Similar Solids

In this lesson, you will explore the relationship between length, surface area, and volume in similar objects. You will learn how changes in scale affect each of these measurementsβ€”for example, how length scales linearly, area scales with the square of the scale factor, and volume scales with the cube of the scale factor.

Through guided examples, you will understand how to apply these relationships and formulas to find unknown lengths, areas, or volumes.

Related Lessons:
  1. S3 Congruency Proving
  2. S3 Similarity Proving

Similar Solids And Its Relationship

When 2 objects are similar, they have the same shape but different size. Since the 2 objects are similar, there are relationships between their length, height, area, volume and mass.

  • \(\frac{π‘™π‘’π‘›π‘”π‘‘β„Ž1}{π‘™π‘’π‘›π‘”π‘‘β„Ž2}=\frac{β„Žπ‘’π‘–π‘”β„Žπ‘‘1}{β„Žπ‘’π‘–π‘”β„Žπ‘‘2}\)
  • \(\frac{area1}{area2}=(\frac{β„Žπ‘’π‘–π‘”β„Žπ‘‘1}{β„Žπ‘’π‘–π‘”β„Žπ‘‘2})^2\)
  • \(\frac{vol1}{vol2}=(\frac{β„Žπ‘’π‘–π‘”β„Žπ‘‘1}{β„Žπ‘’π‘–π‘”β„Žπ‘‘2})^3\)
  • \(\frac{mass1}{mass2}=(\frac{β„Žπ‘’π‘–π‘”β„Žπ‘‘1}{β„Žπ‘’π‘–π‘”β„Žπ‘‘2})^3\)

Similar Solids Example 1

The two vases shown in the diagram are geometrically similar.

The heights of Vase A and Vase B are 15cm and 25cm respectively.

Given that the diameter of the base of Vase A is 7.5cm and the volume of Vase B is 750\(cm^3\), find

  1. the diameter of the base of Vase B
  2. the ratio of the surface areas of Vase A to Vase B
  3. the volume of Vase A.

Similar Solids Example 2

A solid cone, A is cut into two parts, B and C. by a plane parallel to the base as shown in the diagram.

The slant lengths of the two parts are 10cm and 4 cm respectively. Find the ratio of

  1. the diameters of the bases of A and B
  2. the volumes of A and C.
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Related Lessons:

  1. S3 Congruency Proving
  2. S3 Similarity Proving

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