Find The Volume Of The Smaller Figure Calculator

Calculate Volume

Enter the volume and a linear dimension of the larger figure, and the corresponding linear dimension of the smaller similar figure to find the volume of the smaller figure.

Copied!

Understanding the Volume of the Smaller Figure Calculator

What is a Volume of the Smaller Figure Calculator?

A Volume of the Smaller Figure Calculator is a tool used to determine the volume of a smaller geometric figure when it is mathematically similar to a larger figure, and you know the volume of the larger figure and the ratio of their corresponding linear dimensions (like sides, radii, or heights). Two figures are similar if they have the same shape but different sizes, meaning all corresponding angles are equal, and the ratio of all corresponding linear dimensions is constant. This constant ratio is known as the linear scale factor.

This calculator is particularly useful in geometry, architecture, engineering, and model-making, where scaling objects up or down is common. If you know the dimensions or volume of one object, you can easily calculate the volume of a scaled version using the principles of similarity. The core idea is that if the linear scale factor between two similar figures is 'k', the ratio of their areas is 'k²', and the ratio of their volumes is 'k³'.

Who should use it?

• Students: Learning about geometric similarity and scaling in math classes.
• Engineers and Architects: When scaling models and calculating material volumes.
• Model Makers: To determine the volume of scale models based on real-world objects.
• 3D Printing Enthusiasts: To estimate material usage for scaled prints.

Common Misconceptions

A common misconception is that if you halve the linear dimensions of an object, you halve its volume. In reality, if you halve the linear dimensions (scale factor k=0.5), the volume reduces by a factor of 0.5³ = 0.125, meaning the volume becomes one-eighth of the original. Our Volume of the Smaller Figure Calculator accurately applies this cubic relationship.

Volume of the Smaller Figure Calculator Formula and Mathematical Explanation

When two 3D geometric figures are similar, the ratio of their corresponding linear dimensions is constant. Let's call this ratio the linear scale factor, 'k'.

If L_large is a linear dimension (e.g., side, radius, height) of the larger figure, and L_small is the corresponding linear dimension of the smaller figure, then:

k = L_small / L_large

The relationship between the volumes of two similar figures (V_large and V_small) is given by the cube of the linear scale factor:

V_small / V_large = k³ = (L_small / L_large)³

Therefore, to find the volume of the smaller figure (V_small), we use the formula:

V_small = V_large × (L_small / L_large)³

The Volume of the Smaller Figure Calculator uses this exact formula.

Variables Table

Variable	Meaning	Unit	Typical Range
Vlarge	Volume of the larger figure	Cubic units (cm³, m³, in³, etc.)	> 0
Llarge	A linear dimension of the larger figure	Linear units (cm, m, in, etc.)	> 0
Lsmall	The corresponding linear dimension of the smaller figure	Same as Llarge	> 0 (and usually < Llarge for smaller figure)
k	Linear scale factor (Lsmall / Llarge)	Dimensionless	> 0 (and < 1 if smaller)
k^3	Volume scale factor	Dimensionless	> 0 (and < 1 if smaller)
Vsmall	Volume of the smaller figure	Same as Vlarge	> 0

Table explaining the variables used in the volume calculation for similar figures.

Practical Examples (Real-World Use Cases)

Example 1: Scaling a Cube

Suppose you have a large cube with a side length of 10 cm and a volume of 1000 cm³. You want to create a smaller, similar cube whose side length is 2 cm.

• V_large = 1000 cm³
• L_large = 10 cm
• L_small = 2 cm

Using the Volume of the Smaller Figure Calculator or the formula:

k = L_small / L_large = 2 / 10 = 0.2

V_small = V_large × k³ = 1000 × (0.2)³ = 1000 × 0.008 = 8 cm³

The volume of the smaller cube is 8 cm³.

Example 2: Architectural Model

An architect is building a scale model of a building. The actual building will have a volume of 5000 m³. The model is being built at a 1:100 scale (meaning linear dimensions of the model are 1/100th of the actual building).

• V_large = 5000 m³
• L_large = 100 (representing the scale part)
• L_small = 1 (representing the scale part)

Here, k = 1/100 = 0.01.

V_small = 5000 × (0.01)³ = 5000 × 0.000001 = 0.005 m³

The volume of the scale model will be 0.005 m³, or 5000 cm³ (since 1 m³ = 1,000,000 cm³).

