 B S.A. = L.A. + 2 B S.A. = P h + 2 B S.A. = 2 π r h + 2 π r 2 V = B h V = B h V = π r 2 h TABLE OF CONTENTS Section One: Finding Surface Area and Volume of Regular Solids Cylinder....4 Triangular Prism....7 Cube....10 Rectangular Prism...13 Pentagonal Prism...16 Hexagonal Prism...19 Section Two: Challenge Problems Finding Surface Area and Volume of Regular Solids Two Attached Rectangular Prisms...22 Rectangular Prism and Regular Square Pyramid...24 Two Cylinders; Maximizing Lateral Area...26 Cylinder with Two Cones Inserted...28 Rectangular Prism with a Rectangular Hole...30 Rectangular Prism with a Cylindrical Hole....32 3
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Name of the solid: Worksheet 1 diameter = 7 ft height = 20 ft Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 4
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Right Cylinder Name of the solid: Worksheet 1: Answer Key (difficulty medium) diameter = 7 ft height = 20 ft L.A. + 2 B Equation for formula: S.A. = B h Equation for formula: V = circle First, I will solve for the area of the Base, which is a. First, I will find the radius, r. Now I can find the area of the base. r = 1 / 2 diameter B = π r 2 r = 1 / 2 (7) B = π (3.5) 2 r = 3.5 B = 12.25π 12.25 π ft 2, rectangle B = Next, I will solve for the Lateral Area which is a. L.A. = 2 π r h L.A. = 2 π (3.5) (20) L.A. = 140 π 140 π ft 2 L.A. + 2 B B h L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = 140 π + 2 (12.25 π) 12.25 π (20) S.A. = 140 π + 24.5 π Solve the equations. (Show all algebraic steps.) S.A. = 164.5 π ft 2 V = 245 π ft 3 S.A. = 516.79 ft 2 V = 769.69 ft 3 5
Find Surface Area and Volume of the regular solid. Worksheet 2 Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 6
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Name of the solid: Worksheet 3 4 in. 18 in. Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 7
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Triangular Prism Name of the solid: Worksheet 3: Answer Key (difficulty - medium) 4 in. 18 in. Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = S 2 3 4 B = 4 2 3 4 B = 4 3 L.A. + 2 B B h triangle 4 3 in. 2 rectangle B = Next, I will solve for the Lateral Area, which is a. First I will find the Now I can find the perimeter of the base, P. lateral area, L.A. P = 3 S L.A. = P h P = 3 (4) L.A. = 12 (18) P = 12 L.A. = 216 L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 8 216 in. 2 L.A. + 2 B B h 216 + 2 (4 3 ) 4 3 (18) 216 + 8 3 in. 2 72 3 in. 3 S.A. = 229.86 in. 2 V = 124.71 in. 3
Find Surface Area and Volume of the regular solid. Worksheet 4 Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 9
Find Surface Area and Volume of the regular solid. Worksheet 5 Fill in the blanks and show all your work. Name of the solid: 5 cm Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 10
Find Surface Area and Volume of the regular solid. Worksheet 5: Answer Key (difficulty - easy) Fill in the blanks and show all your work. Cube Name of the solid: 5 cm L.A. + 2 B Equation for formula: S.A. = B h Equation for formula: V = square First, I will solve for the area of the Base, which is a. B = S 2 B = 5 2 B = 25 25 cm 2 square B = Next, I will solve for the Lateral Area, which is a. First I will find the Now I can find the perimeter of the base, P. lateral area, L.A. P = 4 S L.A. = P h P = 4 (5) L.A. = 20 (5) P = 20 L.A. = 100 L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. 100 cm 2 L.A. + 2 B B h 100 + 2 (25) 25 (5) S.A. = 100 + 50 150 cm 2 125 cm 3 (Show all algebraic steps.) S.A. = V = 11
Find Surface Area and Volume of the regular solid. Worksheet 6 Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 12
Find Surface Area and Volume of the regular solid. Worksheet 7 Fill in the blanks and show all your work. Name of the solid: 3 m Equation for formula: S.A. = Equation for formula: V = 4 m 15 m First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 13
