← Where two solids meet, both faces disappear · Surface Area and Volume of Solids

Where two solids meet, both faces disappear · 12 practice problems

7.G.B.67.G.B.4

Generated variants — 12

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

Variant 1 easy answer: 301.44 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

3 cm 4 cm radius 5 cm radius 2 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 2 cm and height 4 cm stands on one of radius 5 cm and height 3 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 5 cm, height 3 cm.
  • The upper cylinder: radius 2 cm, height 4 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 5 cm and the top 2 cm.
78.5,12.5678.5,\quad 12.56
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
78.5−12.56=65.9478.5 - 12.56 = 65.94
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
94.2,50.2494.2,\quad 50.24
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
78.5+94.2+65.94+50.24+12.56=301.4478.5 + 94.2 + 65.94 + 50.24 + 12.56 = 301.44
301.44 cm2.
Answer: 301.44 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 326.56 cm2 -- too much by 25.12 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 5 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 2 easy answer: 489.84 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

4 cm 6 cm radius 6 cm radius 3 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 3 cm and height 6 cm stands on one of radius 6 cm and height 4 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 6 cm, height 4 cm.
  • The upper cylinder: radius 3 cm, height 6 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 6 cm and the top 3 cm.
113.04,28.26113.04,\quad 28.26
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
113.04−28.26=84.78113.04 - 28.26 = 84.78
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
150.72,113.04150.72,\quad 113.04
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
113.04+150.72+84.78+113.04+28.26=489.84113.04 + 150.72 + 84.78 + 113.04 + 28.26 = 489.84
489.84 cm2.
Answer: 489.84 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 546.36 cm2 -- too much by 56.52 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 6 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 3 easy answer: 558.92 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

4 cm 3 cm radius 7 cm radius 4 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 4 cm and height 3 cm stands on one of radius 7 cm and height 4 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 7 cm, height 4 cm.
  • The upper cylinder: radius 4 cm, height 3 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 7 cm and the top 4 cm.
153.86,50.24153.86,\quad 50.24
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
153.86−50.24=103.62153.86 - 50.24 = 103.62
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
175.84,75.36175.84,\quad 75.36
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
153.86+175.84+103.62+75.36+50.24=558.92153.86 + 175.84 + 103.62 + 75.36 + 50.24 = 558.92
558.92 cm2.
Answer: 558.92 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 659.4 cm2 -- too much by 100.48 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 7 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 4 easy answer: 778.72 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

5 cm 5 cm radius 8 cm radius 4 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 4 cm and height 5 cm stands on one of radius 8 cm and height 5 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 8 cm, height 5 cm.
  • The upper cylinder: radius 4 cm, height 5 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 8 cm and the top 4 cm.
200.96,50.24200.96,\quad 50.24
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
200.96−50.24=150.72200.96 - 50.24 = 150.72
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
251.2,125.6251.2,\quad 125.6
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
200.96+251.2+150.72+125.6+50.24=778.72200.96 + 251.2 + 150.72 + 125.6 + 50.24 = 778.72
778.72 cm2.
Answer: 778.72 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 879.2 cm2 -- too much by 100.48 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 8 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 5 medium answer: 923.16 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

5 cm 7 cm radius 9 cm radius 3 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 3 cm and height 7 cm stands on one of radius 9 cm and height 5 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 9 cm, height 5 cm.
  • The upper cylinder: radius 3 cm, height 7 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 9 cm and the top 3 cm.
254.34,28.26254.34,\quad 28.26
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
254.34−28.26=226.08254.34 - 28.26 = 226.08
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
282.6,131.88282.6,\quad 131.88
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
254.34+282.6+226.08+131.88+28.26=923.16254.34 + 282.6 + 226.08 + 131.88 + 28.26 = 923.16
923.16 cm2.
Answer: 923.16 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 979.68 cm2 -- too much by 56.52 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 9 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 6 medium answer: 1130.4 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

6 cm 4 cm radius 10 cm radius 5 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 5 cm and height 4 cm stands on one of radius 10 cm and height 6 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 10 cm, height 6 cm.
  • The upper cylinder: radius 5 cm, height 4 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 10 cm and the top 5 cm.
314,78.5314,\quad 78.5
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
314−78.5=235.5314 - 78.5 = 235.5
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
376.8,125.6376.8,\quad 125.6
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
314+376.8+235.5+125.6+78.5=1130.4314 + 376.8 + 235.5 + 125.6 + 78.5 = 1130.4
1130.4 cm2.
Answer: 1130.4 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 1287.4 cm2 -- too much by 157 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 10 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 7 medium answer: 1695.6 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

8 cm 5 cm radius 12 cm radius 6 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 6 cm and height 5 cm stands on one of radius 12 cm and height 8 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 12 cm, height 8 cm.
  • The upper cylinder: radius 6 cm, height 5 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 12 cm and the top 6 cm.
452.16,113.04452.16,\quad 113.04
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
452.16−113.04=339.12452.16 - 113.04 = 339.12
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
602.88,188.4602.88,\quad 188.4
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
452.16+602.88+339.12+188.4+113.04=1695.6452.16 + 602.88 + 339.12 + 188.4 + 113.04 = 1695.6
1695.6 cm2.
Answer: 1695.6 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 1921.68 cm2 -- too much by 226.08 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 12 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 8 medium answer: 2373.84 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

