One set of - Western Illinois...

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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.

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500 Greenview CourtVernon Hills, IL 60061

ETA 75542

This book is dedicated to the memory of my parents, Ray and Marie Errthum, who helped me design and build my first models in their garage. They always encouraged me to create and share. Emily Errthum

1

“Student skills in visualization and reasoning about spatial relationships are fundamental in geometry” (National Council of Teachers of Mathematics, NCTM)

Getting Into Solids is an exciting new tool that has been designed to develop students’ spatial visualization skills by enabling them to examine three-dimensional figures from various perspectives. These exciting Solids allow students to view normally hidden lines that are crucial to foundational understanding and are critical for student success in geometry. Through careful examination, students will be able to identify and visualize the Solids from various perspectives and solve problems involving three-dimensional figures.

Getting Into Solids enables students to visualize the component parts of pyramids and cones. This unique set includes a Cone and the following right pyramids:

Triangular - Square - Rectangular - Pentagonal - Hexagonal

As students begin to visualize the integral parts of these Solids they will become confident at solving problems involving 3-dimensional figures.

Color is used to make key parts stand out especially those that form right triangles. For example, since the apothem and slant height form a right triangle with the height/altitude, they are both red. Similarly, the lateral edge and the radius form a right triangle with the height/altitude, they are both green..

One set of Getting Into Solids will allow 4-6 students to explore the shapes while solving a problem displayed on the board or from their textbook. Once students have had the opportunity to explore the Solids, place them around the room for easy reference and access for students needing additional assistance.

Easy to assemble instructions:1. Insert plain end of brass rods (height/altitude) into narrow end of yellow string attachment caps.2. Screw brass rods into polygon center.3. Slide end of green string (representing lateral edge) into notch in yellow cap. Slide other end into polygon notch indicated with green line(s). 4. Repeat for red strings (slant height).5. Sticky notes: with varying edge, height and/or slant height measures can be folded over strings and/or height rod.

The Getting Into Solids Book is divided into two sections that contain several different types of activities for students.

2

Section One: Contains step-by-step plans for drawing each solid. The ability to draw these solids is critical to student understanding.

Section Two: Contains reproducible worksheets for finding the surface area and volume of each solid. For each solid there are three worksheets for surface area and three for volume. These worksheets were designed to be an introductory tool. They are intended to help the students organize their thinking and mathematical process.

The first worksheet presents students with a sample problem to solve. This worksheet helps students clarify their thinking by asking them to state important formulas, and to name, describe, and then compute missing segments. The second worksheet is a corresponding sheet to the first one with annotated solutions. The solutions used for problems on these sheets may employ the Pythagorean Theorem, special right triangles, and trigonometry. The third worksheet presents students with a drawing of the figure, but no data. This sheet may be used with classroom data or textbook problems for the specific solid that is depicted.

Also included is a single worksheet for surface area and one for volume that does not depict any solid. This sheet will be useful for a variety of activities since students must draw the figure they are working with before beginning to solve problems. It may be used with classroom data or textbook problems for the specific solid that has been drawn.

The final activity worksheet is for students to gather data to be used to develop Euler’s Formula (F+V=E+2).

The Getting Into Solids set will help to develop the most challenged students' spatial visualization of three-dimensional figures and enable them to understand the reasoning behind the processes for computing surface area and volume. In addition, the Getting Into Solids set will help students understand how to apply the Pythagorean Theorem or special right triangles to the task of calculating surface area or volume of three-dimensional solids.

Finally, the spatial visualization skills enhanced by the Getting Into Solids kit will benefit students in subsequent mathematics courses as they continue to study three-dimensional figures and graphical representations of equations.

