Projections of Planes Questions and Answers

Answer: A

Correct answer: 1600 mm\( ^2 \)

A figure parallel to the plane of projection is not foreshortened, so the plan is the true shape at true size: \[ 40 \times 40 = 1600\ \mathrm{mm^2}. \] The plan of a horizontal figure can therefore be measured directly, which is why a drawing places faces parallel to a plane of projection wherever it can.

Answer: B

Correct answer: 800 mm\( ^2 \)

Projected area is the true area times the cosine of the inclination: \[ 1600 \times \cos 60^\circ = 1600 \times 0.5 = 800\ \mathrm{mm^2}. \] Only the dimension across the hinge is shortened; the 40 mm along the hinge is unchanged, so the plan is a 40 by 20 rectangle.

Answer: C

Correct answer: 1414 mm\( ^2 \)

\[ 2000 \times \cos 45^\circ = 2000 \times 0.7071 = 1414\ \mathrm{mm^2}. \] The cosine rule for projected area follows directly from the fact that one dimension is preserved and the perpendicular one is multiplied by the cosine - so the area, being their product, carries the cosine once.

Answer: D

Correct answer: 43.3 mm

The minor axis is the diameter perpendicular to the axis of tilt, foreshortened by the cosine: \[ 50 \cos 30^\circ = 50 \times 0.866 = 43.3\ \mathrm{mm}. \] It is always the shorter of the two axes, so an answer larger than the true diameter must be wrong.

Answer: A

Correct answer: 50 mm

The major axis lies along the axis of tilt and is therefore not foreshortened at all: it remains the true diameter of 50 mm whatever the inclination. This is what makes the construction of the projected ellipse straightforward - the major axis is known at once and only the minor axis needs computing.

Answer: B

Correct answer: 60 degrees

The projected area is the true area times the cosine of the inclination, so \[ \cos\theta = 0.5, \qquad \theta = 60^\circ. \] The distractor of 30 degrees comes from taking the sine instead; at 30 degrees the figure would project at 0.866 of its area, not half.

Answer: C

Correct answer: 60 mm

Being perpendicular to the horizontal plane, the rectangle appears in the plan as a straight line, and the length of that line is the horizontal dimension of the rectangle, which is 60 mm. The inclination to the vertical plane sets the direction of the line but does not shorten it, since the plan shows horizontal lengths at true size.

Answer: D

Correct answer: greater than the true dihedral angle

The true angle is measured in a plane perpendicular to the line of intersection. A section taken parallel to that line cuts both faces obliquely, and cutting a wedge at a slant always opens the angle out, so the measured value exceeds the true one. Only the perpendicular section gives the correct figure.

Answer: A

Correct answer: 55 degrees

The angle a line makes with a plane and the angle it makes with the normal to that plane are complementary: \[ 90 - 35 = 55^\circ. \] The distinction matters because the two are easily interchanged, and a solution that quotes the angle with the normal where the angle with the plane was wanted is out by exactly this amount.

Answer: B

Correct answer: parallel to the horizontal plane

A plane with no horizontal trace never meets the horizontal plane, so it must be parallel to it; and it meets the vertical plane in a line parallel to the reference line, which is the vertical trace described. The mirror case, a plane with only a horizontal trace parallel to the reference line, is parallel to the vertical plane.