Learn on PengiBig Ideas Math, Course 3Chapter 8: Volume and Similar Solids

Lesson 4: Surface Areas and Volumes of Similar Solids

In this Grade 8 lesson from Big Ideas Math, Course 3, students explore similar solids and learn how surface area and volume change when dimensions are scaled by a factor of k. Students apply the rule that the ratio of surface areas equals the square of the ratio of corresponding linear measures, and the ratio of volumes equals the cube of that ratio. The lesson covers identifying similar solids, finding missing measures using proportional reasoning, and solving problems involving cylinders, cones, pyramids, and prisms.

Section 1

Properties of Similar Solids

Property

  • The ratios of all corresponding dimensions in similar objects are equal.
  • The corresponding angles in similar objects are equal.
  • If we scale all the dimensions of a particular object by the same number, the new object will be similar to the old one.

Examples

  • A cube with 2-inch sides is similar to a cube with 6-inch sides. The ratio of their sides is 1:31:3.
  • Two spheres are always similar to each other. A sphere with a 5 cm radius is similar to one with a 10 cm radius.

Section 2

Using Proportions to Find Missing Dimensions in Similar Solids

Property

If two solids are similar, the ratio of their corresponding linear measures is constant. You can set up a proportion to find a missing dimension.

dimension of Solid Acorresponding dimension of Solid B=another dimension of Solid Acorresponding dimension of Solid B\frac{\text{dimension of Solid A}}{\text{corresponding dimension of Solid B}} = \frac{\text{another dimension of Solid A}}{\text{corresponding dimension of Solid B}}

Section 3

Surface Area Ratios of Similar Solids

Property

If two similar solids have a scale factor of a:ba:b, then the ratio of their corresponding surface areas is a2:b2a^2:b^2.

Surface Area of Solid ASurface Area of Solid B=(Corresponding length of ACorresponding length of B)2=a2b2\frac{\text{Surface Area of Solid A}}{\text{Surface Area of Solid B}} = \left(\frac{\text{Corresponding length of A}}{\text{Corresponding length of B}}\right)^2 = \frac{a^2}{b^2}

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Properties of Similar Solids

Property

  • The ratios of all corresponding dimensions in similar objects are equal.
  • The corresponding angles in similar objects are equal.
  • If we scale all the dimensions of a particular object by the same number, the new object will be similar to the old one.

Examples

  • A cube with 2-inch sides is similar to a cube with 6-inch sides. The ratio of their sides is 1:31:3.
  • Two spheres are always similar to each other. A sphere with a 5 cm radius is similar to one with a 10 cm radius.

Section 2

Using Proportions to Find Missing Dimensions in Similar Solids

Property

If two solids are similar, the ratio of their corresponding linear measures is constant. You can set up a proportion to find a missing dimension.

dimension of Solid Acorresponding dimension of Solid B=another dimension of Solid Acorresponding dimension of Solid B\frac{\text{dimension of Solid A}}{\text{corresponding dimension of Solid B}} = \frac{\text{another dimension of Solid A}}{\text{corresponding dimension of Solid B}}

Section 3

Surface Area Ratios of Similar Solids

Property

If two similar solids have a scale factor of a:ba:b, then the ratio of their corresponding surface areas is a2:b2a^2:b^2.

Surface Area of Solid ASurface Area of Solid B=(Corresponding length of ACorresponding length of B)2=a2b2\frac{\text{Surface Area of Solid A}}{\text{Surface Area of Solid B}} = \left(\frac{\text{Corresponding length of A}}{\text{Corresponding length of B}}\right)^2 = \frac{a^2}{b^2}