For more on scaling, see our guide to geometric scaling.

How to Use This Volume of the Smaller Figure Calculator

Using our Volume of the Smaller Figure Calculator is straightforward:

1. Enter Volume of Larger Figure (V_large): Input the known volume of the larger object in the first field.
2. Enter Linear Dimension of Larger Figure (L_large): Input a specific linear dimension (like height, width, or radius) of the larger object.
3. Enter Corresponding Dimension of Smaller Figure (L_small): Input the same type of linear dimension for the smaller, similar object. Ensure you use the same units for L_large and L_small.
4. Calculate: The calculator automatically updates the results as you type or when you click the "Calculate" button.

How to Read Results

The calculator displays:

• Volume of Smaller Figure (V_small): The primary result, shown prominently.
• Linear Scale Factor (k): The ratio L_small / L_large.
• Volume Scale Factor (k³): The cube of the linear scale factor, which is the ratio V_small / V_large.
• Given V_large: The volume you entered for the larger figure.

Check out understanding scale factors for more detail.

Key Factors That Affect Volume of the Smaller Figure Results

The volume of the smaller figure is directly influenced by several factors:

1. Volume of the Larger Figure (V_large): The starting volume. If this is larger, V_small will also be proportionally larger for the same scale factor.
2. Linear Scale Factor (k): This is the ratio L_small / L_large. The volume of the smaller figure is proportional to the CUBE of this factor. A small change in k leads to a much larger change in the volume ratio.
3. Accuracy of Dimension Measurements: Precise measurements of L_large and L_small are crucial. Small errors in these measurements are cubed when calculating the volume ratio, potentially leading to larger errors in V_small.
4. Similarity of the Figures: The formula V_small = V_large × k³ is only valid if the two figures are perfectly similar (all corresponding angles are equal, and all corresponding linear dimensions have the same ratio k).
5. Units Used: Ensure L_large and L_small are in the same units. The units of V_small will be the same as V_large.
6. Choice of Corresponding Dimensions: You must compare corresponding dimensions (e.g., height of large with height of small, radius of large with radius of small). Using non-corresponding dimensions will give an incorrect scale factor. Learn more about similar solids volume.

Frequently Asked Questions (FAQ)

What does it mean for two figures to be similar?

Two geometric figures are similar if they have the same shape but possibly different sizes. This means corresponding angles are equal, and the ratio of corresponding linear dimensions is constant (the scale factor).

Can I use this calculator for 2D figures like squares or circles?

While the principle of scaling is similar (area scales with k²), this calculator is specifically for volumes (3D figures, scaling with k³). For areas, the formula would be Area_small = Area_large × k². See our area scaling calculator.

What if L_small is larger than L_large?

If L_small > L_large, then k > 1, and you are scaling up. The calculator would find the volume of the "larger" figure based on the "smaller" one's volume, though the labels would be reversed from the calculator's perspective. The math still works.

Do the figures have to be regular shapes like cubes or spheres?

No, the figures can be irregular as long as they are mathematically similar. The relationship V_small / V_large = k³ holds for any pair of similar 3D objects.

What units should I use?

You can use any consistent units for volume (cm³, m³, ft³, etc.) and linear dimensions (cm, m, ft, etc.), but ensure L_large and L_small use the SAME linear units, and V_large is in the corresponding cubic units.

How accurate is the Volume of the Smaller Figure Calculator?

The calculator is as accurate as the input values you provide. Ensure your volume and dimension measurements are precise.

What if I know the volumes and want to find the scale factor?

If you know V_small and V_large, the volume scale factor is V_small / V_large, and the linear scale factor k is the cube root of this ratio: k = (V_small / V_large)^1/3.

Where can I learn more about the geometry of similar figures?

Many geometry textbooks and online resources cover the topic of similar figures and their properties in detail. Our section on geometry volume calculations is a good start.

Related Tools and Internal Resources

• Guide to Geometric Scaling: Understand the principles of scaling shapes and objects.
• Understanding Scale Factors: A deep dive into linear, area, and volume scale factors.
• Volume of Similar Solids Calculator: Another tool focusing on the relationship between volumes of similar 3D shapes.
• Area Scaling Calculator: Calculate how areas change with linear scaling for 2D figures.
• Geometry Volume Calculations: Resources for various volume formulas.
• Cube Volume Calculator: Calculate the volume of a cube given its side length.