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India
Class IX

Similar Solids

Lesson

When one shape has been enlarged, we look to find the scale factor of the enlargement. What we need to consider here is the effect of enlargement on 2 dimensional and 3 dimensional shapes. 

To see what happens to a 2D enlargement, try out this applet:

What do you notice about the change in the area scale factor (SF)? You will see that whatever the scale factor, the area scale factor will be the scale factor squared.

$\text{area of enlarged shape}=\text{length}\times\text{SF}\times\text{width}\times\text{SF}$area of enlarged shape=length×SF×width×SF

This can be written as:

$\text{area of enlarged shape}=\text{length}\times\text{width}\times\text{SF}^2$area of enlarged shape=length×width×SF2

What about if you were to increase each side of a 3D shape? Well each length has been increased by the same amount, and to find the volume you need to multiply length by width by height. 

$\text{volume of enlarged solid}=\text{length}\times\text{SF}\times\text{width}\times\text{SF}\times\text{height}\times\text{SF}$volume of enlarged solid=length×SF×width×SF×height×SF

This can be thought of as:$\text{volume of enlarged solid}=\text{length}\times\text{width}\times\text{height}\times\text{SF}^3$volume of enlarged solid=length×width×height×SF3

Remember!

$\text{scale factor for area}=\text{scale factor}^2$scale factor for area=scale factor2

$\text{scale factor for volume}=\text{scale factor}^3$scale factor for volume=scale factor3

Worked Examples

question 1

Consider the rectangular prisms A and B shown.

  1. Find the surface area of Rectangular Prism A.

  2. Find the surface area of Rectangular Prism B.

  3. What is the ratio of the surface area of Rectangular Prism A to Rectangular Prism B?

  4. Is it true that if the matching sides of two similar figures are in the ratio $m$m:$3$3, then their surface areas are in the ratio $m^2$m2:$3^2$32?

    True

    A

    False

    B

 
Question 2

Consider the two similar spheres shown. The smaller sphere has radius $3$3 cm while the larger sphere has a radius of $12$12 cm.

  1. Find the volume of Sphere A, in simplest exact form.

  2. Find the volume of Sphere B, in simplest exact form.

  3. What is the ratio of the volume of Sphere A to Sphere B?

  4. Is it true that if the matching dimensions of two similar figures are in the ratio $m$m:$4$4, then their volumes are in the ratio $m^3$m3:$4^3$43

    True

    A

    False

    B

Outcomes

9.M.SAV.1

Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular cylinders/cones.

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