PLANE SECTIONS OF THREE D FIGURES
INTRODUCTION
The objective for this lesson on Plane Sections of Three Dimensional Figures is, the student will
describe the two-dimensional figures that result from slicing prisms and pyramids.
The skills students should have in order to help them in this lesson include, knowing the
attributes of parallelograms, squares, rectangles, prisms and pyramids.
We will have three essential questions that will be guiding our lesson. Number One, define
plane section. Number Two, explain how a plane section is related to the base of a threedimensional figures if the cut is parallel to the base. And Number Three, explain how a plane
section of a pyramid is related to the lateral sides if the figure is cut through the vertex.
Begin by completing the warm-up for this lesson on identifying two-dimensional and threedimensional figures in order to prepare for the lesson on plane sections of three dimensional
figures.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Tyrese is slicing a block of cheese. He is using the
cheese along with different fruits to make sculptures of animals. He needs a plane section of
cheese in the shape of a parallelogram that is not a rectangle. How should he cut the block of
cheese?
In Step S, we Study the Problem. First we need to identify where the question is located within
the problem and underline the question. The question for this problem is, how should he cut the
block of cheese?
Now that we have identified the question we need to put this question in our own words in the
form of a statement. This problem is asking me to find the way to cut the block of cheese to
make a parallelogram that is not a rectangle.
During this lesson we will learn about plane sections of three-dimensional figures in order to
complete this SOLVE problem at the end of the lesson.
EPLORE TWO- AND THREE-DIMENSIONAL FIGURES
Take just a moment to look through the figures from Questions one through five on the warm-up
page. Decide if the figure is two-dimensional or three-dimensional. Be prepared to explain your
answer.
Which figures did you label as two-dimensional? The trapezoid and the parallelogram.
Explain your answer. These two figures are flat and they only have length and width, but no
depth.
Which figures did you label as three-dimensional? The cube, the rectangle prism and the square
pyramid.
Explain your answer. These three figures all have bases that show length and width but they also
have the third dimension of depth with the height of the figures.
What do you think of when you hear the word “slice?” Cutting something, a piece of a whole, a
slice of bread, a slice of cheese, perhaps an orange slice.
Using an example of a slice with which you are familiar, sketch the whole, larger item in the box
on the left and sketch a slice of that object in the box on the right. We’ve used a wheel of cheese
and then a slice of cheese.
In our drawings do we have any two-dimensional figures? Yes, when looking at the shape of the
slice after it is taken we see a two-dimensional figure.
In our drawings do we have any three-dimensional figures? Yes, the whole solid is threedimensional.
Describe how you found your slice. In the sample answer instead of cutting a slice that would
create two semicircular stacks, the slice was cut similar to a loaf of bread to create a slice that is
circular, like its base.
We can picture whole items and slices of items in many objects around us. Whenever we look at
the shape of the slice, we are seeing the plane section.
PLANE SECTIONS OF RIGHT RECTANGULAR PRISMS PARALLEL AND
PERPENDICULAR TO THE BASE
Cut out Figure A following the thin solid line. For now, ignore the dashed lines.
What solid do you think will be constructed from this net? A rectangular prism.
When folding, we will create two separate prisms.
Fold at all of the dashed lines on the sides of the rectangle with the letter “A.” Tape together all
of the sides except for the rectangle that is at the bottom of the net before it is folded. This
should be the lid of the prism! This should be the lateral side. Fold the final flap closed but do
not tape this flap shut.
What is the shape of the base of the first solid? Rectangle
What is the shape of the base of the second solid? Rectangle
What is the name of the first solid that you built? Rectangular Prism
What is the name of the second solid that you built? Rectangular Prism
When we look at these rectangular prisms, what angles are created at the meeting points of all of
the edges? Ninety degree angles
If all of the angles are ninety degree angles, what type of rectangular prisms have we created?
Right rectangular prisms.
Take a piece of paper and slide it between the lid and the rest of the prism to show our cut into
the first prism. If we kept pushing the paper through like a knife, we would cut off the lid.