Find Surface Area and Volume of the regular solid. Worksheet 7: Answer Key (difficulty - easy) Fill in the blanks and show all your work. Rectangular Prism Name of the solid: L.A. + 2 B Equation for formula: S.A. = B h Equation for formula: V = 4 m 15 m 3 m rectangle First, I will solve for the area of the Base, which is a. B = l w B = 15(4) B = 60 Note: Area of the base, B, could also be 12 m or 45 m. Teachers will need to specify which base to select in a rectangular prism. The surface area and volume results will not be affected. 60 m 2 rectangle B = Next, I will solve for the Lateral Area, which is a. First I will find the Now I can find the perimeter of the base, P. lateral area, L.A. P = 2 l+ 2 w L.A. = P h P = 2 (15) + 2 (4) L.A. = 38 (3) P = 30 + 8 L.A. = 114 P = 38 L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = 234 m 2 V = 180 m 3 14 114 m 2 L.A. + 2 B B h 114 + 2 (60) 60 (3) S.A. = 114 + 120
Find Surface Area and Volume of the regular solid. Worksheet 8 Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 15
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Worksheet 9 regular pentagon base edge = 2 cm apothem = 1.376 cm 13 cm Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 16
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Pentagonal Prism Name of the solid: L.A. + 2 B Equation for formula: S.A. = Worksheet 9 Answer Key: (difficulty - hard) regular pentagon base edge = 2 cm apothem = 1.376 cm 13 cm B h Equation for formula: V = pentagon First, I will solve for the area of the Base, which is a. First I will find the Now I can find the perimeter of the base, P. area of the base, B. P = 5 S B = 1 / 2 Pa P = 5 (2) B = 1 / 2 (10)(1.376) P = 10 B = 6.88 B = Next, I will solve for the Lateral Area, which is a. L.A. = P h L.A. = 10(13) L.A. = 130 6.88 cm 2 rectangle 130 cm 2 L.A. + 2 B B h L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = 130 + 2 (6.88) 6.88 (13) S.A. = 130 + 13.76 Solve the equations. (Show all algebraic steps.) S.A. = 143.76 cm 2 V = 89.44 cm 3 17
Find Surface Area and Volume of the regular solid. Worksheet 10 Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 18
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Worksheet 11 regular hexagon base edge = 10 ft height = 40 ft Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 19
Find Surface Area and Volume of the regular solid. Fill in the blanks and show all your work. Hexagonal Prism Name of the solid: Worksheet 11: Answer Key (difficulty - hard) regular hexagon base edge = 10 ft height = 40 ft Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = L.A. + 2 B B h hexagon First, I will find the Second, I will find the Now I will find the perimeter of the base, P. apothem of the base. area of the base B. P = 6 S 1 /2 the base edge of 10 B = 1 / 2 Pa P = 6 (10) is 5, which is the short B = 1 / 2 (60)(5 3 ) P = 60 leg of a special 30-60 -90 B = 150 3 triangle. So the apothem, long leg, is 5 3. Next, I will solve for the Lateral Area, which is a. L.A. = P h L.A. = 60(40) L.A. = 2,400 5 3 150 3 ft 2 60 5 rectangle 30 L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = 2,400 + 300 3 ft 2 V = 6,000 3 ft 3 20 2,400 ft 2 L.A. + 2 B B h 2,400 + 2 (150 3 ) 150 3 (40) S.A. = 2,919.62 ft 2 V = 10, 392.30 ft 3
Find Surface Area and Volume of the regular solid. Worksheet 12 Fill in the blanks and show all your work. Name of the solid: Equation for formula: S.A. = Equation for formula: V = First, I will solve for the area of the Base, which is a. B = Next, I will solve for the Lateral Area, which is a. L.A. = Write the equations. S.A. = V = Replace variables with their value. S.A. = V = Solve the equations. (Show all algebraic steps.) S.A. = V = 21
6 ft 2 ft 14 ft Worksheet 13 Surface Area Challenge Find the surface area of the right solid. Describe your process through a series of clearly shown mathematical steps. 9 ft 10 ft Solution: 22
2 ft Worksheet 13: Answer Key Surface Area Challenge 6 ft 14 ft Find the surface area of the right solid. Describe your process through a series of clearly shown mathematical steps. 