Plane Sections of 3-D Figures — Practice
Slice a solid with a flat plane and a flat shape appears. The angle of the cut decides which shape you get.
What these questions cover
These questions use 17 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- Cross section — The 2-D face revealed when a plane cuts through a solid.
- Plane — A perfectly flat surface extending forever.
- Net — A 3-D shape unfolded flat.
- Face — A flat surface; a line where two faces meet; a corner point.
- Edge — A flat surface; a line where two faces meet; a corner point.
- Base — The number being multiplied. In 53 the base is 5.
- Radius — Centre to edge. Half the diameter.
- Diameter — All the way across through the centre. d = 2r.
- Cylinder — Two parallel circles joined by a curved surface. V = π r2 h.
- Cone — A circle tapering to a point. V = 13π r2 h.
- Parallel — Same slope, different intercept.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Diagonal — A segment joining two non-adjacent vertices.
- Pyramid — A base tapering to a point. V = 13 Bh.
- Height — The perpendicular distance from the base to the opposite corner.
- Rectangular prism — A box — six rectangular faces meeting at right angles.
Problem 1
A cube is sliced parallel to one of its faces. What shape is the cross section?
- A square
- A triangle
- A circle
- A hexagon
Worked answer
A square
You get a square identical to the face, no matter how deep you cut.
Problem 2
A cylinder is sliced parallel to its base. What shape is the cross section?
- A rectangle
- A circle
- An oval
- A triangle
Worked answer
A circle
A circle, the same size as the base.
Problem 3
How many faces does a cube have?
Worked answer
6
6 faces.
Problem 4
How many edges does a cube have?
Worked answer
12
4 + 4 + 4 = 12 edges.
Problem 5
A cylinder is sliced straight down through its central axis. What shape appears?
- A circle
- A rectangle
- A triangle
- An oval
Worked answer
A rectangle
A rectangle: as tall as the cylinder and as wide as the diameter.
Problem 6
A square pyramid is sliced parallel to its base. What shape is the cross section?
- A triangle
- A square
- A trapezoid
- A pentagon
Worked answer
A square
A square — smaller the higher up you cut.
Problem 7
A square pyramid is sliced vertically through its apex. What shape appears?
- A square
- A trapezoid
- A triangle
- A circle
Worked answer
A triangle
A triangle, with the apex at the top of the cut.
Problem 8
What is the greatest number of sides a cross section of a cube can have?
Worked answer
6
A cube has 6 faces
A plane can be tilted to pass through all six at once
That gives a hexagon — 6 sides.
Problem 9
A cylinder has radius 4 cm and height 10 cm. It is cut vertically through its axis. Find the area of the cross section.
Worked answer
80
Height = 10 cm
Width = 2 × 4 = 8 cm
Area = 10 × 8 = 80 cm2
Problem 10
A cone is sliced parallel to its base. What shape is the cross section?
- A triangle
- A circle
- A trapezoid
- An oval
Worked answer
A circle
A circle — smaller the closer you cut to the tip.
Problem 11
A cube net is a row of four squares A, B, C, D, with E above B and F below B. Which face ends up opposite A?
- B
- C
- D
- E
Worked answer
C
Folding a row of four wraps it into a band
A and C are two apart, so they face each other
Likewise B opposite D, and E opposite F.
Problem 12
A single flat cut slices off one corner of a cube. What shape is the new face?
- A square
- A triangle
- A pentagon
- A hexagon
Worked answer
A triangle
Three faces meet at a corner, so the plane crosses three faces
Three faces cut gives a triangle.
Problem 13
A cube is sliced through the diagonal of its top face and the parallel diagonal of its bottom face. What shape is the cross section?
- A square
- A rectangle
- A triangle
- A hexagon
Worked answer
A rectangle
The plane crosses four faces, so the cross section has four sides
All four corners are right angles, but the horizontal sides are diagonals — longer than the vertical edges
So it is a rectangle, not a square.
Problem 14
A cube is sliced through the three vertices next to one corner. What kind of triangle appears?
- A right triangle
- An equilateral triangle
- A scalene triangle
Worked answer
An equilateral triangle
Each of the three sides is a diagonal of a square face
All the faces are identical, so all three diagonals are equal
The cross section is an equilateral triangle.
Problem 15
A 5 × 4 × 3 rectangular prism is sliced parallel to its 5 × 4 face. Find the area of the cross section.
Worked answer
20
The cross section is a 5 × 4 rectangle
Area = 20, the same at every depth.
Problem 16
How many distinct nets can fold into a cube?
- 6
- 8
- 11
- 14
Worked answer
11
There are exactly 11 distinct cube nets
Six have a row of four squares; the rest use rows of three or a staircase.