Detach the lid to create a plane section.
What is the relationship between the horizontal slice you created and the base? The slice is
parallel to the base because the slice is the same distance away from the base at all points and
never cuts through the base.
What is the shape of the plane section, or slice, that we cut from the top of the first box?
Rectangle. Sketch the shape of the prism’s lid.
Take a piece of paper and slide it between the open side of the prism to show our cut into the
prism. If we kept pushing the paper through like a knife, we would cut off the side. Detach the
side to create a plane section.
What is the relationship between the vertical slice you created and the base? The slice is
perpendicular to the base because it creates ninety degree angles when the slice meets the base.
What is the shape of the plane section, or slice, that we cut from the side of the second box? A
rectangle. Sketch the shape of the prism’s side.
What observations can you make about the plane section from the first cut? The parallel slice is
the same rectangular shape as the bottom or top base.
What observations can you make about the plane section from the second cut? The
perpendicular slice is the same rectangular shape as the side of the prism or the lateral side.
Conclusions:
When we take a horizontal slice parallel to the base of the rectangular prism, the slice is in the
shape of a rectangle and is congruent to the base of the rectangular prism.
When we take a vertical slice perpendicular to the base of the rectangular prism, the slice is in
the shape of a rectangle and is congruent to a side of the rectangular prism.
PLANE SECTIONS OF RECTANGULAR PRISMS ANGLED TOWARDS THE BASE
Cut out each figure following the thin solid line. For now, ignore the dashed lines and the dark
bold lines.
What solid do you think will be constructed for this net? A rectangular prism
Let’s begin constructing the figures. Fold only the dashed lines and create the rectangular prism
that will have different bold lines around the figures.
What is the shape of the base of Figure B? A rectangle
What is the shape of the base of Figure C? A rectangle
What is the name of Figure B that you built? A Rectangular Prism
What is the name of Figure C that you built? Rectangular Prism
When we look at these rectangular prisms, what angles are created at the meeting points of all of
the edges? Ninety degree angles
If all of the angles are ninety degree angles, what type of rectangular prisms have we created?
Right rectangular prisms
What do you notice about the prisms? They have bold lines around them as if a slice is being cut
into them.
Take your scissors and carefully cut the dark bold line until each prism has been separated into
separate pieces. Using the extra paper given, use one of the pieces for tracing.
Place the plane section where the cut was made, face down onto the paper and use a pencil to
trace the outside of the cut. You should be tracing where the dark bold line is on the figure.
If you recreate Figure C again by holding the two pieces together, what do you notice about the
solid black line’s endpoints on the top base? The points are not across from each other. The line
segment connecting opposite edges is on an angle.
How can you describe the direction of the cut? Down on an angle.
If you recreate Figure B again by holding the two pieces together, what do you notice about the
solid black line’s endpoints on the top base? The points are directly across from each other. The
line segment connecting opposite edges in perpendicular to each edge.
What observations can you make about the plane section from the cut in Figure B? The slice is a
rectangle. The slice is taller than the lateral side.
What observations can you make about the plane section fort he cut in Figure C? The slice is a
parallelogram.
PLANE SECTIONS OF RECTANGULAR PRISMS – SUMMARIZED
Take a moment to copy the information to the top four blocks on the page titled: Summary of
Discoveries with Figures A, B and C.
Describe Figure A’s horizontal slice. It is parallel to the base.
What was the sketch of the plane section? A rectangle congruent to the base. Sketch the
rectangle.
Describe Figure A’s vertical slice. It is perpendicular to the base.
What was the sketch of the plane section? A rectangle congruent to the lateral side. Sketch the
rectangle.
Describe Figure B’s slice. It is a line segment perpendicular to opposite sides of the base cut on
a downward angle.
What was the sketch of the plane section? A rectangle. Sketch the rectangle.
Describe Figure C’s slice. The line segment was not perpendicular to opposite sides of the base
cut on a downward angle.
What was the sketch of the plane section? A parallelogram. Sketch the parallelogram.
