Cone volume calculator

What is the volume of a cone?

The volume of a cone is a measure of the space inside the cone. It’s essential for various practical applications, whether in mathematics, physics, engineering, or daily life scenarios, such as determining the amount of a liquid a cone-shaped container can hold. The volume depends on the shape and dimensions of the cone in question—whether it is a right, oblique, or truncated cone.

In understanding how to determine these different volumes, it’s important to become familiar with their definitions and the specific parameters necessary for calculation:

• Right cone: This cone has a circular base and a vertex perpendicular to its center. The height is the perpendicular distance from the base to the vertex.
• Oblique cone: Here, the vertex is not directly above the center of the base, making the cone slanted. The height is still the perpendicular overall height from the base to the cone’s apex.
• Truncated cone (frustum of a cone): This shape arises when a cone is sliced, usually parallel to the base, removing the top portion. It has two bases: the original base and the truncated section base.

For each type of cone, specific formulas are used to calculate volume, accounting for features such as height and base radius.

Formula for cone volume

Right cone

For a right circular cone, the volume $V$ can be calculated using the formula:

$V = \frac{1}{3} \pi r^2 h$

• $r$ is the radius of the base.
• $h$ is the height of the cone.
• $\pi$ is a constant (~3.14159).

Oblique cone

The calculation of an oblique cone theoretically centers around the general cone formula. When the height ($h$) and base radius ($r$) are given from the base center perpendicular to the tip, it employs the same formula:

$V = \frac{1}{3} \pi r^2 h$

Truncated cone

The formula for the volume of a truncated cone computes the space between two bases:

$V = \frac{\pi h}{3} (r_1^2 + r_1r_2 + r_2^2)$

• $r_1$ is the radius of the bottom base.
• $r_2$ is the radius of the top base (cut-off base).
• $h$ is the perpendicular height between the bases.

Examples of cone volume calculations

Example 1: Right cone

Suppose we have a cone with a base radius of 4 cm and a height of 9 cm. What is the volume?

Using the formula for a right cone:

$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (4)^2 (9) = \frac{1}{3} \pi (16) (9) = \frac{1}{3} \pi (144) = 48\pi \approx 150.80 \text{ cm}^3$

Thus, the cone has a volume of 150.80 cm³.

Example 2: Oblique cone

An oblique cone has a height of 5 cm and a base radius of 3 cm.

$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3)^2 (5) = \frac{1}{3} \pi (9) (5) = \frac{1}{3} \pi (45) = 15\pi \approx 47.12 \text{ cm}^3$

In this case, the volume of the oblique cone is 47.12 cm³.

Example 3: Truncated cone

Consider a truncated cone with the bottom base radius of 6 cm and the top base radius of 4 cm. The height is 8 cm.

$V = \frac{\pi h}{3} (r_1^2 + r_1r_2 + r_2^2) = \frac{\pi (8)}{3} ((6)^2 + (6)(4) + (4)^2) = \frac{\pi (8)}{3} (36 + 24 + 16) = \frac{\pi (8)}{3} (76) = \frac{608\pi}{3} \approx 636.7 \text{ cm}^3$

So, the volume of the truncated cone is 636.7 cm³.

Facts about cones

1. Definition: A cone can be defined as a shape formed by rotating a right triangle around one of its sides. The lateral surface of the cone represents a circular sector of this rotation.
2. Base and vertex: A cone consists of a flat base (which is a circle) and a vertex that does not lie within the plane of the base.
3. Height and slant height: The height of a cone is the perpendicular distance from the vertex to the center of the base. The slant height of the cone is the distance from the vertex to any point on the circle of the base.
4. Types of cones: A cone can be classified as a right cone if its vertex is along the perpendicular line drawn from the center of the base, or an oblique cone if the vertex is not on this perpendicular.
5. Sections of a cone: The plane sections of a cone can form various shapes, such as a circle (if the cutting plane is parallel to the base), an ellipse, a parabola, or a hyperbola, forming the foundation of conic section theory.
6. Uses: Cones are frequently encountered in real life and engineering, such as in the shape of paper cups, ice cream cones, or in construction as elements of structures.
7. Sound and acoustics: In acoustics, the cone shape is used in horns and musical instruments to focus or distribute sound.

Frequently asked questions

How to calculate the volume of an oblique cone?

To calculate an oblique cone’s volume, ensure the perpendicular height from the base to the apex is considered, using $V = \frac{1}{3}\pi r^2 h$.

How many liters does a truncated cone with a base radius of 10 cm and top radius of 5 cm and a height of 20 cm hold?

First, calculate the volume using the formula, then convert the cubic centimeters to liters ($1\text{ liter} = 1000 \text{ cm}^3$) if needed:

$V = \frac{\pi (20)}{3} ((10)^2 + (10)(5) + (5)^2) = \frac{\pi (20)}{3} (100 + 50 + 25) = \frac{\pi (20)}{3} (175) = \frac{3500\pi}{3} \approx 3665.19 \text{ cm}^3 = 3.67 \text{ liters }$

A right cone has a volume of 1000 cm³. What is its height if the base radius is 10 cm?

$V = \frac{1}{3} \pi r^2 h = 1000 \text{ cm}^3$

$1000 = \frac{1}{3} \pi (10)^2 h$

$1000 = \frac{1}{3} \pi (100) h$

$1000 = \frac{100}{3} \pi h$

$h = \frac{1000 \times 3}{100 \pi} = \frac{3000}{100 \pi} = \frac{30}{\pi} \approx 9.55 \text{ cm}$

Why is the volume calculation same for the right and oblique cones?

The formula for calculating the volume of both right and oblique cones is the same because the volume depends solely on the area of the base and the height (the perpendicular distance from the vertex to the plane of the base), rather than the inclination of the lateral surface.

To understand this, one can use the Cavalieri’s principle from geometry. This principle states that if two solids have the same area at every level cross-section, then their volumes are equal. Cavalieri’s principle applies to cones through the following steps:

1. Base and height: Both the right and oblique cones have a base that is the same circle with radius $r$, and the height is the perpendicular distance from the vertex to the plane of the base.

2. Parallel sections: If we take a plane parallel to the base, which slices both cones at the same height, the areas of the sections created by this plane will be the same for both cones (they will be similar circles, scaled according to the height).

Because any such parallel plane creates identical sections in both the right and oblique cone, Cavalieri’s principle guarantees that the volumes are the same. Therefore, the volume of any cone, whether it is right or oblique, is calculated using the same formula.

Can the cones’ volumes help in assessing capacities of daily objects?

Yes, the calculation of the volume of liquid that can fit into a container in the shape of a truncated cone or other containers in the shape of a cone, based on the cone volume formula.