A Truncated Cone Volume Calculator is a specialized tool designed to compute the volume of a truncated cone, also known as a frustum.
- A truncated cone with R = 5 cm, r = 3 cm, and h = 10 cm
- A larger frustum with R = 15 m, r = 8 m, and h = 20 m
- A small truncated cone with R = 2 inches, r = 1 inch, and h = 4 inches
This unique shape is formed when a plane parallel to the base cuts off the top portion of a cone, resulting in a structure with two circular bases of different sizes.
This calculator requires input values such as:
- The radius of the larger base (R)
- The radius of the smaller base (r)
- The height of the truncated cone (h)
With these measurements, the calculator applies the appropriate formula to determine the volume of the truncated cone.
Truncated Cone Volume Calculator
| Larger Radius (R) | Smaller Radius (r) | Height (h) | Volume (V) | Conversion Equation |
|---|---|---|---|---|
| 5 cm | 3 cm | 10 cm | 314.16 cm³ | V = (1/3) × π × 10 × (5² + 3² + 5 × 3) |
| 2 m | 1 m | 5 m | 26.18 m³ | V = (1/3) × π × 5 × (2² + 1² + 2 × 1) |
| 8 in | 6 in | 12 in | 1,809.56 in³ | V = (1/3) × π × 12 × (8² + 6² + 8 × 6) |
| 10 ft | 7 ft | 15 ft | 3,436.12 ft³ | V = (1/3) × π × 15 × (10² + 7² + 10 × 7) |
| 1.5 m | 0.5 m | 3 m | 3.93 m³ | V = (1/3) × π × 3 × (1.5² + 0.5² + 1.5 × 0.5) |
Related Tools
Truncated Cone Volume Formula
The formula for calculating the volume of a truncated cone is:
V = (1/3) × π × h × (R² + r² + R × r)
Where:
- V is the volume
- h is the height of the truncated cone
- R is the radius of the larger base
- r is the radius of the smaller base
- π is approximately 3.14159
This formula is derived from the difference between the volumes of two full cones—the larger cone minus the smaller cone that was “cut off.”
Example:
Let’s calculate the volume of a truncated cone with R = 6 cm, r = 4 cm, and h = 8 cm.
$$ V = (1/3) × π × 8 × (6² + 4² + 6 × 4) $$ $$ V = (1/3) × 3.14159 × 8 × (36 + 16 + 24) $$ $$ V = 8.37757 × 76 $$ $$ V ≈ 636.70 \, \text{cm}³ $$
How to Find the Volume of a Truncated Cone?
To find the volume of a truncated cone, follow these simple steps:
- Measure the Dimensions: Determine the radius of the larger base (R), the radius of the smaller base (r), and the height (h) of the truncated cone.
- Apply the Formula: Use the formula V = (1/3) × π × h × (R² + r² + R × r).
- Calculate: Plug in the values and compute the result. Use a calculator for precision.
- Check Units: Ensure your final answer is in cubic units (e.g., cm³, m³, in³).
To calculate the volume of a truncated cone using the formula:
$$ V = \frac{1}{3} \pi h \left( r_1^2 + r_1 r_2 + r_2^2 \right) $$
Given Values
- Larger base radius ($$ r_1 $$): 5 units
- Smaller base radius ($$ r_2 $$): 3 units
- Height ($$ h $$): 10 units
Substituting the values into the formula:
$$ V = \frac{1}{3} \pi \times 10 \left( 5^2 + 5 \times 3 + 3^2 \right) $$
Calculating the terms inside the parentheses:
$$ 5^2 = 25 $$ $$ 5 \times 3 = 15 $$ $$ 3^2 = 9 $$
Now, summing these values:
$$ 25 + 15 + 9 = 49 $$
Substituting back into the volume formula:
$$ V = \frac{1}{3} \pi \times 10 \times 49 $$
Calculating the volume:
$$ V = \frac{490}{3} \pi \approx 490.87 \text{ cubic units} $$
The volume of the truncated cone is approximately 490.87 cubic units.
What is the Formula for the Volume of a Truncated Cylinder?
The formula for the volume of a cylinder (truncated or not) is:
V = π × r² × h
Where:
- V is the volume
- r is the radius of the base
- h is the height of the cylinder
It’s important to note that a truncated cylinder is different from a truncated cone. A truncated cylinder has the same radius for both bases.
How to Calculate Truncated Cone Surface Area?
The formula is:
SA = π × (R² + r²) + π × (R + r) × s
Where:
- SA is the surface area
- R is the radius of the larger base
- r is the radius of the smaller base
- s is the slant height
To find the slant height (s), use the Pythagorean theorem: $$ s² = h² + (R – r)² $$
The surface area of a truncated cone includes the area of both circular bases and the lateral surface area.
What is Truncated Cone Volume?
Truncated Cone Volume refers to the amount of space enclosed within a truncated cone. It represents the capacity of the frustum to contain a substance.
The volume is expressed in cubic units, such as cubic centimeters (cm³), cubic meters (m³), or cubic inches (in³).
Understanding truncated cone volume is crucial in various fields, including:
- Engineering: Designing containers, tanks, or parts
- Architecture: Creating conical structures or elements
- Manufacturing: Producing conical products or molds
- Science: Analyzing geological formations or conducting experiments
Is there a way to calculate the height if we only know the Volume, base radius, and the slant angle?
Surely you can, follow the below steps please:
To calculate the height of a cone when you know the volume, base radius, and slant angle, you can follow these steps:
Volume of a Cone: The volume V of a cone is given by the formula: V = (1/3) pi r^2 * h where r is the radius of the base and h is the height.
Relationship Between Height, Radius, and Slant Height: The slant height l, radius r, and height h are related by the Pythagorean theorem: l^2 = r^2 + h^2 which can be rearranged to find the height: h = sqrt(l^2 – r^2)
Finding Slant Height from the Slant Angle: If you know the slant angle θ, you can express the slant height in terms of the height: l = h / cos(θ)
Steps to Calculate Height
Calculate Slant Height: From the slant angle, you can calculate the slant height l using: l = h / cos(θ)
Substitute in Volume Formula: Rearranging the volume formula gives: h = (3V) / (pi r^2)
Combine Equations: Substitute l into the height equation: h = sqrt((h / cos(θ))^2 – r^2)
Solve for Height: This leads to a quadratic equation in terms of h. You can solve this equation to find the height h.
Example Calculation
If you have a volume V, base radius r, and slant angle θ:
Calculate the height using: h = (3V) / (pi r^2)
Find the slant height using: l = h / cos(θ)
Use the Pythagorean relationship to confirm or adjust h as needed.
This approach allows you to find the height of the cone using the known volume, base radius, and slant angle.