Take a look at the box that asks us to “Generalize the Findings:”
The plane section of every slice taken for a rectangular prism is a parallelogram.
All plane sections of rectangular prisms are more specifically rectangles, EXCEPT when the line
segment connecting opposite edges of the base is on an angle and the cut is also made down on
an angle toward the base.
PRACTICE WITH PLANE SECTIONS OF RECTANGULAR PRISMS
How can you describe the slice that is taken in this picture? The slice is parallel to the bases of
the rectangular prism.
What do we know about slices that are parallel to the base? They will be congruent to the shape
of the base.
How can we describe the plane section? It is a rectangle congruent to the top and bottom bases
of the rectangular prism.
EXPLORING PLANE SECTIONS OF RECTANGULAR PYRAMIDS
Cut out the figures following the thin solid line. For now, ignore the dashed lines and the dark
bold lines.
What solid do you think will be constructed from this net? A rectangular pyramid.
Fold the figures on the dotted lines and tape the edges together.
What is the shape of the base of Figure D? A rectangle
What is the shape of the base of Figure E? A rectangle
What is the name of Figure D that you built? A rectangular pyramid.
What is the name of Figure E that you built? A rectangular pyramid.
What do you notice about these pyramids? They have bold lines around them as if a slice is
being cut into them.
Let’s cut the slice! Take your scissors and carefully cut the dark bold line until each pyramid has
been separated into two pieces.
Take out the paper that you are using to trace the figures. Place the plane section where the cut
was made, face down onto the paper and use a pencil to trace the outside of the cut. This is
essentially where the dark bold line is. Sketch a picture of the slices.
How can you describe the cut out of the Figure D? The slice is perpendicular to the rectangular
base and through the top point, the vertex of the pyramid.
How can you describe the cut out of the Figure E? The slice is parallel to the rectangular base of
the pyramid.
What observations can you make about the plane section from the cut of Figure D? The slice is
the shape of a triangle.
What observations can you make about eh plane section for the cut of Figure E? The slice is in
the shape of a rectangle, similar to the rectangle shape of the base, yet smaller.
Conclusions:
When we take a slice parallel to the base of a rectangular pyramid, the plane section is in the
shape of a rectangle and gradually becomes smaller as the slice moves to the top of the pyramid.
When we take a vertical slice perpendicular to the base of the rectangular pyramid and through
the vertex, the plane section will be in the shape of a triangle.
Cut out the figures following then thin solid line. For now, ignore the dashed lines and the dark
bold lines.
What solid do you think will be constructed from this net? A rectangular pyramid. This is the
same net as the last construction.
Fold the figures on the dotted lines and tape the edges together.
What is the shape of the base of Figure F? A rectangle
What is the shape of the base of Figure G? A rectangle
What is the name of Figure F that you built? A rectangular Pyramid
What is the name of Figure G that you built? A rectangular Pyramid
What do you notice about these pyramids? They have bold lines around them as if a slice is
being cut into them.
Let’s cut the slice! Take your scissors and carefully cut the dark bold line until each pyramid has
been separated into two pieces.
Take out the paper that you are using to trace the figures. Place the plane section where the cut
was made, face down onto the paper and use a pencil to trace the outside of the cut. This is
essentially where the dark bold line is. Sketch a picture of the slices.
How can you describe the cut out of the Figure F? The slice is on the downward angle towards
the base through the lateral sides.
How can you describe the cut out of the Figure G? The slice is perpendicular to the base through
the lateral sides.
What observations can you make about the plane section from the cut of Figure F? The slice is
in the shape of a tall trapezoid.
What observations can you make about the plane section from the cut of Figure G? The slice is
in the shape of a small trapezoid.
Conclusions:
When the slice passes through any of the lateral sides, the plane section will be in the shape of a
trapezoid, no matter if the slice is perpendicular or angled to the base.
