← One number -- the base's sides -- fixes every count a prism has · Net and Solid Structure

One number -- the base's sides -- fixes every count a prism has · 12 practice problems

6.G.A.46.EE.B.7

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Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

A prism has $5$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 5 faces. We want its edges and its vertices added together.

Givens

The prism has 5 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 5 = n + 2.

n = 5 - 2 = 3

It is a triangular prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 3 edges on each base and 3 uprights joining them, and each base contributes 3 corners.

\text{edges} = 3n = 9,\quad \text{vertices} = 2n = 6

9 edges and 6 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

9 + 6 = 15

15 in all.

Answer: 15

4 · Reviewdoes it hold up?

Euler's rule holds: 5 faces minus 9 edges plus 6 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a triangular prism and counting works too, and agrees -- but the drawing gets unreliable well before 3 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 2 easy

A prism has $6$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 6 faces. We want its edges and its vertices added together.

Givens

The prism has 6 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 6 = n + 2.

n = 6 - 2 = 4

It is a square prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 4 edges on each base and 4 uprights joining them, and each base contributes 4 corners.

\text{edges} = 3n = 12,\quad \text{vertices} = 2n = 8

12 edges and 8 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

12 + 8 = 20

20 in all.

Answer: 20

4 · Reviewdoes it hold up?

Euler's rule holds: 6 faces minus 12 edges plus 8 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a square prism and counting works too, and agrees -- but the drawing gets unreliable well before 4 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 3 easy

A prism has $7$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 7 faces. We want its edges and its vertices added together.

Givens

The prism has 7 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 7 = n + 2.

n = 7 - 2 = 5

It is a pentagonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 5 edges on each base and 5 uprights joining them, and each base contributes 5 corners.

\text{edges} = 3n = 15,\quad \text{vertices} = 2n = 10

15 edges and 10 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

15 + 10 = 25

25 in all.

Answer: 25

4 · Reviewdoes it hold up?

Euler's rule holds: 7 faces minus 15 edges plus 10 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a pentagonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 5 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 4 easy

A prism has $8$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 8 faces. We want its edges and its vertices added together.

Givens

The prism has 8 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 8 = n + 2.

n = 8 - 2 = 6

It is a hexagonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 6 edges on each base and 6 uprights joining them, and each base contributes 6 corners.

\text{edges} = 3n = 18,\quad \text{vertices} = 2n = 12

18 edges and 12 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

18 + 12 = 30

30 in all.

Answer: 30

4 · Reviewdoes it hold up?

Euler's rule holds: 8 faces minus 18 edges plus 12 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a hexagonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 6 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 5 medium

A prism has $9$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 9 faces. We want its edges and its vertices added together.

Givens

The prism has 9 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 9 = n + 2.

n = 9 - 2 = 7

It is a heptagonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 7 edges on each base and 7 uprights joining them, and each base contributes 7 corners.

\text{edges} = 3n = 21,\quad \text{vertices} = 2n = 14

21 edges and 14 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

21 + 14 = 35

35 in all.

Answer: 35

4 · Reviewdoes it hold up?

Euler's rule holds: 9 faces minus 21 edges plus 14 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a heptagonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 7 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 6 medium

A prism has $10$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 10 faces. We want its edges and its vertices added together.

Givens

The prism has 10 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 10 = n + 2.

n = 10 - 2 = 8

It is a octagonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 8 edges on each base and 8 uprights joining them, and each base contributes 8 corners.

\text{edges} = 3n = 24,\quad \text{vertices} = 2n = 16

24 edges and 16 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

24 + 16 = 40

40 in all.

Answer: 40

4 · Reviewdoes it hold up?

Euler's rule holds: 10 faces minus 24 edges plus 16 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a octagonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 8 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 7 medium

A prism has $11$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 11 faces. We want its edges and its vertices added together.

Givens

The prism has 11 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 11 = n + 2.

n = 11 - 2 = 9

It is a nonagonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 9 edges on each base and 9 uprights joining them, and each base contributes 9 corners.

\text{edges} = 3n = 27,\quad \text{vertices} = 2n = 18

27 edges and 18 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

27 + 18 = 45

45 in all.

Answer: 45

4 · Reviewdoes it hold up?

