Sections of Solids Questions and Answers

Answer: A

Correct answer: six sides of unequal length

The plane crosses all six faces, so the section has six sides; but each face is met at a different position and the inclination stretches the distances between them unequally, so the hexagon is not regular. The angles between adjacent sides are likewise altered from those of the base.

Answer: B

Seen edge on, the plane is a single line whose angle can be drawn to the stated inclination, and the points where it crosses the solid's edges are read off directly. The true shape still requires an auxiliary view parallel to that line.

Answer: C

Correct answer: 120 mm

The plane is at 30 degrees to the axis, so the major axis is the diameter divided by the sine of that angle: \[ \frac{60}{\sin 30^\circ} = \frac{60}{0.5} = 120\ \mathrm{mm}. \] The shallower the cut relative to the axis, the longer the ellipse becomes - and at 90 degrees it reduces to the circle of 60 mm.

Answer: D

Correct answer: 60 mm

The minor axis is the diameter itself, 60 mm, whatever the inclination. It lies perpendicular to the direction in which the cut is stretched, so the obliquity does not affect it. Only the major axis lengthens, which is why the ellipse becomes more eccentric as the cut is made shallower.

Answer: A

Correct answer: 1200 mm2

A cut through the apex and across the base diameter gives a triangle whose base is the full diameter and whose height is the height of the cone: \[ \frac{1}{2} \times 60 \times 40 = 1200\ \mathrm{mm^2}. \] Using the radius rather than the diameter as the base gives 600, which is the offered trap.

Answer: B

Correct answer: 56.6 mm

The cut is at 45 degrees to the base, so the dimension across the inclined faces is stretched by \( 1/\cos 45^\circ \): \[ \frac{40}{0.7071} = 56.6\ \mathrm{mm}. \] The other side of the rectangle remains 40 mm, being unaffected by the inclination - which is why the true shape is a rectangle rather than a square.

Answer: C

Correct answer: 26.5 mm

By Pythagoras on the radius and the distance from the centre: \[ r = \sqrt{40^2 - 30^2} = \sqrt{700} = 26.5\ \mathrm{mm}. \] The section of a sphere is always a circle, and its radius falls to zero as the plane reaches the surface at 40 mm from the centre.

Answer: D

Correct answer: 86.0 mm

The longest straight line within the cylinder runs from a point on one rim to the diametrically opposite point on the other: \[ \sqrt{50^2 + 70^2} = \sqrt{7400} = 86.0\ \mathrm{mm}. \] It is the diagonal of the rectangular axial section, so it exceeds both the height and the diameter.

Answer: A

Correct answer: 30 mm

At half the height the section is similar to the base with a scale factor of one half: \[ 60 \times \tfrac{1}{2} = 30\ \mathrm{mm}. \] The linear scale is the ratio of the distances from the apex, so a cut at mid-height halves every length in the base.

Answer: B

Correct answer: one quarter

Areas scale as the square of the linear ratio, so a section at half the height has \[ \left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4} \] of the base area. Halving the area would require a cut at \( 1/\sqrt{2} \) of the height from the apex, which is about 71 per cent of the way up rather than half.