Practice · Where two solids meet, both faces disappear · Sensim Math

Variant 1 easy

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

Your answer

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 \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,\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 = 65.94

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,\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 = 301.44

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

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

Your answer

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 \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,\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 = 84.78

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,\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 = 489.84

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

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

Your answer

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 \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,\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 = 103.62

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,\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 = 558.92

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

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

Your answer

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 \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,\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 = 150.72

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,\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 = 778.72

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

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

Your answer

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 \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,\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 = 226.08

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,\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 = 923.16

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

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

Your answer

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 \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,\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 = 235.5

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,\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 = 1130.4

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

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

Your answer

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 \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,\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 = 339.12

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,\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 = 1695.6

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

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

Your answer

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 \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,\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 = 461.58

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,\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 = 2373.84

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

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

Your answer

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 \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,\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 = 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,\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 = 2417.8

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

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

Your answer

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 \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,\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 = 602.88

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,\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 = 3265.6

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

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

Your answer

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 \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,\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 = 763.02

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,\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 = 3391.2

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

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

Your answer

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 \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,\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 = 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,\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 = 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.

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

Generated variants — 12

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