How to Find Volume of Similar Solids Using Scale Factor Formula and Solved 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
| Solid | Volume Formula |
|---|---|
| Cube | \( V = a^3 \) |
| Cuboid | \( V = l \times b \times h \) |
| Sphere | \( V = \frac{4}{3}\pi r^3 \) |
| Cylinder | \( V = \pi r^2 h \) |
| Cone | \( V = \frac{1}{3} \pi r^2 h \) |
| Pyramid | \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \) |
(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:
-
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.
-
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.
FAQs on Volume of Similar Solids and Their Formulas
1. What is the formula for the volume of similar solids?
The formula for the volume of similar solids is that their volumes are in the ratio of the cube of their corresponding linear scale factor. If the linear scale factor is k, then:
Volume scale factor = k³
This means:
- If all lengths are multiplied by k,
- Then the volume is multiplied by k³.
2. How do you find the volume of a similar solid using a scale factor?
To find the volume of a similar solid, multiply the original volume by the cube of the linear scale factor.
Steps:
- Find the linear scale factor (k).
- Cube the scale factor: k³.
- Multiply the original volume by k³.
New volume = 50 × 3³ = 50 × 27 = 1350 cm³.
3. Why is the volume scale factor the cube of the linear scale factor?
The volume scale factor is the cube of the linear scale factor because volume depends on three dimensions: length, width, and height.
If each dimension is multiplied by k:
- Length → k times
- Width → k times
- Height → k times
4. What is the ratio of volumes of two similar solids?
The ratio of volumes of two similar solids is equal to the cube of the ratio of their corresponding sides.
If the ratio of sides is a : b, then:
Volume ratio = a³ : b³
Example: If two cubes have side lengths in the ratio 2 : 5, then their volumes are in the ratio:
2³ : 5³ = 8 : 125.
5. How do you find the linear scale factor from the volume ratio?
To find the linear scale factor from the volume ratio, take the cube root of the volume ratio.
Formula:
Linear scale factor = ∛(Volume ratio)
Example: If the volume ratio is 27 : 1:
- Linear scale factor = ∛27 : ∛1
- = 3 : 1
6. Can you give an example of volume of similar solids with numbers?
Yes, if two similar cones have a linear scale factor of 4, their volumes differ by a factor of 4³ = 64.
Example:
- Small cone volume = 10 cm³
- Scale factor = 4
Large cone volume = 10 × 4³ = 10 × 64 = 640 cm³
This shows how the volume of similar solids increases rapidly when dimensions are enlarged.
7. Does the volume of similar solids increase faster than the surface area?
Yes, the volume increases faster because it depends on k³, while surface area depends on k².
For similar solids:
- Linear scale factor = k
- Surface area scale factor = k²
- Volume scale factor = k³
8. What happens to the volume if the dimensions are halved?
If the dimensions are halved, the volume becomes (1/2)³ = 1/8 of the original volume.
Calculation:
- Linear scale factor = 1/2
- Volume scale factor = (1/2)³
- = 1/8
9. Are the formulas for volume of similar solids the same for cubes, spheres, and cylinders?
Yes, the volume scale factor rule k³ applies to all similar solids, including cubes, spheres, cones, and cylinders.
Although each solid has its own volume formula (for example, cube = a³, sphere = (4/3)πr³), when comparing similar solids:
- All corresponding dimensions scale by k.
- All volumes scale by k³.
10. What are common mistakes when solving volume of similar solids problems?
A common mistake is using the linear scale factor instead of the cube of the scale factor when calculating volume.
Common errors include:
- Forgetting to cube the scale factor.
- Mixing up surface area (k²) and volume (k³).
- Not checking that the solids are truly similar.
- Using incorrect units in final answers.