PLANE SECTIONS OF RECTANGULAR PYRAIDS – SUMMARIZED
What conclusions, if any, can we draw about perpendicular slices? We could not draw
conclusions because Figure D and Figure G were both perpendicular to the base and one resulted
with a plane section of a triangle while the other had a plane section of a trapezoid.
How many lateral sides did the slice in Figure G cut through? Three
How many lateral sides did the slice in Figure D cut through? Two
Explain why Figure D’s plane section is a triangle and Figure G’s is a trapezoid. Figure D
passes through the vertex, which is one point and continues down to the opposite edges to create
a figure with a total of three points.
Figure G cuts through two points at the top of the pyramid on the two edges, and then creates
two more vertices for its shape when it is cut at the base on opposite sides.
Are there any triangular plane sections that do not pass through the vertex of the pyramid? Any
slice that cuts through only two lateral faces will create a triangular plane section.
Take a look at the “Generalize the Findings:” box.
Any slice that is parallel to the base will result in a rectangular plane section.
Any slice, perpendicular or angled, that passes through the vertex or only two lateral faces will
have a triangular plane section.
Any slice, perpendicular or angled, that passes through three lateral faces will have a plane
section in the shape of a trapezoid.
PRACTICE WITH PLANE SECTIONS OF RECTANGULAR PYRAMIDS
How can you describe the slice that is taken in the picture? The slice is through only two lateral
faces.
What do we know about slices cut through only two lateral faces before cutting through the
base? These slices create triangular plane sections.
How can we describe the plane section? As expected, it is a triangle. Sketch the plane section.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson. The
question was, Tyrese is slicing a block of cheese. He is using the cheese along with different
fruits to make sculptures of animals. He needs a plane section of cheese in the shape of a
parallelogram that is not a rectangle. How should he cut the block of cheese?
S, Study the Problem. Underline the question and complete this statement. This problem is
asking me to find the way to cut the block of cheese to make a parallelogram that is not a
rectangle.
In Step O, we Organize the Facts. First we identify the facts. We go back and read our SOLVE
problem and make a vertical line or strike mark after each fact. Tyrese is slicing a block of
cheese. / He is using the cheese along with different fruits to make sculptures of animals. / He
needs a plane section of cheese in the shape of a parallelogram that is not a rectangle. / How
should he cut the block of cheese?
After we identify the facts, we eliminate the unnecessary facts. We do that by drawing a line
through any fact will not help us determine how he should cut the block of cheese.
After we eliminate the unnecessary facts, we list the necessary facts. He has a block of cheese
that is in the shape of a rectangular prism. He needs to have a plane section to be a
parallelogram but not a rectangle.
L, Line Up a Plan. Write in words what your plan of action will be. Draw a prism and visualize
the different cuts that can be made. Think about which will make a parallelogram.
Choose an operation or operations. In this problem we do not have a specific operation.
V, Verify Your Plan with Action. Estimate your answer. We don’t have an estimate here
because we’re not doing any computation.
Carry out your plan. Cut parallel to the base – same as the base – of a rectangle. Cut
perpendicular to the base – it is the same as lateral side – a rectangle. If we cut angled to the
base and segment in between opposite sides of the base are not perpendicular – then we have a
parallelogram.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes we found the cut that
will make a parallelogram that’s not a rectangle.
Is your answer reasonable? Again we did not have an estimate so we can mark that N/A which
means not applicable.
Is your answer accurate? Go back and check your work. Yes.
Write your answer in a complete sentence. Tyrese should make a cut that is angled to the base of
the block of cheese and also angled so that the knife is not perpendicular to both of the opposite
top edges.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, Define plane section. A plane section is a two-dimensional slice of a
three-dimensional figure.
Number Two, explain how a plane section is related to the base of a three-dimensional figure if
the cut is parallel to the base. The plane section will be the same shape as the base – if the
original figure is a prism it will be congruent to the base and if the original figure is a pyramid, it
will be similar to, but smaller than, the base.
Number Three, explain how a plane section of a pyramid is related to the lateral sides if the
figure is cut through the vertex. The plane section will be a triangle. It will start as a point, and
widen as it goes lower to the base.