← Run the surface area backwards to get the edge the volume needs · Surface Area and Volume of Solids

Run the surface area backwards to get the edge the volume needs · 12 practice problems

6.G.A.26.EE.B.76.G.A.4

Generated variants — 12

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

Variant 1 easy answer: 189 cm³

A rectangular prism is 9 cm9\ \text{cm} long and 7 cm7\ \text{cm} tall, and its surface area is 222 cm2222\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 9 cm long and 7 cm tall, with a surface area of 222 cm2. We want its volume.

Givens
  • The length is 9 cm and the height is 7 cm.
  • The surface area is 222 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 9 by 7 and do not involve the missing edge; the other four do.
2×9×7+2×(9+7)×□=2222 \times 9 \times 7 + 2 \times (9 + 7) \times \square = 222
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 9 by 7 faces come to 126 cm2.
222−126=96222 - 126 = 96
96 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 32 times the missing edge.
96÷32=396 \div 32 = 3
The missing edge is 3 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
9×3×7=1899 \times 3 \times 7 = 189
The volume is 189 cm3.
Answer: 189 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(27+63+21)=2222 \times (27 + 63 + 21) = 222, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 222 also reaches 3, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 2 easy answer: 252 cm³

A rectangular prism is 7 cm7\ \text{cm} long and 4 cm4\ \text{cm} tall, and its surface area is 254 cm2254\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 7 cm long and 4 cm tall, with a surface area of 254 cm2. We want its volume.

Givens
  • The length is 7 cm and the height is 4 cm.
  • The surface area is 254 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 7 by 4 and do not involve the missing edge; the other four do.
2×7×4+2×(7+4)×□=2542 \times 7 \times 4 + 2 \times (7 + 4) \times \square = 254
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 7 by 4 faces come to 56 cm2.
254−56=198254 - 56 = 198
198 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 22 times the missing edge.
198÷22=9198 \div 22 = 9
The missing edge is 9 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
7×9×4=2527 \times 9 \times 4 = 252
The volume is 252 cm3.
Answer: 252 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(63+28+36)=2542 \times (63 + 28 + 36) = 254, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 254 also reaches 9, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 3 easy answer: 288 cm³

A rectangular prism is 12 cm12\ \text{cm} long and 6 cm6\ \text{cm} tall, and its surface area is 288 cm2288\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 12 cm long and 6 cm tall, with a surface area of 288 cm2. We want its volume.

Givens
  • The length is 12 cm and the height is 6 cm.
  • The surface area is 288 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 12 by 6 and do not involve the missing edge; the other four do.
2×12×6+2×(12+6)×□=2882 \times 12 \times 6 + 2 \times (12 + 6) \times \square = 288
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 12 by 6 faces come to 144 cm2.
288−144=144288 - 144 = 144
144 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 36 times the missing edge.
144÷36=4144 \div 36 = 4
The missing edge is 4 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
12×4×6=28812 \times 4 \times 6 = 288
The volume is 288 cm3.
Answer: 288 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(48+72+24)=2882 \times (48 + 72 + 24) = 288, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 288 also reaches 4, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 4 easy answer: 400 cm³

A rectangular prism is 10 cm10\ \text{cm} long and 8 cm8\ \text{cm} tall, and its surface area is 340 cm2340\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 10 cm long and 8 cm tall, with a surface area of 340 cm2. We want its volume.

Givens
  • The length is 10 cm and the height is 8 cm.
  • The surface area is 340 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 10 by 8 and do not involve the missing edge; the other four do.
2×10×8+2×(10+8)×□=3402 \times 10 \times 8 + 2 \times (10 + 8) \times \square = 340
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 10 by 8 faces come to 160 cm2.
340−160=180340 - 160 = 180
180 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 36 times the missing edge.
180÷36=5180 \div 36 = 5
The missing edge is 5 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
10×5×8=40010 \times 5 \times 8 = 400
The volume is 400 cm3.
Answer: 400 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(50+80+40)=3402 \times (50 + 80 + 40) = 340, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 340 also reaches 5, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 5 medium answer: 480 cm³

A rectangular prism is 8 cm8\ \text{cm} long and 5 cm5\ \text{cm} tall, and its surface area is 392 cm2392\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 8 cm long and 5 cm tall, with a surface area of 392 cm2. We want its volume.