These models allow students to: A. Identify the different parts of the pyramid by color. 1. Brass rod – height 2. Green cord – lateral edge 3. Red cord – slant height 4. Etched green line on the base – radius 5. Etched red line on the base – apothem

B. Identify different triangles by color. 1. the green triangle (height, radius, and lateral edge) 2. The red triangle (height, apothem, slant height) 3. The multi-colored triangle (slant height, lateral edge, 1/2 base edge)

C. Attach numbers to the model for ease of visualization. 1. Write on the plexiglass base with water markers 2. Attach post-it notes, with unit measure, on the height, lateral edge and slant height

D. Get inside the model and trace the right triangles.

E. Basically, this model employs all learning styles: 1. Visualization 2. Auditory - (when guided by the teacher or students teaching others in groups) 3. Kinesthetic

3

Vocabulary for REGULAR PYRAMIDS and RIGHT CONES

L.A. = lateral area P = perimeter of base r = radius S.A. = surface area B = area of base h = height V = volume l = slant height π = pi

Formulas for REGULAR PYRAMIDS and RIGHT CONESRegular Pyramid

L.A. = 1/2 P l

S.A. = L.A. + B

S.A. = 1/2 P l + B

V = 1/3 B h

Right Cone

L.A. = π r l

S.A. = L.A. + B

S.A. = π r l + π r2

V = 1/3 B h

SECTION 1: DRAWINGDrawing A Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Drawing A Triangular Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Drawing A Square Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Drawing A Rectangular Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Drawing A Pentagonal Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Drawing A Hexagonal Pyramid . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

SECTION 2: WORKSHEETSWorksheet: Surface Area – Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Worksheet: Surface Area – Triangular Pyramid . . . . . . . . . . . . . . . . . . . . . . . 13Worksheet: Surface Area – Square Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . 16Worksheet: Surface Area – Pentagonal Pyramid . . . . . . . . . . . . . . . . . . . . . . 19Worksheet: Surface Area – Hexagonal Pyramid . . . . . . . . . . . . . . . . . . . . . . 22Worksheet: Surface Area – Blank Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Worksheet: Surface Area – Rectangular Pyramid CHALLENGE . . . . . . . . . 26Worksheet: Volume – Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Worksheet: Volume – Triangular Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Worksheet: Volume – Square Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Worksheet: Volume – Rectangular Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . 37Worksheet: Volume – Pentagonal Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Worksheet: Volume – Hexagonal Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Worksheet: Volume – Blank Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Worksheet: Volume – Regular Pentagonal Pyramid CHALLENGE . . . . . . . . 47Extra Surface Area/Volumes Solids, answer key, and difficulty level . . . . . . . 49Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4

Name _______________________________________ Date _______________

Follow the steps below to practice sketching a cone. Complete two sketches.NOTE: Dotted lines represent segments in the solids that are hidden from view.

Step 1: Draw two lateral edges.

Step 3: Complete the base circle.

Sketch 1:

Step 2:Draw the front half of the base circle.

Step 4:Draw the altitude and radius.

Sketch 2:

5

Name _______________________________________ Date _______________

Follow the steps below to practice drawing a triangular pyramid. Complete two sketches.NOTE: Dotted lines represent segments in the solids that are hidden from view.

Step 1: Draw the front triangular face.

Step 3: Draw the remaining lateral edge.

Sketch 1:

Step 2:Draw the remaining base edges.

Step 4:Draw the altitude and apothem.

Sketch 2:

6

Name _______________________________________ Date _______________

Follow the steps below to practice drawing a square pyramid. Complete two sketches.NOTE: Dotted lines represent segments in the solids that are hidden from view.


Step 3: Draw the remaining base edge.

Step 5:Draw the lateral height.

Sketch 1:

Step 2:Draw the two parallel base edges.

Step 4:Draw the two remaining lateral edges.


Sketch 2:

7

Name _______________________________________ Date _______________

Follow the steps below to practice drawing a rectangular pyramid. Complete two sketches.NOTE: Dotted lines represent segments in the solids that are hidden from view.


Step 3: Draw the remaining base edges.

Step 5:Draw the two lateral heights.

Sketch 1:

Step 2:Draw the two visible base edges.

Step 4:Draw the three remaining lateral edges.

Step 6:Draw the altitude and two apothems.

Sketch 2:

8

Name _______________________________________ Date _______________

Follow the steps below to practice drawing a pentagonal pyramid. Complete two sketches.NOTE: Dotted lines represent segments in the solids that are hidden from view.




Sketch 1:

Step 2:Draw the two visible base edges.

Step 4:Draw the three remaining lateral edges.


