Depth · one concept from first principles · step 2 of 4
Surface area is the sum of every face
What is the surface area of the solid at the right, in ?
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — Adding two whole cylinders would count the joint twice -- once from each side -- and both of those faces are hidden. Walk over the outside of the solid instead, naming each surface as you meet it.
1STEP 1
Walk over the outside
From the bottom up: lower base, lower curved side, the ring of the lower top the upper cylinder leaves bare, upper curved side, upper top.
Five surfaces, and the joint is on none of them.
7.G.B.6Visualize Spatial RelationshipsWhere the two cylinders meet, neither face is on the outside.
Why?
Surface area counts only what the outside touches, and the joint is sealed between the two solids.
🧱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?
The narrow cylinder's base and the patch of the wide one's top under it are the same region, so each hides the other exactly.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Measure the two flat circles
The bottom of the whole solid and the top of the narrow cylinder.
8 × 8 × 3.14 = 200.96, 4 × 4 × 3.14 = 50.24
200.96 and 50.24 cm².
7.G.B.4Draw A Diagram3STEP 3
Measure the ring on top of the wide cylinder
It is the wide circle with the narrow one taken out of the middle.
200.96 - 50.24 = 150.72
The ring is 150.72 cm².
7.G.B.4Draw A Diagram4STEP 4
Measure the two curved sides
Each unrolls into a rectangle as wide as its base's circumference and 5 cm tall.
8 × 2 × 3.14 × 5 = 251.2, 4 × 2 × 3.14 × 5 = 125.6
251.2 and 125.6 cm².
7.G.B.4Draw A Diagram5STEP 5
Add the five surfaces
Two flat circles, the ring, and the two curved sides.
200.96 + 50.24 + 150.72 + 251.2 + 125.6 = 778.72
The surface area is 778.72 cm².
7.G.B.6Identify SubproblemsAnswer
778.72 cm²
Check from above: looking straight down, the ring and the narrow top together cover exactly the wide circle, 150.72 + 50.24 = 200.96 cm² -- the same as the bottom, as it should be.
Takeaway
Where two solids meet, both faces disappear -- walk the outside instead of adding the pieces.
- Walk over the outside
- Measure the two flat circles
- Measure the ring on top of the wide cylinder
- Measure the two curved sides
- Add the five surfaces
Where next?
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested
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