The development of a right cylinder's lateral surface is
Answer: C
Its length is the base circumference and its height the axis length, since every generator is equal and perpendicular to the base. Cutting the cylinder obliquely turns the top edge of that rectangle into a curve while the base edge stays straight.
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In a right cone's development, what fixes the angle of the sector at its apex?
Answer: D
The arc must equal the base circumference while the radius is the slant height, so the angle is 360 times the base radius over the slant height. Two cones of quite different size but the same ratio develop as sectors of identical angle.
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How does an oblique cut alter the development of a cylinder?
Answer: A
Each generator is cut at a different height, so the ends of the erected generators trace a curve across the top of the rectangle. The base circle is untouched, so the stretch-out line remains straight and the same length as before.
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What shape does the development of a pyramid's lateral surface take?
Answer: B
Each lateral face is a triangle whose true sides must be found, and the faces are laid out side by side about the common apex. A cone's development is a continuous sector because its generators are infinitely numerous rather than four or six.
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What must be found before a pyramid with an oblique cut can be developed?
Answer: C
The cut crosses each edge at a different distance from the apex, and those distances are foreshortened in the principal views. Each must be brought to true length before it can be stepped off along the corresponding radial line of the pattern.
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A frustum of a cone develops as
Answer: D
The inner arc corresponds to the small end and the outer to the large end, both struck from the apex of the complete cone. The apex must therefore be located by producing the slant sides even though it has been cut away.
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A frustum of a pyramid develops as
Answer: A
Correct answer: a band of trapezia laid side by side
Each face of a pyramid is a triangle, and cutting the top off leaves a trapezium. The frustum’s development is those trapezia laid edge to edge, each drawn to its own true dimensions. Concentric arcs belong to a cone’s frustum, whose surface is curved rather than faceted.
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In the zone method of approximating a sphere's development, the sphere is divided into
Answer: B
Correct answer: horizontal slices treated as frusta of cones
A sphere cannot be developed exactly, so it is approximated. The zone method cuts it into horizontal bands and treats each as a conical frustum, which can be developed. The finer the slices the closer the approximation, and it never becomes exact.
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In the lune method the sphere is divided into
Answer: C
Correct answer: vertical segments between meridian planes
The lune method cuts the sphere into orange-peel segments between meridians and flattens each. It is the alternative approximation to the zone method, and the choice between them is usually settled by how the material is to be cut and joined rather than by accuracy.
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A cylinder of 70 mm diameter and 100 mm height develops as a rectangle whose length is about
Answer: D
Correct answer: 220 mm
The length of the rectangle is the circumference of the cylinder: \[ \pi d = \pi \times 70 = 220\ \mathrm{mm}. \] Using the radius instead gives 110, and using the diameter without pi gives 70 - both natural slips. The width of the rectangle is the height, 100 mm.
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