DISCOVER IT! Euler`s Formula - My Math Class

DISCOVER IT!
Euler’s Formula
A. Go to gcmath.weebly.com and open the “Polyhedron Tool.”
(under Notes/Videos/Handouts)
B. Use the tool to determine the number of faces, edges, and vertices for each polyhedron.
TECH TIPS:
(1) Check the “transparent” box
(2) Use the “zoom level” tool
(3) Click and drag the polyhedron to get a different view
(4) Use different colors for different items
(Example: make all faces blue, vertices red and edges yellow)
Polyhedron
Faces
Vertices
Edges
C. Look for patterns!
Develop a “formula” that shows the relationship between the faces, edges, and
vertices. Once you think you’ve discovered the formula, write it on your record
sheet and check in with Miss Collins.
DISCOVER IT!
Cross Sections
A. Go to gcmath.weebly.com and open the “Cross Section Tool.”
(under Notes/Videos/Handouts)
B. Use the transformation tools (see diagram below) to complete the cross section
chart on your record sheet.
TECH TIP: Use the “Click to show identification of cross section” tool if you are having
trouble identifying the shape of your cross section.
3D-Figure Selection
Transformation Tools
DISCOVER IT!
Cavilieri’s Principle
A. Open the “Cavilieri’s Principle” document on your TI-nspire calculator.
(under My Documents)
B. On page 1.2, you are given a cylinder with a radius of 2 units and a height of 20 units.
1. What do you notice about the volume when you move point A?
C. On page 2.1, you are given a triangular pyramid and 3 different parallel cross sections.
2. What do you notice about the cross sectional areas of the triangle when you move point A?
3. What do you notice about the volume when you move point A?
4. Explore moving point B to create a new triangular pyramids. Do your responses to
questions 2 and 3 change when you do this?
D. On page 3.1 you are given three prisms. Use the given dimensions to answer the
questions below.
5. What is the area of the cross sections for each prism?
6. What is the volume of each prism?
E. Use the observations you made in parts A-D to fill-in the blanks on your record sheet to
complete Cavilieri’s Principle.
DISCOVER IT! ANSWERS
Euler’s Formula
A. Go to gcmath.weebly.com and open the “Polyhedron Tool.”
(under Notes/Videos/Handouts)
B. Use the tool to determine the number of faces, edges, and vertices for each polyhedron.
TECH TIPS:
(1) Check the “transparent” box
(2) Use the “zoom level” tool
(3) Click and drag the polyhedron to get a different view
(4) Use different colors for different items
(Example: make all faces blue, vertices red and edges yellow)
Polyhedron
Faces
Vertices
Edges
6
8
12
4
4
6
8
6
12
12
20
30
20
12
30
C. Look for patterns!
Develop a “formula” that shows the relationship between the faces, edges, and
vertices. Once you think you’ve discovered the formula, write it on your record
sheet and check in with Miss Collins.
(# of faces) + (# of sides) = (# of edges) - 2
DISCOVER IT! ANSWERS

Cross Sections
CROSS SECTIONS:
Describe how each polyhedron has been sliced to form the following cross sections.
Polyhedron
Cross Section
Circle
CONE
Ellipse (Oval)
Triangle
Circle
CYLINDER
Ellipse (Oval)
Part of an Ellipse
Rectangle
TRIANGULAR
PYRAMID
Triangle
Quadrilateral
Description
Parallel to base
Diagonal slice, does not intersect base
Slice through the apex, perpendicular to
the base
Parallel to base
Diagonal slice, does not intersect base
Diagonal slice, intersects base
Perpendicular to base
Answers will vary
Intersects two faces and the base
RECTANGULAR
PYRAMID
Triangle
Answers will vary
Quadrilateral
Answers will vary
Pentagon
Intersects all faces
PENTAGONAL
PYRAMID
Regular Pentagon
Parallel to the base
TRIANGULAR
PRISM
Triangle
Quadrilateral
Triangle
Diagonal slice or parallel to base
Diagonal slice or perpendicular to base
Slice corner
RECTANGULAR
PRISM
Quadrilateral
Answers will vary
Pentagon
Answers will vary
PENTAGONAL
PRISM
Regular Pentagon
Parallel to base
DISCOVER IT!
Cavilieri’s Principle
A. Open the “Cavilieri’s Principle” document on your TI-nspire calculator.
(under My Documents)
B. On page 1.2, you are given a cylinder with a radius of 2 units and a height of 20 units.
1. What do you notice about the volume when you move point A?
The volume does not change
C. On page 2.1, you are given a triangular pyramid and 3 different parallel cross sections.
2. What do you notice about the cross sectional areas of the triangle when you move point A?
The areas do not change
3. What do you notice about the volume when you move point A?
The volume does not change
4. Explore moving point B to create a new triangular pyramids. Do your responses to
questions 2 and 3 change when you do this?
No, volume and area still remain constant.
D. On page 3.1 you are given three prisms. Use the given dimensions to answer the
questions below.
5. What is the area of the cross sections for each prism?
Area = 4in2
(All cross sectional areas are the same)
6. What is the volume of each prism?
Volume = 20 in3
(All volumes are the same)
E. Use the observations you made in parts A-D to fill-in the blanks on your record sheet to
complete Cavilieri’s Principle.
CAVILIERI’S PRINCIPLE states “If two 3-dimensional figures have the same
HEIGHT and the same cross-sectional AREA at every level, then the figures have the
same VOLUME.”