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AMC 8 Geometry Math Olympiad USA Math Olympiad

Total surface area of a cube | AMC-8, 2009 | Problem 25

Try this beautiful problem from Geometry: Total surface area of a cube. You may use sequential hints to solve the problem

Try this beautiful problem from Geometry: Total surface area of a cube

Total surface area of a cube – AMC-8, 2009- Problem 25


A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is \(\frac{1}{2}\) foot from the top face. The second cut is  \(\frac{1}{3}\) foot below the first cut, and the third cut is \(\frac{1}{17}\) foot below the second cut. From the top to the bottom the pieces are labeled A,B,C and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?

 a cube
stair
  • $10$
  • $11$
  • $12$

Key Concepts


Geometry

Cube

surface area

Check the Answer


Answer:$11$

AMC-8 (2009) Problem 25

Pre College Mathematics

Try with Hints


Calculate the surface area side by side

Can you now finish the problem ……….

The total height be 1

can you finish the problem……..

total surface of a cube
stair

Clearly The tops of A,B,C, and D in the figure such that 1+1+1+1=4 as do the bottoms .

Thus the total surface area is 8.

 Now, one of the sides has area one, since it combines all of the heights of A,B,C and D which is 1

The other side also satisfies this. Thus the total area now is  10.

Now From the front, the surface area is half, because if you looked at it straight from the front it would look exactly like A, with surface area half. From the back it is the same thing. Thus, the total is \(10+\frac{1}{2}+\frac{1}{2}\)=11

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