A cone of 70 mm base diameter is cut by a plane parallel to its axis and 15 mm from it. The section obtained is
Answer: C
A plane parallel to the axis is shallower to the base than any generator, which is the hyperbolic condition, and only one branch falls within a single finite cone. Moving the plane onto the axis would give a triangle instead.
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A hexagonal pyramid is cut by a plane through its apex. The number of sides in the section is
Answer: D
Any plane through the apex cuts two lateral faces and the base, giving a triangle whatever the base polygon. Cutting across all six lateral faces without passing through the apex would give a hexagonal section instead.
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Match the section plane with the curve it cuts from a cone.
1. Parallel to the base 2. Parallel to a generator 3. Through the apex
P. A triangle Q. A circle R. A parabola
Answer: A
The inclination of the plane relative to the generator decides the curve entirely, which is why one solid yields the whole family of conics. Passing through the apex gives the degenerate triangular form rather than a curve.
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Match the solid and cut with the section produced.
1. Cylinder cut perpendicular to its axis 2. Cylinder cut containing its axis 3. Cylinder cut obliquely to its axis
P. A rectangle Q. A circle R. An ellipse
Answer: B
Correct answer: 1-Q, 2-P, 3-R
The three cuts give the three sections a cylinder can produce. Perpendicular to the axis reproduces the base circle. Containing the axis gives the rectangle of diameter by height. And any oblique cut clear of the ends gives an ellipse, whose minor axis is always the diameter.
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Put the steps of finding the true shape of an inclined section in order.
1. Project an auxiliary view on a plane parallel to the section plane 2. Draw the solid's plan and elevation 3. Mark where the section line crosses the solid's edges 4. Set the section plane as a line in one view at the stated inclination
Answer: C
Correct answer: 2, 4, 3, 1
The plan and elevation must exist before anything can be marked on them; the section plane is then drawn as a line at the stated inclination; the points where it crosses the edges are identified in both views; and the auxiliary view is projected last, on a plane parallel to the section, where the true shape appears.
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Consider the following statements about sections of solids.
1. Every plane section of a sphere is a circle 2. A plane through the apex of a cone gives a triangle 3. A cut parallel to a pyramid's base gives a figure identical to the base
Answer: D
Correct answer: 1 and 2 only are correct
Statements 1 and 2 are correct: every plane section of a sphere is a circle, and a plane through the apex of a cone gives a triangle. Statement 3 is the standard error - a cut parallel to a pyramid’s base is similar to it but smaller, and only a prism gives a section identical to its base.
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One cut of a right circular cone yields no curve at all. Which is it?
Answer: A
Correct answer: one passing through the apex
A plane through the apex meets the curved surface along two straight generators, so the boundary is entirely straight. Every other cut produces one of the conics: a circle parallel to the base, a parabola parallel to a generator, and a hyperbola when steeper than one.
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An auxiliary view is needed for the true shape in all but one of these cases. Which one?
Answer: B
Correct answer: a cylinder cut perpendicular to its axis
A cut perpendicular to a vertical axis is horizontal, and the plan is already a view of a horizontal plane, so the section appears there at true size with no further projection. Each of the other three cuts is inclined to both principal planes and is foreshortened in each, so an auxiliary view is required.
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A student hatches the near portion of a solid rather than the retained far portion. The resulting view
Answer: C
Correct answer: contradicts the arrows on the cutting plane line
The arrows state the direction of viewing, and what is drawn is the part remaining on the far side of the plane from the observer. Hatching the near portion instead shows the piece that has been removed, so the view and the arrows describe different halves of the solid.
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A section is measured directly from the elevation of an obliquely cut prism and reported as the true shape. The figure will be
Answer: D
Correct answer: too small in the direction of the inclination
An inclined face is foreshortened in a principal view: the elevation shows its projected length rather than its real one, and the projection is always the shorter. Measuring from the elevation therefore understates the section in the direction of the inclination, which is precisely why an auxiliary view is projected.
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