Sketch 2:

9

Name _______________________________________ Date _______________

Follow the steps below to practice drawing a hexagonal pyramid. Complete two sketches.NOTE: Dotted lines represent segments in the solids that are hidden from view.




Sketch 1:

Step 2:Draw the visible base edges.

Step 4:Draw all the remaining lateral edges.


Sketch 2:

Fill in the blanks and show all your work.

Name of the solid:

Equation for formula: S.A. =

The FIRST variable I will solve for is .Describe what the variable represents and which colored triangle or base you will work with._________________________________________________________________________________________________________

The SECOND variable I will solve for is .Describe what the variable represents and which colored triangle or base you will work with._________________________________________________________________________________________________________

The THIRD variable I will solve for is .Describe what the variable represents and which colored triangle or base you will work with._________________________________________________________________________________________________________

Solution:

Lateral Area:L.A. =

Area of Base:B =

Radius = 7 cm Height = 7 cm

Find Surface Area of theright solid.

Name: GETTING INTO SOLIDS Worksheet: 1

Standard Formula:S.A. = (substitute values)

10


Name of the solid: Cone

Equation for formula: S. A. = π r l + B

The FIRST variable I will solve for is l .Describe what the variable represents and which colored triangle or base you will work with.l represents the length of the slant height. I will use the green triangle.

The green triangle is a special 45° - 45° - 90° triangle, therefore,l = 7 2 The SECOND variable I will solve for is B .Describe what the variable represents and which colored triangle or base you will work with.B represents the area of the base, which is a circle. B = π r2

B = π (7)2

B = 49 π


Solution:S.A. = 49 2 π + 49 π cm2

S.A. = 371.64 cm2

Lateral Area:L.A. = 49 2 π cm2

L.A. = 217.70 cm2

Area of Base:B = 49 π cm2

B = 153.94 cm2

Radius = 7 cm Height = 7 cm


Name: GETTING INTO SOLIDS Worksheet: 1 (difficulty - Easy)

l7

7

Standard Formula:S.A. = π r l + B (substitute values)

S.A. = π (7)(7 2) + 49π

11


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




12


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =

Find Surface Area of theregular solid.



Slant Height = 4 in. Base Edge = 6 in.

13


Name of the solid: Triangular Pyramid

Equation for formula: S.A. = 1/2 Pl + B

The FIRST variable I will solve for is P .Describe what the variable represents and which colored triangle or base you will work with.

P represents the perimeter of the base. I will work with the base.

P = 3 S P = 3(6) P = 18

The SECOND variable I will solve for is l .Describe what the variable represents and which colored triangle or base you will work with.

l represents slant height. Slant height is given to be 4 in.

The THIRD variable I will solve for is B .Describe what the variable represents and which colored triangle or base you will work with.

B represents the area of the base. I will work with the base which is an equilateral triangle.

Solution:S.A. = 36 + 9 3 in.2

S.A. = 51.59 in.2

Lateral Area:L.A. = 36 in.2

Area of Base:B = 9 3 in.2

15.59 in.2


Name: GETTING INTO SOLIDS Worksheet: 3 (difficulty - Medium)

Standard Formula:S.A. = 1/2 P l + B (substitute values)

S.A. = 1/2 (18)(4) + 9 3

B = S2 3 4B = 62 3 4B = 9 3

Slant Height = 4 in. Base Edge = 6 in.

14


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




15


Name of the solid:




The THIRD variable I will solve for is .Describe what the variable represents and which colored triangle or base you will work with.

Solution:

Lateral Area:L.A. =

Area of Base:B =




Height = 4 ft. Base Edge = 6 ft.

16


Name of the solid: Square Pyramid

Equation for formula: S.A. = 1/2 P l + B


P represents the perimeter of the base. I will use the base to find P.

P = 4S P = 4(6) P = 24


l represents the length of the slant height. I will use the red triangle to solve for l.


B represents the area of the base. The base is a square. _________________________________________________________________

S.A. Solution: S.A. = 96 ft.2

Lateral Area:L.A. = 60 ft.2

Area of Base:B = 36 ft.2



Standard Formula:S.A. = 1/2 P l + B (substitute values)

S.A. = 1/2 (24)(5) + 36 S.A. = 60 + 36

Height = 4 ft. Base Edge = 6 ft.

