Volume of Similar Solids: Formulas, Ratios & Examples

The concept of Volume of Similar Solids Formulas is an essential geometry topic for students, especially in upper middle school and high school. It relates to comparing volumes of 3D shapes like cubes, spheres, cylinders, and cones when their dimensions are in equal ratio—helpful for questions in CBSE, ICSE, JEE, NEET, and daily practical applications.

Understanding the Volume of Similar Solids

Similar solids are three-dimensional figures (like cubes, prisms, pyramids, cylinders, cones, and spheres) that have the same shape but different sizes. This means their corresponding sides are proportional, but the solids are not necessarily congruent. For two solids to be similar, all corresponding linear measures (length, height, radius, slant height, etc.) must be in the same ratio—called the scale factor.

If the scale factor between corresponding linear dimensions of two similar solids is a:b, then the ratios of their surface areas and volumes are related as:

• Surface area ratio: \( a^2 : b^2 \)
• Volume ratio: \( a^3 : b^3 \)

Understanding these ratios helps solve exam questions efficiently and is commonly found in geometry, mensuration, and practical model questions.

Formula for Volume of Similar Solids

The formula for comparing the volume of two similar solids is:

If the scale factor of two similar solids is k (i.e., the ratio of corresponding sides is k), then
  \( \frac{\text{Volume}_1}{\text{Volume}_2} = \left(\frac{\text{Side}_1}{\text{Side}_2}\right)^3 = k^3 \)

This means if you know the linear scale factor, you can easily find the ratio of their volumes by cubing the scale factor. This relationship holds for all types of similar solids, including cubes, cuboids, spheres, cones, cylinders, and pyramids.

Common Volume Formulas for Solids

(You can explore more about solid geometry at Solid Shapes and Properties on Vedantu.)

Worked Example: Volume Ratio for Similar Solids

Let's see how this formula is applied in real problems:

1. Two similar cubes have side lengths of 4 cm and 6 cm. What is the ratio of their volumes?

   • Step 1: Find the linear scale factor: \( \frac{4}{6} = \frac{2}{3} \).
   • Step 2: Raise it to the power of 3: \( \left(\frac{2}{3}\right)^3 = \frac{8}{27} \).
   • Step 3: So, the ratio of their volumes is 8:27.

2. A small spherical ball has radius 2 cm. A larger, similar ball has a radius of 5 cm. If the small ball’s volume is 33.51 cm³, find the volume of the larger ball (round your answer appropriately).

   • Scale factor = \( \frac{2}{5} \)
   • Let the volume of the larger ball be \( V \).
   • \( \frac{33.51}{V} = \left(\frac{2}{5}\right)^3 = \frac{8}{125} \)
   • So, \( V = \frac{33.51 \times 125}{8} = 523.125 \) cm³

For extra practice on cubes, cylinders, and combinations, see Volume of Cube, Cuboid and Cylinder.

Practice Problems

• Two similar cones have heights in the ratio 1 : 3. What is the ratio of their volumes?
• The radii of two similar cylinders are 5 cm and 15 cm. If the smaller cylinder has a volume of 500 cm³, find the volume of the larger cylinder.
• Two spheres have surface areas in the ratio 16 : 9. What is the ratio of their volumes?
• A model car has a scale of 1 : 10. If its actual volume is 2,000,000 cm³, what is the volume of the model?
• If two similar pyramids have base edges in the ratio 3 : 5, what is the ratio of their surface areas and their volumes?

(Find answers at the bottom or test yourself with Vedantu’s downloadable worksheets.)

Common Mistakes to Avoid

• Confusing similarity (shapes are same, sizes are proportional) with congruence (shapes and sizes are identical).
• Using the square instead of the cube of the scale factor when working with volumes.
• Assuming formulas only work for cubes—remember, the cube law applies for all similar solids.
• Forgetting to compare matching quantities (e.g., height-to-height, not height-to-radius).

Real-World Applications

The volume of similar solids concept is found everywhere: designers making scale models, architects scaling up building plans, manufacturers creating various sizes of bottles or cans, or in biomedicine when scaling drug dosages from test animals to humans. Even in sports—like comparing football sizes across leagues—the same formulas apply. At Vedantu, we help you connect these concepts to both exam and real-life problem-solving.

In this topic, we explored how to use Volume of Similar Solids Formulas and their relationships with scale factors, surface areas, and ratios. Knowing how to apply these formulas helps you solve a wide range of geometry and modeling problems efficiently—an important skill for board exams and entrance tests. For more practice and deeper understanding, explore other related concept pages on Vedantu.