What figure results from cutting a cylinder by a plane perpendicular to its axis?
Answer: C
A plane square across the axis meets the curved surface everywhere at the same distance from it, so the cut repeats the base circle. Tilting the plane away from perpendicular begins immediately to stretch it into an ellipse.
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What figure results from cutting a cylinder by a plane inclined to its axis but not meeting the bases?
Answer: D
The diameter across the axis of tilt is unchanged while the one along it lengthens, which is precisely the definition of an ellipse. Its minor axis equals the cylinder diameter and its major axis grows as the tilt increases.
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A cylinder is cut by a plane containing its axis. The resulting section is
Answer: A
Correct answer: a rectangle
A plane containing the axis cuts both circular ends along a diameter and the curved surface along two generators, which for a right cylinder are parallel straight lines. The figure is therefore a rectangle of the diameter by the height - the largest plane section the cylinder possesses.
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A cone is cut by a plane parallel to one of its generators. The section obtained is
Answer: B
Correct answer: a parabola
The conic obtained depends on how the cutting plane’s inclination compares with a generator. Less steep gives an ellipse, steeper gives a hyperbola, and exactly parallel to a generator gives the parabola - the transitional case, and the only one whose curve does not close.
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A cone is cut by a plane parallel to its axis, clear of the apex. What curve appears?
Answer: C
Correct answer: a hyperbola
A plane parallel to the axis is steeper than any generator, so it cuts the surface in a hyperbola. Only one branch appears on a single cone; the second branch belongs to the opposite nappe, which a solid cone does not have.
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Which section of a cone is bounded by two straight lines?
Answer: D
Correct answer: one taken through the apex
A plane through the apex meets the curved surface in two straight generators, so the section is a triangle and contains no curve. Every other plane cuts the curved surface obliquely and produces a conic - circle, ellipse, parabola or hyperbola according to its inclination.
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A cone standing on its base is cut by a plane perpendicular to the axis. The true shape of the section appears in
Answer: A
Correct answer: the plan
A cut perpendicular to a vertical axis is a horizontal plane, and the plan is the view looking down on a horizontal plane - so the section appears there undistorted. This is the one case in which no auxiliary view is needed, because a principal view already lies parallel to the section.
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Every plane section of a sphere, whatever its position, is
Answer: B
The surface is equidistant from the centre in every direction, so any plane cuts it in a curve of constant radius about the foot of the perpendicular from the centre. Only the radius of that circle changes with the plane's distance from the centre.
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What is the largest circle obtainable as a section of a sphere?
Answer: C
A plane through the centre gives a circle of the sphere's own radius, and any other plane gives a smaller one. Its radius diminishes as the plane moves away from the centre until the plane becomes tangent and the circle vanishes.
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What determines the longest possible straight cut across a right circular cylinder standing on its base?
Answer: D
The greatest chord runs diagonally from one base circle to the opposite point on the top, combining the full height with the full diameter. Its length is the hypotenuse on those two, which exceeds both the height and the diameter.
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