- A solid shape is made by joining three cubes together with the largest cube on the bottom and the smallest on the top. Where the faces of two cubes join, the corners of the smaller cube are at the midpoints of the sides of the larger cube.
The sides of the smallest cube have a length of 1 cm. What is the total surface area of the shape?
Answer:
20) D
It is advisable to skip such long and tricky questions and finish the other simpler ones first.
We need to use a lot of Pythagoras in this question.
Side of smaller cube = 1cm
Side of 2nd cube = A
Side of 3rd cube = B
5 faces of the smallest cube can be seen.
Surface area of the smallest cube = 1 x 1 x 5 = 5 cm²
The vertices of the smallest cube are at the midpoints of A.
You can see that multiple right-angled triangles are formed.
1² = (½A)² + (½A)²
1 = ¼A² + ¼a²
1 = ½a²
2 = A²
A = √2
4 faces of the 2nd cube can be seen and the top face is partially covered by the smallest cube.
Surface area = 5 x √2 x √2 = 10cm²
Area covered by one of the faces of the smaller cube = 1 x 1 = 1cm²
So total surface area of the 2nd cube = 10 – 1 = 9cm²
Again the vertices of the second cube are at the midpoints of B.
You can see that multiple right-angled triangles are formed.
(√2)² = (½B)² + (½B)²
2 = ¼B² + ¼B²
2 = ½B²
4 = B²
B = √4 = 2cm
5 faces of the largest cube are uncovered and 1 is partially covered.
Surface area = 6 x 2 x 2 = 24cm²
Again the top face is partially covered by the 2nd cube.
Surface area of one of the faces of the 2nd cube = √2 x √2 = 2cm²
Total surface area of largest cube = 24 – 2 = 22cm²
Total surface area of solid = 22 + 9 + 5 = 36 cm²
