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Maths

Congruent and similar shapes

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Similar areas and volumes

We already know that if two shapes are similar their corresponding sides are in the same ratio, and their corresponding angles are equal.

Look at the two cubes below:

image: two cubes: left cube: (a) is equals in lengths for width, height and depth. Right cube:(b) is equal in length for width, height and depth.

The cubes are similar, and the ratio of their lengths is a:b or a over b

Question

What is the ratio of:

  1. the area of their faces?
  2. their volumes?

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Answer
  1. Cube 'a' has a face area of a2

    Cube 'b' has a face area of b2

    The ratio of their areas is a2 :b2 or a to the power of 2 over b to the power of 2

  2. Cube 'a' has a volume of a3

    Cube 'b' has a volume of b3

    The ratio of their volumes is a3 :b3 or a to the power of 3 over b to the power of 3

For any pair of similar shapes, the following is true:

Ratio of lengths = a:b or a over b

Ratio of areas = a2:b2 or a to the power of 2 over b to the power of 2

Ratio of volumes = a3:b3 or a to the power of 3 over b to the power of 3

Question

These two shapes are similar. What is the length of x?

image: two 6 sides shape (hexagon): left hexagon: area equal to 25cm squared, bottom right length:2cm. Right hexagon: area:36cm squared, bottom right length: (x)

Answer

The ratio of the areas is 25:36 (a2:b2)

The ratio of the lengths is a:b Therefore, we find the square roots of 25 and 36. Ratio of lengths = 5:6

5:6 = 2:x

So 6 over 5 = x over 2

(multiply both sides by 2) 12 over 5 = x

x = 2.4cm

Now you try one.

Question

Two similar pyramids have volumes of 64cm3 and 343cm3. What is the ratio of their surface areas?

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Answer

The answer is 16:49. Here is how to work it out. Try to fill in the blanks below:

Ratio of volumes = 64:?

To find the ratio of the lengths, we find the cube roots of 64 and ?

Therefore, the ratio of the lengths is ?:7

To find the ratio of the areas, we square ? and 7

The ratio of the areas is 16:49.

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