Slicing Solids

A cross section is the challenge you acquire when you make one slice v an object. Below is a sample slice v a cube, showing among the overcome sections you can get.

You are watching: Is it possible for a cross section of a cube to be an octagon

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The polygon developed by the slice is the cross section. The cross ar cannot contain any type of piece of the initial face; it all originates from “inside” the solid. In this picture, just the gray piece is a cross section.

Use the Interactive activity below to occupational on difficulties C1 and C2. Because that a non-interactive version of the activity, and to occupational on difficulties C3 and also C4, you might want to use clay solids and also dental floss come derive your answers. Alternatively, you might want to use colored water in plastic solids. Note 3


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How may faces does a cube have? every side of her cross section comes from cutting v a face of her cube.


Problem C1

Try to do the adhering to cross sections by slicing a cube:

a.a square
b.an it is intended triangle
c.a rectangle the is no a square
d.a triangle that is not equilateral
e.a pentagon
f.a hexagon
g.an octagon
h.a parallel that is not a rectangle
i.a circle

Record which of the shapes you were able to create and also how you did it. The Interactive activity provides you with one way to make each the the forms that girlfriend can, in fact, make together a cross section.

Problem C2

A couple of the forms on the list in problem C1 are impossible to make by slicing a cube. Explain what renders them impossible.

Problem C3

Find a way to part a tetrahedron to make a square overcome section. How did you perform it?

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Problem C4

What overcome sections can you gain from every of these figures?

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Polygon Shadows


You’ve looked at few of the “two-dimensional” nature of three-dimensional figures: your surfaces (nets) and also their cross sections. Solid figures likewise cast two-dimensional shadows. Do you think you can recreate a heavy by learning what shadows that casts?

Take the Further

Problem C5

A solid object casts a circular zero on the floor. As soon as it is lit indigenous the front, it casts a square shadow on the back wall. Try to construct the object out of clay. Deserve to you surname it?

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Problem C6

A solid thing casts a circular shadow on the floor. Once it is lit from the left, it casts a triangular shadow on the back wall. Shot to construct the object out of clay. Have the right to you surname it?

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Problem C7

Suppose the object casts a circular shadow on the floor, a square shadow when lit indigenous the front, and a triangular shadow once lit from the left. Shot to build the object out of clay. Deserve to you surname it? Note 4

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Notes


Note 3

To usage clay solids and also dental floss come derive her answers, mold a cube native the clay. Then usage the dental floss or piano wire to do a right slice through the cube, creating a overcome section. Once making a cut, friend may end up with “bent” cuts, but what you want to shot to do is a planar slice. For example, think about slicing straight through a bread of bread.

Alternatively, you might want to use colored water in plastic solids. Fill the plastic solids component way, and also then tilt them at various angles to view the encounters of cross sections created by an imaginary slice along the surface ar of the water.

Note 4

You may want to more explore difficulties C5-C7 ~ above your own using a flashlight and also objects defined in remedies to these problems.


Solutions


Problem C1

a.A square overcome section have the right to be developed by slicing the cube by a aircraft parallel to one of its sides.
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b.An equilateral triangle overcome section deserve to be derived by cut the cube by a airplane defined by the midpoints that the three edges emanating from any type of one vertex.
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c.One method to achieve a rectangle the is not a square is by cut the cube through a aircraft perpendicular to one of its encounters (but no perpendicular to the edges of the face), and also parallel come the four, in this case, vertical edges.
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d.Pick a vertex, let’s say A, and also consider the three edges meeting at the vertex. Construct a aircraft that consists of a suggest near a vertex (other 보다 vertex A) on among the 3 edges, a point in the middle of another one the the edges, and also a 3rd point that is no in the center nor coinciding with the first point. Slicing the cube through this plane creates a cross ar that is a triangle, however not an it is intended triangle; that is a scalene triangle. Notification that if any kind of two selected points are equidistant indigenous the original vertex, the cross ar would be an isosceles triangle.
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e.To obtain a pentagon, slice through a plane going through five of the six faces of the cube.
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f.To acquire a hexagon, slice through a airplane going with all six encounters of the cube.
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g.It is not possible to produce an octagonal cross ar of a cube.h.To develop a non-rectangular parallelogram, slice v a plane from the top confront to the bottom. The part cannot be parallel to any kind of side the the top face, and also the slice should not it is in vertical. This allows the cut to type no 90° angles. One example is to cut through the top confront at a corner and a midpoint the a non-adjacent side, and cut come a different corner and also midpoint in the bottom face.
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i.It is not feasible to create a one cross ar of a cube.

Problem C2

Whenever we cut the cube with a plane, every edge that the cross section corresponds to one intersection of among the cube’s faces with the plane. Since the cube has actually only 6 faces, the is difficult to cut it with one plane and produce an octagonal cross section. Also, since the cube has no bent faces, a aircraft will not be able to intersect a cube and also create a cross ar with a curved segment in the perimeter.

Problem C3

One way to create a square cross ar in a tetrahedron is to cut at the midpoints of 4 edges.

Alternatively, you can start with a network for the tetrahedron together as:

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We then connect the midpoints that the sides through segments of same length: EF, FG, GH, and HE.

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When us fold the net into a tetrahedron, the point out E, F, G, and H room on the very same plane, and also they specify a square overcome section once that plane cuts the tetrahedron.

Problem C4

a.Any cross ar of a round will be a circle.
b.Possible overcome sections space circles (cut parallel to the base), rectangles, and ellipses.
c.Possible cross sections are circles (cut parallel come the one base), ellipses (cut at an angle, not parallel to the circular base and not intersecting the base of the cone), parabolas (cut parallel come the leaf of the cone, no intersecting the vertex yet intersecting the base), and hyperbolas (cut perpendicular come the base, however not intersecting the vertex).

Problem C5

A appropriate square cylinder, i.e., a cylinder who height amounts to the diameter of its base.

Problem C6

A appropriate circular cone.

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Problem C7

It’s a solid the looks favor a “triangular” filter because that a coffee maker, or the head (not handle) that a flathead screwdriver.