9 ft 10 ft Step 1: I will find the surface area of the large rectangular prism less the surface area of the 2 x 9 area. First, I will find the perimeter Second, I will find the Third, I will find the of the base, P, using the formula lateral area, L.A. area of the base, B, P = 2 l + 2 w L.A. = P h using the formula P = 2 (10) + 2 (9) L.A. = 38 (6) B = l w P = 20 + 18 L.A. = 228 B = 10 (9) P = 38 B = 90 Fourth, total surface area of Finally, I will subtract 18 the large rectangular prism is (the area of the 2 x 9 area) S.A. = L.A. + 2 B because it is hidden by the S.A. = 228 + 2 (90) top rectangular prism. S.A. = 228 + 180 S.A. = 408-18 S.A. = 408 S.A. = 390 Step 2: I will find the surface area of the small rectangular prism less the surface area of the 2 x 9 area. I will find the perimeter of the Second, I will find the Third, I will find the base, P, using the formula lateral area, L.A. area of the base, B, P = 2 l + 2 w L.A. = P h using the formula P = 2 (2) + 2 (9) L.A. = 22 (8) B = l w P = 4 + 18 L.A. = 176 B = 2 (9) P = 22 B = 18 Fourth, total surface area of Finally, I will subtract 18 (the area of the 2 x 9 area) the small rectangular prism is because it is hidden by the bottom rectangular prism. S.A. = L.A. + 2 B S.A. = 212-18 S.A. = 176 + 2 (18) S.A. = 194 S.A. = 176 + 36 S.A. = 212 Step 3: The surface area of large rectangular prism + surface area of small rectangular prism will equal the total surface area. S.A. = 390 + 194 S.A. = 584 ft 2 Solution: 584 ft 2 23
8 in. 8 in. 9 in. 9 in. 10 in. 12 in. Worksheet 14 Volume Challenge Find the surface area of the right solid. Describe your process through a series of clearly shown mathematical steps. 24 Solution:
8 in. 8 in. 9 in. 9 in. 10 in. 12 in. Worksheet 14: Answer Key Volume Challenge Find the surface area of the right solid. Describe your process through a series of clearly shown mathematical steps. Step 1: I will find the volume of the square pyramid. You will need to reference the first book entitled, Getting Into Solids Pyramids. First, I will find the area of Second, I will use the formula the base, B. for the volume of a regular pyramid. B = S 2 V = 1 / 3 B h B = 9 2 V = 1 / 3 (81)(10) B = 81 V = 270 Step 2: I will find the volume of the square prism. Note: Since the base of the Now I will find the volume of the square pyramid is exactly square prism using the formula the same as the base of the V = B h square prism, the area of the V = 81 (12) bases will be the same. V = 972 Therefore, B = 81 Step 3: I will now add the volume of the square pyramid to the volume of the rectangular prism to find the total volume. V = 270 + 972 V = 1,242 in. 3 V = 1,242 in. 3 Solution: 25
Worksheet 15 11.2 cm 3.5 cm 4 cm 8.4 cm Surface Area Challenge Your company, Premium Sauce, is going to purchase one of the two 16-ounce cans shown and fill them with your secret tomato sauce products. Your task is to determine which can will minimize the paper that will go around the can to advertise the secret sauce. Write a memo to the advertising department informing them of the dimensions. Describe your process through a series of clearly shown mathematical steps. Solution: 26
Worksheet 15: Answer Key 11.2 cm 3.5 cm 4 cm 8.4 cm Surface Area Challenge Your company, Premium Sauce, is going to purchase one of the two 16 ounce cans shown and fill them with your secret tomato sauce products. Your task is to determine which can will minimize the paper that will go around the can to advertise the secret sauce. Write a memo to the advertising department informing them of the dimensions. Describe your process through a series of clearly shown mathematical steps. Step 1: I will find the lateral area of the tall right cylinder. L.A. = 2 π r h L.A. = 2 π (3.5)(11.2) L.A. = 78.4 π L.A. = 246.301 cm 2 Step 2: I will find the lateral area of the short right cylinder. L.A. = 2 π r h L.A. = 2 π (4)(8.4) L.A. = 67.2 π L.A. = 211.115 cm 2 Step 3: The lateral area that yields the minimum paper is the short right cylinder. There will be a savings of 35.186 cm 2 of paper for each can. Step 4: I will now find the dimensions of the lateral area, which is a rectangle. The circumference of the circle will be the length. I will find circumference using the formula C = 2 π r C = 2 π (4) C = 8 π C = 25.133 Step 5: The dimensions of the rectangle will be circumference x height. The dimensions are 25.1 cm x 8.4 cm (The memo written to the advertising department will vary. However, the dimensions should be the same.) 25.1 cm x 8.4 cm Solution: 27
21 in. 8 in. 9 in. 9 in. Worksheet 16 Volume Challenge Find the volume of the cylinder if the two cones do not contain any substance. Describe your process through a series of clearly shown mathematical steps. 28 Solution:
