Measures and solids · Ages 12–13

Euler's formula and regular polyhedra worksheet

Part of: 3-D figures

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Measures and solids · Ages 12–13

Numsprite
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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.A polyhedron has 14 faces and 9 vertices. How many edges does it have?
Answer:
2.A polyhedron has 12 vertices and 18 edges. How many faces does it have?
Answer:
3.A polyhedron has 6 faces and 8 vertices. How many edges does it have?
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4.How many edges does the regular polyhedron in the picture have?
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5.A polyhedron has 14 faces and 21 edges. By Euler's formula, its vertices V = 2 − + = .
Fill in each blank
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6.A polyhedron's faces are all triangles, 20 of them. Edges = 20 × 3 ÷ 2 = , vertices = 2 − 20 + = .
Fill in each blank
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7.A corner is cut off a prism on a 3-sided base, the cut a triangle. How many edges does the solid left have?
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8.A corner is cut off a prism on a 10-sided base, the cut a triangle. How many edges does the solid left have?
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9.What is the regular polyhedron in the picture called?
- A. A regular tetrahedron
- B. A regular dodecahedron
- C. A regular icosahedron
- D. A regular octahedron
Choose the right answer
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10.A polyhedron has 5 faces, 6 vertices and 7 edges, and these numbers fit Euler's formula.
True or false?
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11.How many kinds of regular polyhedron are there?
- A. 4
- B. 8
- C. 5
- D. 6
Choose the right answer
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12.3 equal regular polygons with 5 sides each meet at every vertex. This makes a regular polyhedron.
True or false?
Answer:

13.A gem has 10 triangular faces, two pyramids stuck base to base. How many vertices does it have?
Working:
Answer:
14.A polyhedron has 17 faces and 32 edges. Find its vertices by Euler's formula: put the steps in order.
- A. V = 2 − 17 + 32
- B. 17 + V − 32 = 2
- C. V = 17
- D. F + V − E = 2
Write the letters in the right order
Working:
Answer:
15.A toy house is a prism on a 8-sided base with a pyramid on top as its roof. How many faces does the house have?
Working:
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16.A polyhedron model is made of 20 triangular plastic pieces joined edge to edge. How many vertices does it have?
Working:
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17.A football is sewn from 12 pentagons and 20 hexagons of leather, 5 cm of stitching along each edge. How many cm of stitching in all?
Working:
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18.Every vertex of a prism on a 3-sided base is cut off (each cut a triangle, none touching). The new solid has faces, vertices and edges.
Fill in each blank
Working:
Answer:

Show the answers and working

Euler's formula and regular polyhedra worksheet: Answers and working

1. 21 edges
F + V − E = 2, so E = F + V − 2 = 14 + 9 − 2 = 21.
2. 8 faces
F + V − E = 2, so F = 2 − V + E = 2 − 12 + 18 = 8.
3. 12 edges
F + V − E = 2, so E = F + V − 2 = 6 + 8 − 2 = 12.
4. 30 edges
It has 20 faces of 3 sides each, and each edge belongs to two faces: 20 × 3 ÷ 2 = 30. Or count them, dashed ones too.
5. 14, 21, 9
F + V − E = 2 rearranges to V = 2 − F + E = 2 − 14 + 21 = 9.
6. 30, 30, 12
20 triangles have 20 × 3 sides, each edge counted twice, so there are 30 edges; from F + V − E = 2, V = 2 − 20 + 30 = 12.
7. 12 edges
Cutting the corner adds one face, the cut: 6; the one vertex becomes the cut's 3: 8; so E = F + V − 2 = 12.
8. 33 edges
Cutting the corner adds one face, the cut: 13; the one vertex becomes the cut's 3: 22; so E = F + V − 2 = 33.

9. B. A regular dodecahedron
Look at the faces and how many meet at a vertex: triangles 3 at a time make a tetrahedron, 4 an octahedron, 5 an icosahedron; pentagons make a dodecahedron.
10. False
Put them in F + V − E: 5 + 6 − 7 must equal 2. The edges that fit are 5 + 6 − 2 = 9.
11. C. 5
At least 3 faces meet at a vertex, their angles adding to under 360°: 3, 4 or 5 triangles, 3 squares, 3 pentagons. 5 kinds.
12. True
The angles at each vertex must add to under 360°: a triangle's 60°, so 3, 4 or 5; a square's 90° and a pentagon's 108°, only 3; a hexagon's 120°, and 3 already make 360°.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 7 vertices
Each pyramid has 10 ÷ 2 = 5 triangles on a 5-sided base: its 5 base corners and two tips make 7. Or by Euler: edges 10 × 3 ÷ 2 = 15, V = 2 − 10 + 15 = 7.
14. D → B → A → C
Write the formula, put in F = 17 and E = 32, rearrange, then work it out.
15. 17 faces
8 walls, 1 floor and 8 roof triangles; where the walls meet the roof is no face: 8 + 1 + 8 = 17.
16. 12 vertices
The triangles have 20 × 3 sides, two to an edge: 30 edges; V = 2 − F + E = 2 − 20 + 30 = 12.
17. 450 cm
The sides total 12 × 5 + 20 × 6 = 180, two to an edge: 90 edges, 90 × 5 = 450 cm.
18. 11, 18, 27
The 3 + 2 faces plus 6 cuts: 11; each vertex becomes a cut's 3: 3 × 6 = 18; E = F + V − 2 = 27.

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