HS GEO.S.GMD.3.
Cross Sections -
I DO 1
A deli had a piece of lunchmeat in the shape of a cylinder as shown below. The clerk in the deli cut
the lunchmeat horizontally along the dotted line. What was the shape of the lunchmeat where he
made his cut?
In this question, they are asked to find the shape of lunch meat.
To find the answer, we will imagine a cylinder.
The cylinder is a solid object with two parallel circular bases If you want to slice through this
cylinder horizontally at any height, we will get a cross-section of the cylinder at that height. That
would be a circle.
This intersection is because of the plane with the cylinder.
When the plane is parallel to the basis, it forms a circle.
Therefore, the clerk at Delhi cut the lunch meat horizontally along the dotted line.
So, the shape of the lunch meat where he cut it would be a circle.
Therefore, the cross-section will be a circle.
CFU
When a cylinder is cut horizontally, what shape does the cross-section reveal?
A circle.
What determines the shape formed when a cylinder is cut along a horizontal plane?
A horizontal cut intersects with the cylinder's curved surface to create a circular cross-section.
Is the cross-section of a cylinder after a horizontal cut a circle or a square?
A circle.
WE DO 1
Suppose you have a hexagonal pyramid. Name the figure that shows a cross-section of the
pyramid parallel to its base.
We’re asked to find the name of the figure.
Suppose you have a hexagonal pyramid.
Name the figure that shows a cross-section of the pyramid parallel to its base.
This question asks us to find the figure showing the pyramid's cross-section parallel to the base.
A hexagonal pyramid has a base that is a hexagon, which means it has six sides.
When you cut the pyramid parallel to its base, the cutting plane passes through all six sides of the
hexagon.
The resulting cross-section of the pyramid at this level will be a hexagon.
This hexagon is the same shape and size as the pyramid's base.
Therefore, a hexagon is a cross-section of the hexagonal pyramid parallel to its base.
CFU
What shape is the base of a hexagonal pyramid?
A hexagon.
What shape does the cross-section form when a hexagonal pyramid is cut parallel to its base?
A hexagon.
Does the cross-section of a hexagonal pyramid, taken parallel to its base, resemble its base in
shape?
Yes.
YOU DO 1
Paul bought a piece of cheese that was in the shape of a cylinder, as shown below.
He unwrapped the cheese and cut a thin slice off of the end. What would be the shape of the
cheese slice?
Responses: circle:triangle:rectangle: square
In this question, we are asked to find the shape of the cheese slice.
The cheese is initially in the shape of a cylinder, which has a circular cross-section.
When Paul cuts a thin slice off the end of the cylinder, he makes a perpendicular cut through the
cylinder.
The shape of the cross-section, which Paul cut, will be circular.
This is because a cylinder creates circular cross-sections when cut perpendicular to its axis.
Therefore, the shape of the cheese slice after cutting off the end of the cylinder is a circle.
CFU
What shape is the cross-section of the cheese after Paul cuts off a thin slice from the end of the
cylinder?
A circle.
When a cylinder is cut perpendicular to its axis, what shape does the resulting cross-section
typically have?
A circle.
Does the shape of the cheese slice, after cutting off the end of the cylinder, resemble the shape of
the cylinder's base?
Yes.
YOU DO 2
A deli clerk has lunchmeat formed in the shape of a cylinder with a radius of 2 inches and a height
of 4 inches. He divides the lunchmeat into two equal sections by making a vertical cut as shown
below.
What is the shape of the cross-section formed by the vertical cut?
In this question, we are asked to find the shape of the cross-section, which is formed by the
vertical cut.
A cylinder has a circular base.
When you make a vertical cut through a cylinder, the cross-section you see will be the shape of its
base.
The base of the cylinder is a circle because the cylinder is circular in shape.
Therefore, the shape of the cross-section formed by the vertical cut is a circle.
CFU
Is the lunchmeat in the problem described as having a cylindrical shape?
Yes, the lunchmeat is formed in the shape of a cylinder.
What is the radius of the cylinder mentioned in the problem?
The radius of the cylinder is 2 inches.
What shape is the cross-section formed by the vertical cut through the cylinder?
The cross-section formed by the vertical cut through the cylinder is a circle.
YOU DO 3
A rectangle has vertices at (0, 0), (5, 0), (5, 6), and (0, 6).
If the rectangle is revolved around the y-axis, what 3-dimensional solid is formed?
In this question, we are asked to find a 3 dimensional solid formed by the points.
The rectangle's vertices are at (0, 0), (5, 0), (5, 6), and (0, 6).
Its width is 5 units (distance between (0, 0) and (5, 0)).
Its height is 6 units (distance between (0, 0) and (0, 6)).
When the rectangle is rotated around the y-axis:
It forms a cylinder.
The cylinder's diameter equals the rectangle's width of 5 units.
The cylinder's height equals the height of the rectangle, which is 6 units.
