← Write both counts in the same unknown and the condition solves itself · Net and Solid Structure

Write both counts in the same unknown and the condition solves itself · 12 practice problems

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

Generated variants — 12

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

Variant 1 easy answer: 4

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=10(\text{number of faces}) + (\text{number of edges}) = 10

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 10. We want how many vertices it has.

Givens
  • faces + edges = 10.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 10, which gives one equation in n.
(n+1)+2n=10⇒3n=9(n + 1) + 2n = 10 \Rightarrow 3n = 9
So n is 3.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 3 corners plus the apex.
vertices=n+1=4\text{vertices} = n + 1 = 4
A triangular pyramid has 4 vertices.
Answer: 4
4 · Reviewdoes it hold up?

Check the condition with n = 3: 4 faces and 6 edges add to 10, which is the 10 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same triangular one, after 1 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 2 easy answer: 5

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=13(\text{number of faces}) + (\text{number of edges}) = 13

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 13. We want how many vertices it has.

Givens
  • faces + edges = 13.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 13, which gives one equation in n.
(n+1)+2n=13⇒3n=12(n + 1) + 2n = 13 \Rightarrow 3n = 12
So n is 4.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 4 corners plus the apex.
vertices=n+1=5\text{vertices} = n + 1 = 5
A square pyramid has 5 vertices.
Answer: 5
4 · Reviewdoes it hold up?

Check the condition with n = 4: 5 faces and 8 edges add to 13, which is the 13 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same square one, after 2 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 3 easy answer: 6

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=16(\text{number of faces}) + (\text{number of edges}) = 16

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 16. We want how many vertices it has.

Givens
  • faces + edges = 16.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 16, which gives one equation in n.
(n+1)+2n=16⇒3n=15(n + 1) + 2n = 16 \Rightarrow 3n = 15
So n is 5.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 5 corners plus the apex.
vertices=n+1=6\text{vertices} = n + 1 = 6
A pentagonal pyramid has 6 vertices.
Answer: 6
4 · Reviewdoes it hold up?

Check the condition with n = 5: 6 faces and 10 edges add to 16, which is the 16 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same pentagonal one, after 3 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 4 easy answer: 7

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=19(\text{number of faces}) + (\text{number of edges}) = 19

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 19. We want how many vertices it has.

Givens
  • faces + edges = 19.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 19, which gives one equation in n.
(n+1)+2n=19⇒3n=18(n + 1) + 2n = 19 \Rightarrow 3n = 18
So n is 6.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 6 corners plus the apex.
vertices=n+1=7\text{vertices} = n + 1 = 7
A hexagonal pyramid has 7 vertices.
Answer: 7
4 · Reviewdoes it hold up?

Check the condition with n = 6: 7 faces and 12 edges add to 19, which is the 19 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same hexagonal one, after 4 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 5 medium answer: 8

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=22(\text{number of faces}) + (\text{number of edges}) = 22

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 22. We want how many vertices it has.

Givens
  • faces + edges = 22.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 22, which gives one equation in n.
(n+1)+2n=22⇒3n=21(n + 1) + 2n = 22 \Rightarrow 3n = 21
So n is 7.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 7 corners plus the apex.
vertices=n+1=8\text{vertices} = n + 1 = 8
A heptagonal pyramid has 8 vertices.
Answer: 8
4 · Reviewdoes it hold up?

Check the condition with n = 7: 8 faces and 14 edges add to 22, which is the 22 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same heptagonal one, after 5 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 6 medium answer: 9

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=25(\text{number of faces}) + (\text{number of edges}) = 25

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 25. We want how many vertices it has.

Givens
  • faces + edges = 25.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 25, which gives one equation in n.
(n+1)+2n=25⇒3n=24(n + 1) + 2n = 25 \Rightarrow 3n = 24
So n is 8.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 8 corners plus the apex.
vertices=n+1=9\text{vertices} = n + 1 = 9
A octagonal pyramid has 9 vertices.
Answer: 9
4 · Reviewdoes it hold up?

Check the condition with n = 8: 9 faces and 16 edges add to 25, which is the 25 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same octagonal one, after 6 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 7 medium answer: 10

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=28(\text{number of faces}) + (\text{number of edges}) = 28

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 28. We want how many vertices it has.

Givens
  • faces + edges = 28.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 28, which gives one equation in n.
(n+1)+2n=28⇒3n=27(n + 1) + 2n = 28 \Rightarrow 3n = 27
So n is 9.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 9 corners plus the apex.
vertices=n+1=10\text{vertices} = n + 1 = 10
A nonagonal pyramid has 10 vertices.
Answer: 10
4 · Reviewdoes it hold up?