21 in. 8 in. 9 in. 9 in. Worksheet 16: Answer Key Volume Challenge Find the volume of the cylinder if the two cones do not contain any substance. Describe your process through a series of clearly shown mathematical steps. Step 1: I will find the volume of the two right cones. You will need to reference the first book entitled, Getting Into Solids Pyramids. First, I will find the area of the base, B, of the cone, which is a circle. B = π r 2 B = π (8) 2 B = 64 π Second, I will find the volume of the right cone using the formula V = 1 / 3 B h V = 1 / 3 (64 π)(9) V = 192 π Third, since the two right cones are the same, I will double the volume to find the total volume of both cones. V of both cones = 192 π(2) V of both cones = 384 π Step 2: I will find the volume of the right cylinder. Note: Since the base of the right Now I will find the volume of the cylinder is exactly the same as the right cylinder using the formula, base of the right cones, the area V = B h of the bases will be the same. V = 64π (21) Therefore, B = 64 π V = 1,344 π Step 3: I will find the volume of the solid by subtracting the volume of the two right cones from the volume of the right cylinder. V = volume of the cylinder - volume of the 2 cones V = 1,344 π - 384 π V = 960 π in. 3 960 π in. 3 (or) 3,015.93 in. 3 Solution: 29
Worksheet 17 20 cm 20 cm 10 cm Surface Area Challenge Find the surface area of the rectangular prism with a 10 cm x 12 cm hole through the center. Describe your process through a series of clearly shown mathematical steps. Solution: 30
20 cm 20 cm 10 cm Worksheet 17: Answer Key Surface Area Challenge Find the surface area of the rectangular prism with a 10 cm x 12 cm hole through the center. Describe your process through a series of clearly shown mathematical steps. Step 1: I will find the surface area of the large rectangular prism and subtract the area of the two 10 cm x 12 cm regions. First, I will find the Second, I will find the L.A. Third, I will find the area perimeter of the base, P. L.A. = P h of the base, B. P = 2 w + 2 l L.A. = 80(10) B = l w P = 2(20) + 2(20) L.A. = 800 B = 20(20) P = 80 B = 400 Fourth, I will calculate Fifth, I will find the area Finally, the surface area of the surface area of the of the two open regions that the outside region of the large rectangular prism using 10 cm x 12 cm. rectangular prism is the S.A. less the S.A. formula. A = 2(lw) the area of the two open regions. S.A. = L.A. + 2 B A = 2(10)(12) S.A. = 1,600-240 S.A. = 800 + 2(400) A = 240 S.A. = 1,360 S.A. = 1,600 Step 2: I will calculate the lateral area of the inside rectangular prism with 10 cm x 12 cm dimensions. First, I will find the perimeter of Second, I will find the lateral area, L.A. the base, P, of the 12 x 10 region. L.A. = P h P = 2 l + 2 w L.A. = 44(10) P = 2(12) + 2(10) L.A. = 440 P = 44 Step 3: Iʼll combine the S.A. of the outside region plus the L.A. of the inside region. S.A. = Surface area of the outside region + inside lateral area S.A. = 1,360 + 440 S.A. = 1,800 cm 2 1,800 cm 2 Solution: 31
Worksheet 18 32 in. Volume Challenge Find the volume of the rectangular prism with a cylindrical hole through the prism. The diameter of the cylinder is 10 inches. Describe your process through a series of clearly shown mathematical steps. 25 in. 29 in. Solution: 32
32 in. Worksheet 18: Answer Key Volume Challenge Find the volume of the rectangular prism with a cylindrical hole through the prism. The diameter of the cylinder is 10 inches. Describe your process through a series of clearly shown mathematical steps. 25 in. 29 in. Step 1: I will find the volume of the rectangular prism. First, I will find the area Second, I will find the volume of the base, B. of the rectangular prism. B = l w V = B h B = 25(29) V = 725 (32) B = 725 V = 23,200 Step 2: I will find the volume of the cylinder with a diameter of 10. First, I will find the radius Second, I will find the volume of the circle, r. of the right cylinder. d = 2 r V = π r 2 h 10 = 2 r V = π (5) 2 (32) 5 = r V = 800 π (or 2,513.274) Step 3: I will subtract the volume of the right cylinder from the volume of the rectangular prism. V = Volume of rectangular prism - volume of the right cylinder V = 23,200-800 π in. 3 V = 20,686.726 in. 3 V = 20,686.726 in. 3 Solution: 33