Projections of Solids Questions and Answers

Answer: A

Correct answer: plane surfaces

A polyhedron is bounded entirely by flat faces - prisms, pyramids, cubes and the five regular solids. The moment a curved surface appears the solid belongs to another class: a cylinder and a cone are solids of revolution, and a sphere is bounded by a single curved surface. The distinction matters in drawing because plane faces project as straight-edged figures.

Answer: B

Correct answer: a fixed axis

A solid of revolution is generated by turning a plane figure through a full revolution about a fixed line in its own plane. A rectangle gives a cylinder, a right triangle a cone, and a semicircle a sphere. Every section perpendicular to that axis is therefore a circle, which is what makes such solids easy to recognise from their views.

Answer: C

Tetrahedron, cube, octahedron, dodecahedron and icosahedron exhaust the possibilities, since no other arrangement of identical regular faces closes on itself. The proof is a matter of the angles meeting at each vertex.

Answer: D

Correct answer: two parallel identical bases

A prism has two parallel and identical bases joined by lateral faces; a pyramid has a single base and all its lateral faces meeting at an apex. Both are polyhedra with plane faces, and both may be right or oblique - the difference is in the number of bases, and hence in whether the lateral edges are parallel or convergent.

Answer: A

Correct answer: a plane parallel to the base

A frustum is what remains when the top of a pyramid or cone is cut off by a plane parallel to the base. Because the cut is parallel, the two ends are similar figures, which is what makes the frustum easy to draw and to develop. A cut inclined to the base gives a truncated solid instead, whose ends are dissimilar.

Answer: B

Correct answer: inclined to the base rather than parallel to it

Both are the remainder after the top is removed, and the difference is the attitude of the cutting plane. A frustum is cut parallel to the base, so its two ends are similar and the section is the same shape as the base. A truncated solid is cut at an angle, so the section is a different figure - an ellipse on a cone, for instance.

Answer: C

Correct answer: perpendicular to its base

In a right solid the axis, which joins the centre of the base to the apex or to the centre of the other base, stands at right angles to the base. If it is inclined instead the solid is oblique. The distinction governs the drawing: a right solid standing on its base gives a plan of true size and an axis projecting as a point.

Answer: D

Correct answer: a point on the base circle

The slant height is the distance from the apex to any point on the circumference of the base, measured along the surface of the cone, and by symmetry it is the same for every such point. It is the hypotenuse of a right triangle whose other two sides are the vertical height and the base radius, so it always exceeds both.

Answer: A

Correct answer: a circle

Looking down on a cylinder standing on its base, every generator projects into the base circle and the axis into its centre, so the plan is simply a circle of the true base diameter. The elevation would be a rectangle of the base diameter by the axis length - the two views together identifying the solid unambiguously.

Answer: B

Correct answer: cone

A cone standing on its base gives a circle in plan, of the true base diameter, and a triangle in elevation, its base the diameter and its apex on the axis. A cylinder would give a rectangle in elevation, a sphere a circle in both views, and a square pyramid a square in plan.