Givens
  • The length is 8 cm and the height is 5 cm.
  • The surface area is 392 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 8 by 5 and do not involve the missing edge; the other four do.
2×8×5+2×(8+5)×□=3922 \times 8 \times 5 + 2 \times (8 + 5) \times \square = 392
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 8 by 5 faces come to 80 cm2.
392−80=312392 - 80 = 312
312 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 26 times the missing edge.
312÷26=12312 \div 26 = 12
The missing edge is 12 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
8×12×5=4808 \times 12 \times 5 = 480
The volume is 480 cm3.
Answer: 480 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(96+40+60)=3922 \times (96 + 40 + 60) = 392, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 392 also reaches 12, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 6 medium answer: 528 cm³

A rectangular prism is 11 cm11\ \text{cm} long and 6 cm6\ \text{cm} tall, and its surface area is 404 cm2404\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 11 cm long and 6 cm tall, with a surface area of 404 cm2. We want its volume.

Givens
  • The length is 11 cm and the height is 6 cm.
  • The surface area is 404 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 11 by 6 and do not involve the missing edge; the other four do.
2×11×6+2×(11+6)×□=4042 \times 11 \times 6 + 2 \times (11 + 6) \times \square = 404
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 11 by 6 faces come to 132 cm2.
404−132=272404 - 132 = 272
272 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 34 times the missing edge.
272÷34=8272 \div 34 = 8
The missing edge is 8 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
11×8×6=52811 \times 8 \times 6 = 528
The volume is 528 cm3.
Answer: 528 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(88+66+48)=4042 \times (88 + 66 + 48) = 404, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 404 also reaches 8, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 7 medium answer: 504 cm³

A rectangular prism is 14 cm14\ \text{cm} long and 9 cm9\ \text{cm} tall, and its surface area is 436 cm2436\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 14 cm long and 9 cm tall, with a surface area of 436 cm2. We want its volume.

Givens
  • The length is 14 cm and the height is 9 cm.
  • The surface area is 436 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 14 by 9 and do not involve the missing edge; the other four do.
2×14×9+2×(14+9)×□=4362 \times 14 \times 9 + 2 \times (14 + 9) \times \square = 436
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 14 by 9 faces come to 252 cm2.
436−252=184436 - 252 = 184
184 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 46 times the missing edge.
184÷46=4184 \div 46 = 4
The missing edge is 4 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
14×4×9=50414 \times 4 \times 9 = 504
The volume is 504 cm3.
Answer: 504 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(56+126+36)=4362 \times (56 + 126 + 36) = 436, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 436 also reaches 4, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 8 medium answer: 624 cm³

A rectangular prism is 13 cm13\ \text{cm} long and 8 cm8\ \text{cm} tall, and its surface area is 460 cm2460\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 13 cm long and 8 cm tall, with a surface area of 460 cm2. We want its volume.

Givens
  • The length is 13 cm and the height is 8 cm.
  • The surface area is 460 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 13 by 8 and do not involve the missing edge; the other four do.
2×13×8+2×(13+8)×□=4602 \times 13 \times 8 + 2 \times (13 + 8) \times \square = 460
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 13 by 8 faces come to 208 cm2.
460−208=252460 - 208 = 252
252 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 42 times the missing edge.
252÷42=6252 \div 42 = 6
The missing edge is 6 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
13×6×8=62413 \times 6 \times 8 = 624
The volume is 624 cm3.
Answer: 624 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(78+104+48)=4602 \times (78 + 104 + 48) = 460, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 460 also reaches 6, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 9 hard answer: 900 cm³

A rectangular prism is 15 cm15\ \text{cm} long and 10 cm10\ \text{cm} tall, and its surface area is 600 cm2600\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 15 cm long and 10 cm tall, with a surface area of 600 cm2. We want its volume.

Givens
  • The length is 15 cm and the height is 10 cm.
  • The surface area is 600 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 15 by 10 and do not involve the missing edge; the other four do.
2×15×10+2×(15+10)×□=6002 \times 15 \times 10 + 2 \times (15 + 10) \times \square = 600
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 15 by 10 faces come to 300 cm2.
600−300=300600 - 300 = 300
300 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 50 times the missing edge.
300÷50=6300 \div 50 = 6
The missing edge is 6 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
15×6×10=90015 \times 6 \times 10 = 900
The volume is 900 cm3.
Answer: 900 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(90+150+60)=6002 \times (90 + 150 + 60) = 600, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 600 also reaches 6, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 10 hard answer: 1680 cm³

A rectangular prism is 20 cm20\ \text{cm} long and 12 cm12\ \text{cm} tall, and its surface area is 928 cm2928\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 20 cm long and 12 cm tall, with a surface area of 928 cm2. We want its volume.

