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A series of questions and solutions related to geometry, focusing on the concepts of length, area, and volume. The questions involve calculating the new lengths, areas, and volumes when the given dimensions are doubled. This resource is ideal for students studying geometry, algebra, or mathematics, as it offers practical examples and explanations.
Typology: Study Guides, Projects, Research
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Supported by UBC Teaching and Learning Enhancement Fund 2012-
F AC U L T Y O F E D U C AT I O NF AC U L T Y O F E D U C AT I O N D e p a r t m e n t o f C u r r i c u l u m a n d P e d a g o g y
F A C U L T Y O F E D U C A T I O N
Answer: A
Justification: The length of the line is doubled. Therefore, we must double the original length to obtain the new, expanded length.
The side length of a square is x and therefore has a perimeter of 4x. Each of the side lengths in the square are then doubled. What is the perimeter of the new square?
x
E.16x
D.8x
C.4x
B.2x
A.x
A. (4x)^2
B. 4x^2
C. 2x^2
D. x^2
E. 4x
A square with side length x has area x^2. Each of the side lengths in the square are then doubled. What is the area of the new square?
x
Answer: B
Justification: The doubled side length is 2x. The area of the square is side length squared:
Therefore, 4x^2 is the area of the new square. Notice,
x x
When each side length is doubled, you can “fill” the larger square with four squares the same size as the original.
2 x^ ^2 ^2 x^ ^2 x^ ^4 x^2
4 x^ 2 ^4 x^^2 , because:^ 4 x^ ^2 ^4 x^ ^4 x^ ^16 x^2
x
Answer: C
Justification: Each of the edge lengths doubles, so the total length of all the edges together must also double.
Alternatively, each side length doubles, and there are 12 sides, so:
A. 48x^2
B. 36x^2
C. 28x^2
D. 24x^2
E. 24x
The edge length of a cube is x, and each face has an area of x^2. Each of the edge lengths in the cube are then doubled. What is the total surface area of the new cube?
x
2x
A. 9x^3
B. 8x^3
C. 6x^3
D. 8x^2
E. 2x^3
The volume of a cube with side length x is x^3. Each of the edge lengths in the cube are then doubled. What is the volume of the new cube?
x
2x
Answer: D
Justification: The volume of a cube is the edge length cubed. This gives:
As seen in the diagram below, when the side length doubles, 8 of the original cubes can fit inside the new, larger cube.
x
2 x^ 3 ^ 2 x^ ^ 2 x^ ^ 2 x^ ^8 x^3
2x
Answer: B
Justification: Think about the previous questions.
The surface area of a cube of length x is 6x^2. If the side length increases by a factor of F, the side length becomes Fx. The surface area then becomes 6(Fx)^2 , which is F^2 times larger than the original surface area:
You can check this by thinking about the question in which we doubled the side length. The surface area of the original cube was 6x^2. When the side lengths were doubled, the surface area became 6(2x)^2 = 24x^2
The edge length of a cube is increased by a factor of F. By how many times will the volume increase?
x
A. F^3
B. F^2
C. 2F^2
D. 4F^2
E. F
Fx
A. 16
B. 8
C. 4
D. 2
E. No idea
The edge length of a tesseract (cube of 4 spatial dimensions) is doubled. What is the factor that the 3D volume increases by?
x 2x
Answer: B
Justification: As we have deduced before, the number of dimensions of the object does not affect the scaling of its constituent parts, as in, squares and cubes both scale by a factor of F^2 in terms of their surface area. Both cubes and the 3D volume on the tesseract scale the same (F^3 ) when we talk about 3D volume, and therefore the answer is 2^3 =8.