First, the apothem is 1/2 an edge so the apothem is 3.

l4

3

42 + 32 = l 2

16 + 9 = l 2

25 = l 2

5 = l

B = S2

B = 62

B = 36

17


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




18


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




Apothem = 4.82 cm Base Edge = 7 cmHeight = 6.63 cm

19


Name of the solid: Pentagonal Pyramid



P represents the perimeter of the base. I will use the base to find the area.


l represents the length of the slant height. I will use the red triangle.


B represents the area of the base. I will use the base to find its area.


Name: GETTING INTO SOLIDS Worksheet: 7 (difficulty - Hard)

Apothem = 4.82 cm Base Edge = 7 cmHeight = 6.63 cm

P = 5SP = 5(7)P = 35

B = 1/2 Pa

B = 1/2 (35)(4.82)

B = 84.35

l6.63

4.82

h2 + a2 = l 2

(6.63)2 + (4.82)2 = l 2

43.9569 + 23.2324 = l 2

67.1893 = l 2

8.1969 = l

Solution: 227.80 cm

Lateral Area:L.A. = 143.45 cm

Area of Base:B = 84.35 cm

Standard Formula:S.A. = 1/2 Pl + B (substitute values)

S.A. = 1/2 (35)(8.1969) + 84.35

2 2 2

20


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




21


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




Base Edge = 12 m Height = 6 m

22


Name of the solid: Hexagonal Pyramid



P represents the perimeter of the base. I will use the base to find P.

P = 6 S P = 6(12) P = 72

The SECOND variable I will solve for is B .Describe what the variable represents and which colored triangle or base you will work with.

B represents the area of the base. I will use the base to find area.

The THIRD variable I will solve for is l .Describe what the variable represents and which colored triangle or base you will work with.

l represents the length of the slant height. I will use the red triangle to find l.

Solution: S.A. = 432 + 216 32

S.A. = 806.12 m2

Lateral Area:L.A. = 432 m

Area of Base:B = 216 3 m B = 374.12 m2



Standard Formula:S.A. = 1/2 Pl + B (substitute values)

S.A. = 1/2 (72)(12) + 216 3

S.A. = 432 + 216 3

Base Edge = 12 m Height = 6 m

B = 1/2 PaB = 1/2 (72)(6 3)B = 216 3

First I will need to find the apothem of the base.

1/2 the base edge of 12 is 6,which is the short legof a special 30˚- 60˚- 90˚ triangle. So the apothem, long leg, is 6 3

62 + (6 3) 2 = l 2

36 + 108 = l 2

144 = l 2

12 = l

660˚

a

l6

6 3

2 2

23


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




24


Name of the solid:





Solution:

Lateral Area:L.A. =

Area of Base:B =




25

Solution:

Surface Area

Challenge


Write a description explaining how you will solve for the surface area of this pyramid. Then organize and show your work in detail.

The base of a rectangular pyramid (not a regular pyramid) is 30 feet by 12 feet. The height is 8 feet. All lateral edges are congruent. Find the surface area. Below is part of a flat view for this pyramid.

26

Write a description explaining how you will solve for the surface area of this pyramid. Then organize and show your work in detail.

Solution: S.A. = 864 ft.

Surface Area

Challenge


The base of a rectangular pyramid (not a regular pyramid) is 30 feet by 12 feet. The height is 8 feet. All lateral edges are congruent. Find the surface area. Below is part of a flat view for this pyramid.

1st I will find slant height of the red triangle with base of 15, call it l1.

2nd I will find slant height of the red triangle with base of 6, call it l2.

3rd I will find area of green isosceles triangle with height of l1.

4th I will find area of green isosceles triangle with height of l2.

5th I will find area of the rectangular base.

6th to find surface area, I will add the area of all five sides together. Two pairs of triangles have the same area.

l8

6

2

82 + 152 = l12

64 + 225 = l12

289 = l12

17 = l12

82 + 62 = l22

64 + 36 = l22

100 = l22

10 = l2

A = 1/2 b h A = 1/2 (12)(17)A = 102

A = 1/2 b h A = 1/2 (30)(10)A = 150

A = l wA = 30(12)A = 360

S.A. = 2(102) + 2(150) + 360S.A. = 864

2

27


Name of the solid:

Equation for formula: V =



Solution:

Area of Base:B =

Circumference = 20 π in. Height = 7 in.

