Plane Sections of 3-D Figures — Challenge
Slice a solid with a flat plane and a flat shape appears. The angle of the cut decides which shape you get.
What these problems 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 cylinder is cut by a plane parallel to its circular base. What shape is the cross-section?
- A circle
- A square
- A triangle
- A rectangle
- An ellipse
Worked answer
A circle
A plane parallel to the base meets the curved surface all the way round at the same distance from the axis.
That set of points is a circle, the same size as the base.
Tilting the plane would stretch the cross-section into an ellipse.
Problem 2
A cylinder is cut by a plane containing its axis. What shape is the cross-section?
- A circle
- A rectangle
- A triangle
- An ellipse
- A square
Worked answer
A rectangle
Such a plane cuts straight down through the middle of the cylinder.
It meets each flat end along a straight line equal to the diameter, and the curved surface along two straight vertical lines.
Four straight sides meeting at right angles give a rectangle, of width the diameter and height the cylinder's height.
Problem 3
A cube is cut by a plane parallel to one of its faces. What shape is the cross-section?
- A square
- A triangle
- A circle
- A hexagon
- A pentagon
Worked answer
A square
A plane parallel to a face meets the four side faces in four straight lines.
Those lines have the same lengths and right angles as the face itself.
So the cross-section is a square identical to that face. Cutting at an angle through a cube can give other shapes, including a hexagon.
Problem 4
A sphere is cut by any flat plane that passes through it. What shape is the cross-section?
- A circle
- An ellipse
- A parabola
- A square
- A single point
Worked answer
A circle
A sphere looks the same from every direction, so the orientation of the plane cannot matter.
Every point of the cross-section is the same distance from the centre of that section, which makes it a circle.
Only a plane that just touches the sphere gives a single point instead.
Problem 5
A regular octahedron has 8 triangular faces. Each edge belongs to two faces. How many edges does it have?
- 6
- 8
- 12
- 16
- 24
Worked answer
12
The faces account for 8·3=24 edge appearances.
Each edge appears twice, so there are 24/2=12 edges.
Problem 6
A cone is cut by a vertical plane through its apex. What shape is the cross-section?
- A circle
- A triangle
- A rectangle
- An ellipse
- A square
Worked answer
A triangle
The plane passes through the single point at the top and cuts straight down to the base.
It meets the base along a straight line and the curved surface along two straight lines meeting at the apex.
Three straight sides give a triangle.
Problem 7
A cone is cut by a plane parallel to its base, above the base and below the apex. What shape is the cross-section?
- A circle
- A triangle
- A square
- A rectangle
- A hexagon
Worked answer
A circle
A plane parallel to the base meets the curved surface at a constant distance from the axis.
That gives a circle, smaller than the base and shrinking as the plane rises towards the apex.
At the apex itself the circle has shrunk to a single point.