Problem
Surface area is the sum of every face
A solid has congruent L-shaped faces at top and bottom, with flat sides joining them. We need its surface area.
Geometry
Your answer
How to solve
Strategy Identify Subproblems — This is a prism whose base happens to be an L rather than a rectangle, so the same rule works: two bases plus a band wrapped round the outside. The band's area is the base's perimeter times the height, so the L's inside corner costs nothing extra.
1STEP 1
Split the L into two rectangles
Cut across the L: one rectangle runs the full 18 cm and is 5 cm wide; what is left below it is 6 cm wide and reaches the remaining 10 - 5 = 5 cm.
18 × 5 = 90, 6 × 5 = 30
One L-shaped face covers 120 cm².
6.G.A.1Draw A Diagram2STEP 2
Measure the way round the L
Walk the outline: 18 along the top, 5 down the far end, 12 back along the underside of the long arm, 5 down the step, 6 along the bottom, then 10 back up the near side.
18 + 5 + 12 + 5 + 6 + 10 = 56
The L's outline is 56 cm long.
6.G.A.4Visualize Spatial Relationships3STEP 3
Wrap a band round it
Every side face is a rectangle 20 cm tall, and laid flat side by side they make one band as long as the outline.
56 × 20 = 1120
The sides come to 1120 cm².
6.G.A.4Draw A DiagramThe side faces laid flat make one band as long as the L's outline.
Why?
Each side face stands on one straight piece of the L's outline and rises the same , so the faces and the outline's pieces pair off one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Laid side by side those faces leave no gap and overlap nowhere, so their widths add to the whole way round the L.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
Add the two L faces
The top and bottom are congruent, so count that area twice.
120 × 2 = 240, 240 + 1120 = 1360
The surface area is 1360 cm².
6.G.A.4Identify SubproblemsAnswer
1360 cm²
Count the six side faces one at a time: 18, 5, 12, 5, 6 and 10 centimeters wide, all 20 cm tall, giving 360 + 100 + 240 + 100 + 120 + 200 = 1120 cm².
Takeaway
Any prism's surface is two bases plus a band as long as the base's outline.
- Split the L into two rectangles
- Measure the way round the L
- Wrap a band round it
- Add the two L faces