Problem
Work back from volume or surface area to a missing edge
Two of a rectangular prism's edges are known along with its surface area. We need its volume.
GeometryExpressions
Your answer
How to solve
Strategy Convert to Algebra — The volume cannot be reached without the third edge, and the surface area is the only thing that mentions it. Write the surface area as a sentence about that unknown edge and the sentence solves.
1STEP 1
Write the surface area with the unknown in it
Call the missing edge ■. Two faces are 10 by 8, two are 10 by ■, and two are 8 by ■.
10 × 8 × 2 = 160
The two known faces already use 160 cm².
6.G.A.4Convert To Algebra2STEP 2
See what is left for the other four faces
Take the known faces off the total surface area.
340 - 160 = 180
The four side faces come to 180 cm².
6.G.A.4Identify SubproblemsThe four side faces are all that is left of the surface area.
Why?
A rectangular prism has six faces and no more, so taking the two known ones off the total leaves exactly the other four.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Those four all run the full height of the prism, so laid flat side by side they make one band whose width is the way round the prism.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Solve for the missing edge
Those four faces are 10, 10, 8 and 8 centimeters wide, all ■ tall, so together they are 36 times ■.
10 + 10 + 8 + 8 = 36, 180 ÷ 36 = 5
The missing edge is 5 cm.
6.EE.B.7Work Backwards4STEP 4
Multiply the three edges
Now all three are known, so the volume follows.
10 × 5 × 8 = 400
The volume is 400 cm³.
6.G.A.2Identify SubproblemsAnswer
400 cm³
Check the surface area with all three edges: 2 × (10 × 8 + 10 × 5 + 8 × 5) = 340 cm², as given.
Takeaway
When a quantity you need is missing, look for the given that mentions it and turn that into an equation.
- Write the surface area with the unknown in it
- See what is left for the other four faces
- Solve for the missing edge
- Multiply the three edges