Solids: Volume, Sections & Rotations — Challenge
Every solid here is a flat shape that has been stacked, tapered or spun.
What these problems cover
These questions use 16 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.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Cross section — The 2-D face revealed when a plane cuts through a solid.
- Volume — Space inside, in cubic units.
- Surface area — Total area of all the outside faces, in square units.
- Prism — A solid with two identical parallel bases joined by rectangles.
- Base — The number being multiplied. In 53 the base is 5.
- Factor — A number that divides evenly into another.
- Radius — Centre to edge. Half the diameter.
- Scale — What one grid step is worth — and where the axis starts.
- Cylinder — Two parallel circles joined by a curved surface. V = π r2 h.
- Cone — A circle tapering to a point. V = 13π r2 h.
- Sphere — A ball. V = 43π r3.
- Parallel — Same slope, different intercept.
- Pyramid — A base tapering to a point. V = 13 Bh.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
How many faces does a cube have?
- 4
- 6
- 8
- 12
- 24
Worked answer
6
A cube is a box with all sides equal, so it has a top and bottom and four walls.
That is 6 faces, each a square.
Problem 2
How many edges does a cube have?
- 6
- 8
- 12
- 16
- 24
Worked answer
12
Each of the 6 square faces has 4 edges, giving 24 edge-sides in total.
Every edge is shared by exactly two faces, so each has been counted twice.
The number of edges is 242 = 12.
Problem 3
How many vertices does a cube have?
- 4
- 6
- 8
- 12
- 16
Worked answer
8
A cube has a square top and a square bottom.
Each square has 4 corners, so there are 4+4 = 8 vertices.
Problem 4
For a cube, what is the value of F+V-E, where F, V and E are the numbers of faces, vertices and edges?
- 0
- 1
- 2
- 3
- 26
Worked answer
2
A cube has 6 faces, 8 vertices and 12 edges.
6+8-12 = 2.
This is Euler's formula, and the answer is 2 for every convex polyhedron, not just the cube.
Problem 5
How many faces does a triangular prism have?
- 3
- 4
- 5
- 6
- 9
Worked answer
5
A triangular prism has two triangular ends.
Joining them are three rectangular faces, one for each side of the triangle.
That gives 2+3 = 5 faces.
Problem 6
What is the surface area of a cube with side 2?
- 8
- 12
- 16
- 24
- 48
Worked answer
24
Each face is a square of area 2· 2 = 4.
There are 6 faces, so the surface area is 6· 4 = 24 square units.
The volume would be 23 = 8, which is the other value worth keeping distinct.
Problem 7
How many flat faces does a cylinder have?
- 0
- 1
- 2
- 3
- 4
Worked answer
2
A cylinder has a circular face at each end.
The curved surface joining them is not flat, so it does not count.
That gives 2 flat faces.
Problem 8
How many faces does a square pyramid have?
- 4
- 5
- 6
- 8
- 9
Worked answer
5
The base is one square face.
Each of the four sides of the square carries a triangular face meeting at the apex.
That gives 1+4 = 5 faces.