Therefore, the 3-dimensional solid formed is a cylinder with a diameter of 10 units and a height of
6 units.
CFU
What are the coordinates of the vertices of the rectangle in question?
(0, 0), (5, 0), (5, 6), and (0, 6).
What is the width of the rectangle in units?
5 units.
What is the shape of the 3-dimensional solid formed when the rectangle is revolved around the y-
axis?
A cylinder with a diameter of 10 units and a height of 6 units.
EXIT TICKET 1
Points A and C lie on the surface of this right cone. Point B is in the interior of the cone.
Which statement is true?
This question asks us to find the true statement from the given options.
The truth statement forgiven question is option c.
Let's analyze the option C.
the plane is positioned relative to the cone's apex.
When a plane intersects a cone such that it includes two points (like Points A and C) on the cone's
surface, and these points are not directly above the apex and are not on the same line from the
apex to the base (generatrix), the resulting shape is an ellipse.
The reason for this is that the intersection of a cone and a plane under these conditions forms an
ellipse.
This is a type of conic section characterized by its closed and bounded shape.
CFU
What shape does the intersection of a cone and a plane, including Points A and C, create?
An ellipse.
Where should Points A and C be located for the intersection of the cone and a plane to create an
ellipse?
It is on the cone's surface but not aligned with the apex and not on the same generatrix.
EXIT TICKET 2
A right circular cylinder is intersected by a plane at the oblique angle shown.
What geometric shape is formed by the planar cross section?
In this question, we are asked to find the shape of the cross-section.
A right circular cylinder is intersected by a plane at an oblique angle.
The geometric shape formed by the planar cross section is Ellipse.
When a plane intersects a right circular cylinder at an oblique angle.
The resulting cross-section is an ellipse.
This occurs because the intersection of a cylinder and a plane at an angle forms a conic section.
The conic section is known as an ellipse.
CFU
What shape is formed when a plane intersects a right circular cylinder at an oblique angle?
An ellipse.
How does the angle of intersection affect the shape of the cross-section in a right circular cylinder?
A plane intersecting at an oblique angle forms an elliptical cross-section.
EXIT TICKET 3
Gina made a gelatin dessert in a cylindrical mold. She took it out of the mold, placed it on a plate,
and sliced it down the middle, as shown below.
What shape did Gina expose on each piece of the dessert where her cross-sectional cut was made?
In this question, we are asked to find the cross-section shape.
When Gina slides the cylinder from the middle, the shape formed on the cross-sectional cut is a
rectangle.
The cross-sectional cut is the point from which the original shape was cut.
The original shape is a cylinder.
The shape that is asked to be identified is the one on the face of the new shape formed after the
cut.
Therefore, each piece of the gelatin dessert that Gina exposed by cutting it down the middle
would reveal a rectangle.
CFU
What was the name of the initial ship?
The name of the initial shape is a cylinder.
How did she cut the cylinder?
She cut the cylinder vertically.
BONUS 1
In the diagram below, a right circular cylinder with a radius of 3 inches is intersected vertically by
a plane passing through Points A and B, the centers of the circular bases.
What is the perimeter of the figure formed by the intersection of the plane and the cylinder?
In this question, we are asked to find the perimeter.
Identify that the cross-section formed by the intersection of the plane and the cylinder is a
rectangle
2×3 inches, or 6 inches
Since the cylinder is intersected vertically, the rectangle's height is the same as the height of the
cylinder.
Calculate the perimeter of the rectangle using the formula.
P=2(base+height)
Substitute the base and height into the formula to get
P=2(6+14) inches
Perform the addition inside the parentheses:
6+14=20
Multiply the sum by 2 to find the perimeter:
2×20=40 inches.
So, the perimeter of the cross-section 40 inches.
CFU
What is the shape of the given question?
The shape of the given question is a cylinder.
What is the value of 2 times 3?
The value of 2 * 3 is 6.
What's the value of 4 + 16?
The value of 4 + 16 is 20.
BONUS 2
In the figure below, points A, B, and C are midpoints of the edges of a cube.
If the cube is intersected by a plane at points A, B, and C, which of the following best describes the
intersection?
In this question, we are asked to analyze the intersection.
Points A, B, and C are midpoints of the cube's edges.
This means each point divides one of the cube's edges into two equal halves.
When a plane passes through these three points (A, B, and C), it intersects the cube in a manner
that forms a triangle.
Since A, B, and C are equidistant from each other (each being the midpoint of an edge), and the
cube is symmetric, the triangle formed will have all sides of equal length.
Therefore, the intersection shape is an equilateral triangle, with all sides of equal length and all
angles equal (60 degrees).
CFU
How many midpoints are given in the question?
The total three midpoints are given in the question.
What is the shape of the given question?
The shape of the given question is a cube.
What is the equilateral triangle?
The equilateral triangle is the one which has 60 degrees in each vertex