Find Volume of theright solid.


Standard Formula:V = (substitute values)

28


Name of the solid: Cone

Equation for formula: V = 1/3 B h

The FIRST variable I will solve for is B .Describe what the variable represents and which colored triangle or base you will work with.

B represents the area of the base. The base is a circle.

The SECOND variable I will solve for is h .Describe what the variable represents and which colored triangle or base you will work with.

h represents the height of the cone

Solution: V = 300π in.3 V = 942.48 in.3

Area of Base:B = 100π in. B = 314.16 in.2

Circumference = 20 π in. Height = 9 in.


Name: GETTING INTO SOLIDS Worksheet: 13 (difficulty -Medium)

Standard Formula:V = 1/3 B h (substitute values)

V = 1/3 (100π)(9)

1st I will need to find the radius of the circle using the circumference formula.

The height is given to be 9 in.

Now I will find the area of the circle (base).

C = 2 π r20 π = 2 π r10 = r

B = π r2

B = π(10)2

B = 100 π

2

29


Name of the solid:




Solution:

Area of Base:B =




30


Name of the solid:




Solution:

Area of Base:B =

Slant height = 10 m Base Edge = 12 m Apothem = 2 3 m

Find Volume of theregular solid.



31


Name of the solid: Triangular Pyramid


The FIRST variable I will solve for is h .Describe what the variable represents and which colored triangle or base you will work with.

h represents the height of the pyramid. I will use the red triangle to find h.

The SECOND variable I will solve for is B .Describe what the variable represents and which colored triangle or base you will work with.

B represents the the area of the base. I will use the triangular base to find its area.

Solution: V = 24 66 m V = 194.98 m3

Area of Base:B = 36 3 m B = 62.35m2




V = 1/3 (36 3)(2 22)

h2 + (2 3)

2 = 10

2

h2 + 12 = 100

h2 = 88

h = 88

h = 2 22

1st, I will need to find the perimeter of the base.

P = 3S

P = 3(12) P = 36

Now I can find area of the base.

B = 1/2 Pa B = 1/2 (36)(2 3)

B = 36 3

2

Slant height = 10 m Base Edge = 12 m Apothem = 2 3 m

3

32


Name of the solid:




Solution:

Area of Base:B =




33


Name of the solid:




Solution:

Area of Base:B =




Lateral Edge = 18 ft. Radius = 12 ft.Base Edge = 288 ft.

34


Name of the solid: Square Pyramid



B represents the area of the base. I will use the base of a square.


h represents the height of the pyramid. I will use the green triangle to find h.

B = S2

B = ( 288)2

B = 288

Solution: V = 576 5 ft. V = 1,287.98 ft.3

Area of Base:B = 288 ft.




V = 1/3 (288)(6 5)

Lateral Edge = 18 ft. Radius = 12 ft.Base Edge = 288 ft.

h2 + 12

2 = 18

2

h2 + 144 = 324

h2 = 180

h = 180h = 6 5

h 18

12

3 2

35


Name of the solid:




Solution:

Area of Base:B =




36


Name of the solid:




Solution:

Area of Base:B =

Length of Base = 9 in. Width of Base = 6 in. Slant Height = 9.18 in.

Find Volume of the solid.



37


Name of the solid: Rectangular Pyramid



B represents the the area of the base. The base is a rectangle.


h represents the the height of the pyramid. I will use the red triangle to find height.

Solution: V =144.03 in.

Area of Base:B = 54 in.

Length of Base = 9 in. Width of Base = 6 in. Slant Height = 9.18 in.




V = 1/3 (54)(8.0014)

B = l wB = 9(6)B = 54

(4.5)2 + h2

= (9.18)2

20.25 + h2 = 84.2724

h2 = 64.0224

h = 8.0014

3 2

38


Name of the solid:




Solution:

Area of Base:B =




39


Name of the solid:




Solution:

Area of Base:B =

Area of Base = 139.3 mSlant Height = 8 m Apothem = 6.19 m




40


Name of the solid: Pentagonal Pyramid



B represents the the area of the base.


h represents the height of the pyramid. I will use the red triangle to find height.

