Projections of Solids Questions and Answers

Answer: C

Correct answer: 60 mm

The plan of a cone standing on its base is the base circle itself, projected without foreshortening because the base lies flat on the horizontal plane. So its diameter is the true base diameter, 60 mm, and the axis length plays no part - the height appears in the elevation instead, as the height of the triangle.

Answer: D

Correct answer: 60.6 mm

A line inclined to the horizontal plane projects into the plan foreshortened by the cosine of that inclination: \[ 70 \cos 30^\circ = 70 \times 0.866 = 60.6\ \mathrm{mm}. \] Using the sine instead gives 35 mm, which would be the height the axis rises rather than the length it covers on plan.

Answer: A

Correct answer: perpendicular to the vertical plane

A line perpendicular to a plane of projection appears on it as a point, so an axis showing as a point in the elevation must be perpendicular to the vertical plane. It would then appear in true length in the plan. The corresponding case in the other view is the axis perpendicular to the horizontal plane, which shows as a point in the plan.

Answer: B

Correct answer: an equilateral triangle of 50 mm side

A regular tetrahedron has four equilateral triangular faces of equal edge, so resting on any face puts that face flat on the horizontal plane, where it projects at true size - an equilateral triangle of 50 mm side. The fourth vertex projects into the centroid of that triangle, and the three edges rising to it appear as lines drawn to it.

Answer: C

Correct answer: 1-Q, 2-P, 3-R

Each plan is the base figure at true size, with whatever edges or generators are visible drawn on it. A cylinder gives a plain circle. A square pyramid gives a square with its diagonals, the diagonals being the four lateral edges seen from above. And a cone gives a circle on which radial generators may be drawn to indicate the surface.

Answer: D

Correct answer: 1-Q, 2-R, 3-P

The three resting positions differ in how much of the base touches the plane. On its base the whole face is in contact, so the base is parallel to the horizontal plane. On an edge the contact is along a line, so the base is inclined. On a corner the contact is at a single point, and the base is inclined more steeply still.

Answer: A

Correct answer: 2, 4, 1, 3

The order is fixed by the need to draw true shapes before distorting them. Begin with the simple position, base flat and axis vertical, where the base appears at true size. Tilt that drawing to give the inclination to the first plane. Then turn the result to give the inclination to the second. Only when the final views are complete can dimensions be read from them.

Answer: B

Correct answer: 1, 2 and 3 are all correct

All three hold. There are exactly five regular polyhedra - tetrahedron, cube, octahedron, dodecahedron and icosahedron. The slant height of a right cone is \( \sqrt{r^2+h^2} \) and so always exceeds the vertical height. And a sphere presents the same circle of its own diameter whatever the direction of viewing.

Answer: C

Correct answer: a hexagonal prism

A solid of revolution is generated by turning a plane figure about an axis, so its sections perpendicular to that axis are circles. A cone, a cylinder and a sphere all qualify. A hexagonal prism is a polyhedron with plane faces and could not be produced by revolving anything, since a revolution cannot generate flat lateral faces.

Answer: D

Correct answer: the cutting plane is inclined to the base

A frustum is cut parallel to the base; an inclined cut produces a truncated solid instead. The other three all describe a frustum correctly: the cut is parallel, the apex has been removed, and because the section is parallel to the base the two ends are similar figures.