Concept 6 of 9
3D Solids: F · E · V
Today's big idea
Count faces, edges and vertices, then check with F + V − E = 2.
Cube: 6F, 12E, 8V. Cone: 2F, 1E, 1V. Sphere: 1F, 0E, 0V.
How it works
👀 What's going on?
A drawing of a box is flat, but the box itself is not. To describe a solid you need three counts, not one. Once you can find them, any solid becomes easy to talk about.
💡 Here's the trick
Look at the diagram above. A face is a surface you could paint, an edge is the line where two faces meet, and a vertex is a sharp corner. When every face of a solid is flat, the three counts obey one rule: F + V − E = 2.
🪄 How to do it
- 1.Find your solid in the diagram above, then count its flat surfaces.
- 2.Count the lines where two surfaces meet, then count the corners.
- 3.Try your three numbers in the rule F + V − E = 2.
✍️ Try it
A square pyramid has a square base and four triangular sides. How many faces and how many vertices does it have? Then: a cylinder has a flat circle at each end. How many edges does it have?
The pyramid has 1 + 4 = 5 faces and 4 + 1 = 5 vertices, plus 8 edges, and 5 + 5 − 8 = 2. The cylinder has one rim at each end, so it has 2 edges.
⚠️ Watch out
You might be thinking that this rule works for every solid you meet. It only holds when every face is flat, so a cube and a pyramid both obey it. A cylinder gives 3 + 0 − 2 = 1, because its curved surface breaks the rule.
🌟 Why this matters
Wrapping a gift, folding a carton or building with blocks all rest on knowing the faces and edges of a solid. A box maker counts edges to work out how much tape each carton needs. The counts also show how a flat net folds up into a shape.
Exam tip: Count the top and the bottom apart from the sides, then add, so nothing gets missed.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“How many EDGES does a cube have?”
- 1
Count the top & bottom edges
4 + 4 = 8. 🧊
- 2
Add the four vertical edges
8 + 4 = 12.
- 3
Verify with Euler
F + V − E = 6 + 8 − 12 = 2 ✓.
- 4
Answer ✓
A cube has 12 edges.
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap once to count the faces, again for the edges, and again for the corners.
SOLID · ROUND 1 OF 3
Cuboid
Quick check
✅ VERIFICATION
A solid has 4 flat faces and 4 vertices. Using F + V − E = 2, how many edges does it have?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Parts of a Circle
Diameter = 2 × radius. Longest chord = diameter.