Solution: V = 235.32 m

Area of Base:B = 139.3 m

Area of Base = 139.3 mSlant Height = 8 m Apothem = 6.19 m




V = 1/3 (139.3)(5.06793)

The area of the base is given.

B = 139.3

h2 + (6.19)

2 = 8

2

h2 + 38.3161 = 64

h2 = 25.6839

h = 5.06793

3 2

41


Name of the solid:




Solution:

Area of Base:B =




42


Name of the solid:




Solution:

Area of Base:B =

Apothem = 12 cm Slant Height = 14 cm Base Edge = 8 3 cm




43


Name of the solid: Hexagonal



B represents the area of the base I will use the hexagon to find its area.


h represents height. I will use the red triangle to find height.

h2 + 12

2 = 14

2

h2 + 144

2 = 196

h2 = 52

h = 52 h = 2 13

Solution: V = 192 39 cm V = 1,999.04 cm3

Area of Base:B = 288 3 cm B = 498.83 cm2

Apothem = 12 cm Slant Height = 14 cm Base Edge = 8 3 cm




V = 1/3 (288 3)(2 13)

First, I will need to find the perimeter of the base.

P = 6S

P = 6(8 3) P = 48 3

Now I can find the area of the base.

B = 1/2 Pa B = 1/2 (48 3)(12) B = 288 3

3 2

44


Name of the solid:




Solution:

Area of Base:B =




45


Name of the solid:




Solution:

Area of Base:B =




46

Volume

Challenge


Find the volume of a regular pentagonal pyramid. All edges are 8 ft. and the slant height is 4 3 ft. Draw the picture and write a description of how you will find the volume. ORGANIZE and show your work in detail.

Solution:

47

Volume

Challenge


The first variable I will solve for is B. B represents the area of the base.

In the multi-colored triangle of the base, I will find

the apothem.

Next, I will find the perimeter of the base.

P = 5s

P = 5(8)

P = 40

Finally, the area of the base is:

B = 1/2 Pa B = 1/2 (40)(5.50553) B = 110.1106

The second variable, I will solve for is h.

h represents the height of the pyramid.

I will use the red triangle to find height.

h2 + (5.50553)2 = (4 3)2

h2 + 30.31086 = 48

h2 = 17.68916 h = 4.20585

Now the formula:

V = 1/3 B hV = 1/3 (110.1106)(4.20585)

tan 36° = 4/aa tan 36° = 4

a = 4/tan 36°

a = 5.50553

Find the volume of a regular pentagonal pyramid. All edges are 8 ft. and the slant height is 4 3 ft. Draw the picture and write a description of how you will find the volume. ORGANIZE and show your work in detail.

Solution: V = 154.37 ft.3

48

1. 2. 3.

5.

8. 1.

4.3.2.

7.

4.

12 cm 6 cm

5 cm

12 cm

8 cm

Base Edge = 6 Height = 4

Base Edge = 12 Height = 6

Diameter of Base = 24Slant Height = 15

SURFACE AREA & VOLUME SOLIDSfor REGULAR PYRAMIDS & RIGHT CONES

10 in.

3 in

8.2 cm12

15 cm

9 cm5

7 cm

10 13

26 mm

20 mm

6.

Lateral Edge = 5 Base Edge = 6 Height = 13

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KEY for SURFACE AREA of REGULAR PYRAMIDSLevel of Difficulty: 1 - Easy 2 - Medium 3 - Hard 4 - Challenge

NAME *P l *B *LA *SA Level

1. Square Pyramid 48cm 8cm 144cm2 192cm

2 336cm

2 1

2. Square Pyramid 24u *5u 36u2 60u

2 96u

2 2

3. Square Pyramid 24cm *4cm 36cm2 48cm

2 84cm

2 2-3

4. Hexagonal Pyramid 18in 10in 13.5 3in2 90in

2 90 +13.5 3in

2 3-4

5. Hexagonal Pyramid 72u *12u 216 3u2 432u

2 432 + 216 3u

2 4

6. Triangular Pyramid 18u *4u 9 3u2 36u

2 36 + 9 3u

2 4

7. Pentagonal Pyramid 35cm 8.2cm 84.30cm2 143.50cm

2 227.80cm

2 4

8. Pentagonal Pyramid 25u *11.74u 43.01u2 146.71u

2 189.72u

2 4

* Indicates the variable must be calculated.

