For that same cylinder, the area of the lateral development is about
Answer: A
Correct answer: 22 000 mm2
\[ 220 \times 100 = 22{,}000\ \mathrm{mm^2}. \] The lateral area of a cylinder is simply the circumference times the height, which is exactly what the development makes visible - the curved surface laid flat is a plain rectangle.
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A cone of 60 mm base diameter and 80 mm slant height develops as a sector whose angle is about
Answer: B
Correct answer: 135 degrees
The sector angle is the base circumference divided by the circumference of a full circle of the slant height: \[ \theta = \frac{r}{L} \times 360 = \frac{30}{80} \times 360 = 135^\circ. \] Note the base radius is used, not the diameter - the 60 mm given is the diameter.
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A cone of 50 mm base radius and 100 mm slant height develops as a sector whose angle is
Answer: C
Correct answer: 180 degrees
\[ \theta = \frac{r}{L} \times 360 = \frac{50}{100} \times 360 = 180^\circ. \] When the slant height is twice the base radius the development is exactly a semicircle, which is a useful case to recognise - it corresponds to a half-angle of 30 degrees at the apex.
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A cone of base radius 30 mm and vertical height 40 mm has a development struck with a radius of
Answer: D
The slant height is the hypotenuse on the radius and the height, that is the square root of 900 plus 1600. Using the 40 millimetre vertical height as the radius would give a sector far too small to reach round the base.
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A square prism of 40 mm side develops laterally as a rectangle whose length is
Answer: A
Correct answer: 160 mm
The lateral development of a prism is a rectangle whose length is the perimeter of the base: \[ 4 \times 40 = 160\ \mathrm{mm}. \] The four faces are laid side by side, and the fold lines between them are the vertical edges of the prism.
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A hexagonal pyramid of 30 mm base side has a development whose total arc-wise base length is
Answer: B
Correct answer: 180 mm
The base edges of the six triangular faces lie end to end along the bottom of the pattern, so their total is the base perimeter: \[ 6 \times 30 = 180\ \mathrm{mm}. \] It is called an arc-wise length because the faces fan about the apex, so the base line of the pattern is not straight.
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A cylinder of 80 mm diameter is cut obliquely so that its generators vary from 60 mm to 100 mm. The development's stretch-out line measures about
Answer: C
Correct answer: 251 mm
The stretch-out line is the base circumference, unaffected by the oblique cut at the top: \[ \pi \times 80 = 251\ \mathrm{mm}. \] The cut changes the shape of the upper boundary into a sinusoidal curve, and changes nothing about the length of the base.
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For that same truncated cylinder, the mean generator length is
Answer: D
Correct answer: 80 mm
\[ \frac{60 + 100}{2} = 80\ \mathrm{mm}. \] The generators vary linearly round the cylinder for a plane oblique cut, so the mean is the average of the extremes - and the lateral area is that mean multiplied by the circumference, exactly as for an uncut cylinder of 80 mm height.
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A frustum of a cone with end radii 20 mm and 50 mm and slant side 40 mm develops between arcs whose radii differ by
Answer: A
Correct answer: 40 mm
The two arcs bounding a frustum’s development are concentric about the apex of the complete cone, and the radial distance between them is the slant side of the frustum itself - here 40 mm. The end radii determine the two arc lengths rather than the gap between the arcs.
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A sphere of 100 mm diameter approximated by six horizontal zones has each zone spanning a height of about
Answer: B
Dividing the 100 millimetre diameter into six equal heights gives 16.67 millimetres each. Each zone is then treated as a conical frustum, whose development is exact even though the assembly only approximates the sphere.
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