Check the condition with n = 9: 10 faces and 18 edges add to 28, which is the 28 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same nonagonal one, after 7 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 8 medium answer: 11

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=31(\text{number of faces}) + (\text{number of edges}) = 31

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 31. We want how many vertices it has.

Givens
  • faces + edges = 31.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 31, which gives one equation in n.
(n+1)+2n=31⇒3n=30(n + 1) + 2n = 31 \Rightarrow 3n = 30
So n is 10.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 10 corners plus the apex.
vertices=n+1=11\text{vertices} = n + 1 = 11
A decagonal pyramid has 11 vertices.
Answer: 11
4 · Reviewdoes it hold up?

Check the condition with n = 10: 11 faces and 20 edges add to 31, which is the 31 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same decagonal one, after 8 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 9 hard answer: 12

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=34(\text{number of faces}) + (\text{number of edges}) = 34

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 34. We want how many vertices it has.

Givens
  • faces + edges = 34.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 34, which gives one equation in n.
(n+1)+2n=34⇒3n=33(n + 1) + 2n = 34 \Rightarrow 3n = 33
So n is 11.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 11 corners plus the apex.
vertices=n+1=12\text{vertices} = n + 1 = 12
A 11-gonal pyramid has 12 vertices.
Answer: 12
4 · Reviewdoes it hold up?

Check the condition with n = 11: 12 faces and 22 edges add to 34, which is the 34 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same 11-gonal one, after 9 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 10 hard answer: 13

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=37(\text{number of faces}) + (\text{number of edges}) = 37

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 37. We want how many vertices it has.

Givens
  • faces + edges = 37.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 37, which gives one equation in n.
(n+1)+2n=37⇒3n=36(n + 1) + 2n = 37 \Rightarrow 3n = 36
So n is 12.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 12 corners plus the apex.
vertices=n+1=13\text{vertices} = n + 1 = 13
A 12-gonal pyramid has 13 vertices.
Answer: 13
4 · Reviewdoes it hold up?

Check the condition with n = 12: 13 faces and 24 edges add to 37, which is the 37 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same 12-gonal one, after 10 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 11 hard answer: 16

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=46(\text{number of faces}) + (\text{number of edges}) = 46

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 46. We want how many vertices it has.

Givens
  • faces + edges = 46.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 46, which gives one equation in n.
(n+1)+2n=46⇒3n=45(n + 1) + 2n = 46 \Rightarrow 3n = 45
So n is 15.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 15 corners plus the apex.
vertices=n+1=16\text{vertices} = n + 1 = 16
A 15-gonal pyramid has 16 vertices.
Answer: 16
4 · Reviewdoes it hold up?

Check the condition with n = 15: 16 faces and 30 edges add to 46, which is the 46 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same 15-gonal one, after 13 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.
Variant 12 hard answer: 21

A pyramid satisfies the condition below. How many vertices does it have?

(number of faces)+(number of edges)=61(\text{number of faces}) + (\text{number of edges}) = 61

Show solution
1 · Understandwhat's really being asked

For some pyramid, the faces and the edges add to 61. We want how many vertices it has.

Givens
  • faces + edges = 61.
  • The solid is a pyramid.
Unknowns
  • The number of vertices.
Constraints
  • A pyramid has one base and one apex, and every base edge runs up to that apex.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #17 Visualize Spatial Relationships#7 Identify Subproblems

The number given is not either count on its own, so neither can be read off. Write both in terms of the base's sides and the condition becomes one equation with one unknown.

3 · Execute4 carry out the plan

1Count the faces in terms of the base

#17 Visualize Spatial Relationships 6.G.A.4
One triangle per base edge, plus the base itself.
faces=n+1\text{faces} = n + 1
Side faces meet at the apex.

2Count the edges the same way

#17 Visualize Spatial Relationships 6.G.A.4
The base has n edges, and n more run up to the apex.
edges=2n\text{edges} = 2n
Twice the base's sides.

3Put them into the condition

#13 Convert to Algebra 6.EE.B.7
Their sum is 61, which gives one equation in n.
(n+1)+2n=61⇒3n=60(n + 1) + 2n = 61 \Rightarrow 3n = 60
So n is 20.

4Count the vertices

#7 Identify Subproblems 6.G.A.4
The base's 20 corners plus the apex.
vertices=n+1=21\text{vertices} = n + 1 = 21
A 20-gonal pyramid has 21 vertices.
Answer: 21
4 · Reviewdoes it hold up?

Check the condition with n = 20: 21 faces and 40 edges add to 61, which is the 61 we were given.

Another way: Trying pyramids one at a time -- triangular, square, pentagonal -- lands on the same 20-gonal one, after 18 wrong guesses.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a pyramid's faces, edges and vertices to its base.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving the stated condition for the number of base sides.
💡Takeaway. When you are given a total of two things you do not know, write both in the same letter first.