10 cm 6 cm radius 14 cm radius 7 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 7 cm and height 6 cm stands on one of radius 14 cm and height 10 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 14 cm, height 10 cm.
  • The upper cylinder: radius 7 cm, height 6 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 14 cm and the top 7 cm.
615.44,153.86615.44,\quad 153.86
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
615.44−153.86=461.58615.44 - 153.86 = 461.58
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
879.2,263.76879.2,\quad 263.76
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
615.44+879.2+461.58+263.76+153.86=2373.84615.44 + 879.2 + 461.58 + 263.76 + 153.86 = 2373.84
2373.84 cm2.
Answer: 2373.84 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 2681.56 cm2 -- too much by 307.72 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 14 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 9 hard answer: 2417.8 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

9 cm 5 cm radius 15 cm radius 5 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 5 cm and height 5 cm stands on one of radius 15 cm and height 9 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 15 cm, height 9 cm.
  • The upper cylinder: radius 5 cm, height 5 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 15 cm and the top 5 cm.
706.5,78.5706.5,\quad 78.5
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
706.5−78.5=628706.5 - 78.5 = 628
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
847.8,157847.8,\quad 157
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
706.5+847.8+628+157+78.5=2417.8706.5 + 847.8 + 628 + 157 + 78.5 = 2417.8
2417.8 cm2.
Answer: 2417.8 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 2574.8 cm2 -- too much by 157 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 15 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 10 hard answer: 3265.6 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

12 cm 9 cm radius 16 cm radius 8 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 8 cm and height 9 cm stands on one of radius 16 cm and height 12 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 16 cm, height 12 cm.
  • The upper cylinder: radius 8 cm, height 9 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 16 cm and the top 8 cm.
803.84,200.96803.84,\quad 200.96
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
803.84−200.96=602.88803.84 - 200.96 = 602.88
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
1205.76,452.161205.76,\quad 452.16
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
803.84+1205.76+602.88+452.16+200.96=3265.6803.84 + 1205.76 + 602.88 + 452.16 + 200.96 = 3265.6
3265.6 cm2.
Answer: 3265.6 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 3667.52 cm2 -- too much by 401.92 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 16 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 11 hard answer: 3391.2 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

7 cm 10 cm radius 18 cm radius 9 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 9 cm and height 10 cm stands on one of radius 18 cm and height 7 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 18 cm, height 7 cm.
  • The upper cylinder: radius 9 cm, height 10 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 18 cm and the top 9 cm.
1017.36,254.341017.36,\quad 254.34
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
1017.36−254.34=763.021017.36 - 254.34 = 763.02
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
791.28,565.2791.28,\quad 565.2
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
1017.36+791.28+763.02+565.2+254.34=3391.21017.36 + 791.28 + 763.02 + 565.2 + 254.34 = 3391.2
3391.2 cm2.
Answer: 3391.2 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 3899.88 cm2 -- too much by 508.68 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 18 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.
Variant 12 hard answer: 3768 cm²

What is the surface area of the solid at the right, in cm2\text{cm}^2?

6 cm 8 cm radius 20 cm radius 10 cm
Show solution
1 · Understandwhat's really being asked

A cylinder of radius 10 cm and height 8 cm stands on one of radius 20 cm and height 6 cm. We want the whole solid's outside surface.

Givens
  • The lower cylinder: radius 20 cm, height 6 cm.
  • The upper cylinder: radius 10 cm, height 8 cm.
  • The smaller one stands centred on the larger.
Unknowns
  • The surface area of the whole solid.
Constraints
  • Where the two meet, neither face is on the outside.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#16 Count the Complement

Adding the two cylinders' surface areas counts the join twice over. Walk the outside instead and name each surface as you meet it -- the top of the lower cylinder turns out to be a ring, not a circle.

3 · Execute5 carry out the plan

1Walk the outside

#17 Visualize Spatial Relationships 7.G.B.6
Bottom circle, the wide curved side, the ring left showing on top of it, the narrow curved side, and the top circle.
5 surfaces5 \text{ surfaces}
Five, not six.

2The two flat circles

#2 Make a Systematic List 7.G.B.4
The bottom has radius 20 cm and the top 10 cm.
1256,3141256,\quad 314
Two ordinary circles.

3The ring between them

#16 Count the Complement 7.G.B.4
The lower cylinder's top with the upper one's base taken out of it.
1256−314=9421256 - 314 = 942
A ring, not a circle.

4The two curved sides

#2 Make a Systematic List 7.G.B.6
Each unrolls into a rectangle as wide as its base's circumference.
753.6,502.4753.6,\quad 502.4
Two rectangles.

5Add the five

#2 Make a Systematic List 7.G.B.6
Nothing overlaps, so they simply add.
1256+753.6+942+502.4+314=37681256 + 753.6 + 942 + 502.4 + 314 = 3768
3768 cm2.
Answer: 3768 cm²
4 · Reviewdoes it hold up?

Adding the two cylinders' own surface areas would give 4396 cm2 -- too much by 628 cm2, which is exactly the two faces hidden at the join.

Another way: Counting the bottom and the ring together as one circle of radius 20 cm taken twice, less the join, is the same sum grouped differently.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the surfaces of a solid made of two cylinders.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle, ring and curved side.
💡Takeaway. Two solids stuck together lose two faces, not none. Walk the outside and count only what you can touch.