Faces, edges and vertices

Nine solids with their three counts, every hidden edge drawn in — and one piece of arithmetic underneath that tells a child whether they counted right.

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Faces, edges and vertices

Solid	Faces	Edges	Vertices	F + V − E
Cube	6	12	8	6 + 8 − 12 = 2
Cuboid	6	12	8	6 + 8 − 12 = 2
Triangular prism	5	9	6	5 + 6 − 9 = 2
Hexagonal prism	8	18	12	8 + 12 − 18 = 2
Square pyramid	5	8	5	5 + 5 − 8 = 2
Triangular pyramid	4	6	4	4 + 4 − 6 = 2
Cylinder	2 flat	2 curved	0	—
Cone	1 flat	1 curved	1 point	—
Sphere	0 flat	0	0	—

Count the ones you cannot see. Every dashed line is a hidden edge. For any solid with FLAT faces F + V − E = 2, which checks that you counted them all. The last three have a curved surface, which is not a face and is not counted as one here — so the check has nothing to say about them.

What's on this chart

Counting is where this goes wrong, and it goes wrong in a way that a table of numbers cannot fix: a child counts the faces they can see. A drawn cube shows three of them and the answer is six, so a page that hides the back half teaches the mistake it exists to prevent. Every solid here carries its hidden edges as dashed lines, which makes the drawing the method — count what is dashed as well, and the count comes out right.

Why this chart helps

Euler's formula is the reason this chart is worth having on a wall rather than looking up once. Faces plus vertices, take away edges, is 2 for every flat-faced solid on this page and for every other one a child will meet. That makes it the only self-check available at this age: count three numbers and the arithmetic says whether you counted them all. A cube gives 6 + 8 − 12 = 2, a tetrahedron gives 4 + 4 − 6 = 2, and a hexagonal prism — which nobody counts right first time — gives 8 + 12 − 18 = 2.

Faces, edges and vertices

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