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Cone Volume Calculator

Calculate the volume, surface area, and other properties of a cone given its radius and height.

What this tool does

Calculate the volume, surface area, and other properties of a cone given its radius and height. You fill in these fields: Radius of the base and Height. It shows volume, surface area, lateral surface area, base area, and slant height.

Inputs

  • Radius of the base: A number. Must be greater than 0. Distance from the centre of the base to its edge. Use the same length unit for radius and height, such as cm or inches.
  • Height: A number. Must be greater than 0. Straight distance from the centre of the base to the tip, at a right angle to the base.

How to use

  1. Enter the radius of the base.
  2. Enter the height.
  3. Read the result: volume, surface area, lateral surface area, base area, and slant height. It appears once every required field has a valid value.

Worked examples

Example 1. Inputs:

  • Radius of the base: 3
  • Height: 4

Results:

  • Volume: 37.699112 cubic units
  • Surface area: 75.398224 square units
  • Lateral surface area: 47.12389 square units
  • Base area: 28.274334 square units
  • Slant height: 5 units

How this is worked out: Volume = (1/3) x pi x r^2 x h = (1/3) x pi x 9 x 4 = 12 pi = 37.69911184. Slant height = sqrt(r^2 + h^2) = sqrt(9 + 16) = 5. Base = pi x 9 = 28.27433388. Lateral = pi x r x s = pi x 3 x 5 = 15 pi = 47.1238898. Surface = base + lateral = 24 pi = 75.39822369.

Example 2. Inputs:

  • Radius of the base: 1
  • Height: 1

Results:

  • Volume: 1.047198 cubic units
  • Surface area: 7.584476 square units
  • Lateral surface area: 4.442883 square units
  • Base area: 3.141593 square units
  • Slant height: 1.414214 units

How this is worked out: Volume = (1/3) x pi x 1 x 1 = pi / 3 = 1.04719755. Slant height = sqrt(1 + 1) = sqrt(2) = 1.41421356. Base = pi = 3.14159265. Lateral = pi x 1 x sqrt(2) = 4.44288294. Surface = pi + pi sqrt(2) = pi (1 + sqrt 2) = 7.58447559.

Example 3. Inputs:

  • Radius of the base: 2
  • Height: 6

Results:

  • Volume: 25.132741 cubic units
  • Surface area: 52.304724 square units
  • Lateral surface area: 39.738353 square units
  • Base area: 12.566371 square units
  • Slant height: 6.324555 units

How this is worked out: Volume = (1/3) x pi x 4 x 6 = 8 pi = 25.13274123. Slant height = sqrt(4 + 36) = sqrt(40) = 6.32455532. Base = 4 pi = 12.56637061. Lateral = pi x 2 x sqrt(40) = 2 pi x 6.32455532 = 39.73835305. Surface = 12.56637061 + 39.73835305 = 52.30472366.

Limitations

  • Assumes a right circular cone, with the tip directly above the centre of a round base.
  • Assumes an ideal shape. It does not allow for a rounded tip, a truncated top, or a hollow shell.
  • Results are exact formulas, not measurements, so the accuracy depends on the dimensions you enter.

FAQs

Q: What is the slant height?
A: It is the distance from the edge of the base up the side to the tip. It is found from the radius and height with the Pythagorean theorem, and it is used for the lateral surface area.
Q: What units do the results use?
A: The results use whatever unit you type. Volume is in that unit cubed and areas are in that unit squared, so inches give cubic inches and square inches.
Q: Does the surface area include the base?
A: Yes. The surface area is the lateral area plus the base area. The lateral surface area is shown on its own as well.
Q: Can I use a diameter instead of a radius?
A: Not directly. Divide the diameter by 2 to get the radius before you enter it.

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