Givens
  • The length is 20 cm and the height is 12 cm.
  • The surface area is 928 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 20 by 12 and do not involve the missing edge; the other four do.
2×20×12+2×(20+12)×□=9282 \times 20 \times 12 + 2 \times (20 + 12) \times \square = 928
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 20 by 12 faces come to 480 cm2.
928−480=448928 - 480 = 448
448 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 64 times the missing edge.
448÷64=7448 \div 64 = 7
The missing edge is 7 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
20×7×12=168020 \times 7 \times 12 = 1680
The volume is 1680 cm3.
Answer: 1680 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(140+240+84)=9282 \times (140 + 240 + 84) = 928, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 928 also reaches 7, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 11 hard answer: 2700 cm³

A rectangular prism is 18 cm18\ \text{cm} long and 15 cm15\ \text{cm} tall, and its surface area is 1200 cm21200\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 18 cm long and 15 cm tall, with a surface area of 1200 cm2. We want its volume.

Givens
  • The length is 18 cm and the height is 15 cm.
  • The surface area is 1200 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 18 by 15 and do not involve the missing edge; the other four do.
2×18×15+2×(18+15)×□=12002 \times 18 \times 15 + 2 \times (18 + 15) \times \square = 1200
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 18 by 15 faces come to 540 cm2.
1200−540=6601200 - 540 = 660
660 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 66 times the missing edge.
660÷66=10660 \div 66 = 10
The missing edge is 10 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
18×10×15=270018 \times 10 \times 15 = 2700
The volume is 2700 cm3.
Answer: 2700 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(180+270+150)=12002 \times (180 + 270 + 150) = 1200, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 1200 also reaches 10, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.
Variant 12 hard answer: 4800 cm³

A rectangular prism is 25 cm25\ \text{cm} long and 16 cm16\ \text{cm} tall, and its surface area is 1784 cm21784\ \text{cm}^2. What is its volume, in cm3\text{cm}^3?

Show solution
1 · Understandwhat's really being asked

A prism is 25 cm long and 16 cm tall, with a surface area of 1784 cm2. We want its volume.

Givens
  • The length is 25 cm and the height is 16 cm.
  • The surface area is 1784 cm2.
Unknowns
  • The volume of the prism.
Constraints
  • Volume needs all three edges, and only two are given.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #11 Work Backwards#7 Identify Subproblems

The volume is blocked by one missing edge, so go and get it. The surface area is an equation in that edge -- gather the terms it appears in and it comes out in one division.

3 · Execute4 carry out the plan

1Write the surface area with the unknown in it

#13 Convert to Algebra 6.G.A.4
Two faces are 25 by 16 and do not involve the missing edge; the other four do.
2×25×16+2×(25+16)×□=17842 \times 25 \times 16 + 2 \times (25 + 16) \times \square = 1784
One unknown, appearing in four faces.

2Take off the part that is already known

#11 Work Backwards 6.EE.B.7
The two 25 by 16 faces come to 800 cm2.
1784−800=9841784 - 800 = 984
984 cm2 belongs to the other four faces.

3Divide by what multiplies the unknown

#13 Convert to Algebra 6.EE.B.7
Those four faces come to 82 times the missing edge.
984÷82=12984 \div 82 = 12
The missing edge is 12 cm.

4Multiply the three edges

#7 Identify Subproblems 6.G.A.2
Now the volume is available.
25×12×16=480025 \times 12 \times 16 = 4800
The volume is 4800 cm3.
Answer: 4800 cm³
4 · Reviewdoes it hold up?

Checking the surface area with all three edges: 2×(300+400+192)=17842 \times (300 + 400 + 192) = 1784, which is the figure we were given.

Another way: Trying whole numbers for the missing edge until the surface area comes to 1784 also reaches 12, after as many attempts as the answer is large.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Writing the surface area as a sum of six faces.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving that equation for the missing edge.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Multiplying the three edges for the volume.
💡Takeaway. When a formula is missing one number, the other formula you were given is usually where to find it.