Euler's rule holds: 11 faces minus 27 edges plus 18 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a nonagonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 9 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 8 medium

A prism has $12$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 12 faces. We want its edges and its vertices added together.

Givens

The prism has 12 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 12 = n + 2.

n = 12 - 2 = 10

It is a decagonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 10 edges on each base and 10 uprights joining them, and each base contributes 10 corners.

\text{edges} = 3n = 30,\quad \text{vertices} = 2n = 20

30 edges and 20 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

30 + 20 = 50

50 in all.

Answer: 50

4 · Reviewdoes it hold up?

Euler's rule holds: 12 faces minus 30 edges plus 20 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a decagonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 10 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 9 hard

A prism has $14$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 14 faces. We want its edges and its vertices added together.

Givens

The prism has 14 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 14 = n + 2.

n = 14 - 2 = 12

It is a 12-gonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 12 edges on each base and 12 uprights joining them, and each base contributes 12 corners.

\text{edges} = 3n = 36,\quad \text{vertices} = 2n = 24

36 edges and 24 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

36 + 24 = 60

60 in all.

Answer: 60

4 · Reviewdoes it hold up?

Euler's rule holds: 14 faces minus 36 edges plus 24 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a 12-gonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 12 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 10 hard

A prism has $16$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 16 faces. We want its edges and its vertices added together.

Givens

The prism has 16 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 16 = n + 2.

n = 16 - 2 = 14

It is a 14-gonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 14 edges on each base and 14 uprights joining them, and each base contributes 14 corners.

\text{edges} = 3n = 42,\quad \text{vertices} = 2n = 28

42 edges and 28 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

42 + 28 = 70

70 in all.

Answer: 70

4 · Reviewdoes it hold up?

Euler's rule holds: 16 faces minus 42 edges plus 28 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a 14-gonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 14 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 11 hard

A prism has $20$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 20 faces. We want its edges and its vertices added together.

Givens

The prism has 20 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 20 = n + 2.

n = 20 - 2 = 18

It is a 18-gonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 18 edges on each base and 18 uprights joining them, and each base contributes 18 corners.

\text{edges} = 3n = 54,\quad \text{vertices} = 2n = 36

54 edges and 36 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

54 + 36 = 90

90 in all.

Answer: 90

4 · Reviewdoes it hold up?

Euler's rule holds: 20 faces minus 54 edges plus 36 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a 18-gonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 18 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Variant 12 hard

A prism has $27$ faces. How many edges and vertices does it have in all?

Your answer

Show solution

1 · Understandwhat's really being asked

A prism has 27 faces. We want its edges and its vertices added together.

Givens

The prism has 27 faces.

Unknowns

The number of edges plus the number of vertices.

Constraints

It is a prism, so it has two congruent bases joined by rectangles.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Do not try to picture the solid first. Every count a prism has is a formula in the number of sides on its base, so get that number from the faces and the rest follows by substitution.

3 · Execute4 carry out the plan

1Write the faces in terms of the base

#13 Convert to Algebra 6.G.A.4

A prism has one side face per base edge, plus the two bases.

\text{faces} = n + 2

The base's sides decide everything.

2Find the base

#13 Convert to Algebra 6.EE.B.7

Solve 27 = n + 2.

n = 27 - 2 = 25

It is a 25-gonal prism.

3Count edges and vertices

#17 Visualize Spatial Relationships 6.G.A.4

There are 25 edges on each base and 25 uprights joining them, and each base contributes 25 corners.

\text{edges} = 3n = 75,\quad \text{vertices} = 2n = 50

75 edges and 50 vertices.

4Add the two

#5 Look for a Pattern 6.EE.B.7

The question asks for both together.

75 + 50 = 125

125 in all.

Answer: 125

4 · Reviewdoes it hold up?

Euler's rule holds: 27 faces minus 75 edges plus 50 vertices is 2, which is 2 for every solid of this kind.

Another way: Drawing a 25-gonal prism and counting works too, and agrees -- but the drawing gets unreliable well before 25 sides.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a prism's faces, edges and vertices to its base.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the number of base sides, then substituting it.

💡Takeaway. Find the one number a shape is built from, and every other count is a formula away.

Where next?

← Back to the problem: One number -- the base's sides -- fixes every count a prism has