Practice · Run the surface area backwards to get the edge the volume needs · Sensim Math

Variant 1 easy

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

Your answer

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 \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 = 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 \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 \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: $2 \times (27 + 63 + 21) = 222$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (63 + 28 + 36) = 254$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (48 + 72 + 24) = 288$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (50 + 80 + 40) = 340$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (96 + 40 + 60) = 392$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (88 + 66 + 48) = 404$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (56 + 126 + 36) = 436$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (78 + 104 + 48) = 460$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (90 + 150 + 60) = 600$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (140 + 240 + 84) = 928$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (180 + 270 + 150) = 1200$ , 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

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

Your answer

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 \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 = 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 \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 \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: $2 \times (300 + 400 + 192) = 1784$ , 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.

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

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