KEY for SURFACE AREA of RIGHT CONESLevel of Difficulty: 1 - Easy 2 - Medium 3 - Hard 4 - Challenge

NAME r l *B *LA *SA Level

1. Right Cone 9cm 15cm 81π cm2 135π cm

2 216π cm

2 1

2. Right Cone *10mm 26mm 100π mm2 260π mm

2 360π mm

2 2

3. Right Cone *12u 15u 144π u2 180π u

2 324π u

2 2

4. Right Cone *5u * 194u 25π u2 5 194π u

2 5 194π + 25π u

2 2-3


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KEY for VOLUME of REGULAR PYRAMIDSLevel of Difficulty: 1 - Easy 2 - Medium 3 - Hard 4 - Challenge

NAME *B h *V Level

1. Square Pyramid 144cm2 *2 7cm 96 7cm

3 2

2. Square Pyramid 36u2 4u 48u

3 2

3. Square Pyramid 36cm2 * 7cm 12 7cm

3 3

4. Hexagonal Pyramid 13.5 3in2 *9.66in 75.27 in.

3 4

5. Hexagonal Pyramid 216 3u2 6u 432 3u

3 3-4

6. Triangular Pyramid 9 3u2 13u 3 39u

3 3-4

7. Pentagonal Pyramid 84.30cm2 *6.64cm 186.47cm

3 4

8. Pentagonal Pyramid 43.01u2 *11.22u 160.87u

3 4


KEY for VOLUME of RIGHT CONESLevel of Difficulty: 1 - Easy 2 - Medium 3 - Hard 4 - Challenge

NAME *B h *V Level

1. Right Cone 81π cm2 *12cm 324π cm

3 2-3

2. Right Cone 100π mm2 *24mm 800π mm

3 2-3

3. Right Cone 144π u2 *9u 432π u

3 2-3

4. Right Cone 25π u2 13u (325/3)π u

3 1-2


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Leonhard Euler was the most famous mathematician of his time. He made so many discoveries in mathematical that over 80 books were needed to contain his descriptions of them. One discovery Euler made involves polyhedrons like the pyramids you can build with the Getting Into Solids set. He discovered that there is a relationship between the number of faces, vertex points, and edges in every pyramid. In fact, the relationship he discovered is true for any polyhedron, no matter how irregular its shape. The relationship is called Euler's Formula in his honor. Collect data from the Getting Into Solids shapes and record the data in the table below. Once you have filled in the data table, use your findings to write out Euler's formula.

Triangular Pyramid

Square Pyramid

Rectangular Pyramid

Pentagonal Pyramid

Hexagonal Pyramid

Polyhedron Number of faces (F) Number of vertex points (V) Number of edges (E)


Euler’s Formula

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Triangular Pyramid 4 4 6

Square Pyramid 5 5 8

Rectangular Pyramid 5 5 8

Pentagonal Pyramid 6 6 10

Hexagonal Pyramid 7 7 12

Polyhedron Number of faces (F) Number of vertex points (V) Number of edges (E)

Swiss mathematician Leonhard Euler (1707-1783) made many discoveries over a vast range of mathematical topics, including geometry, calculus, number theory and infinitesimals. He was also a prolific writer, producing more than 800 books, pamphlets, and articles during his career. When he died 56 pages were needed to list all his works in his obituary. One of Euler's discoveries was the relationship between the number of faces, edges, and vertex points for all polyhedrons. Euler's Formula is F + V = E + 2, where F represents the number of faces, V represents the number of vertex points, and E represents the number of edges. In this activity students are directed to collect data about the various pyramids in the Getting Into Solids set and then look for a relationship between the number of faces, vertex points, and edges. The table below shows the data students should collect and record from the Getting Into Solids pyramids to help them derive Euler's formula.


Euler’s Formula F + V = E + 2

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