Problem
Build the prism from its top, front and side views
Three rectangles show what a rectangular prism looks like from above, from the front and from the side. We need its surface area.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — Each view is a face of the prism, so each shows two of the three edges. Every edge turns up in two different views, and that overlap is what pins all three lengths down at once.
1STEP 1
Match the views up
The 5 appears in the top and front views, the 6 in the top and side views, and the 7 in the front and side views. Each number is one edge seen from two directions.
The prism is 5 by 6 by 7 centimeters.
6.G.A.2Visualize Spatial RelationshipsThe three views give the prism's three edge lengths.
Why?
Looking straight at a face of a box shows that face at its true size, so each view is one face and shows two of the three edges.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Each edge belongs to two of the three faces, so each length turns up in exactly two of the views and the three views name three lengths in all.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Find the three different face areas
A rectangular prism has three pairs of identical faces, one pair for each way of choosing two edges.
5 × 6 = 30, 5 × 7 = 35, 6 × 7 = 42
Three different faces, each appearing twice.
6.G.A.4Draw A Diagram3STEP 3
Double the total
Each of those three faces appears on both sides of the prism.
30 + 35 + 42 = 107, 107 × 2 = 214
The surface area is 214 cm².
6.G.A.4Identify SubproblemsAnswer
214 cm²
Check the views again: a 5 × 6 × 7 prism does look 5 by 6 from above, 5 by 7 from the front and 6 by 7 from the side.
Takeaway
Three views of a box overlap: every edge appears in two of them.
- Match the views up
- Find the three different face areas
- Double the total