Enter the radius or diameter and the height of a right circular cone to get its volume, slant height, lateral and total surface area, with the formulas worked out. You can start from the slant height instead, and see how much a point-down cone holds when it is only partly full.
Last updated:
I know the
And the
%
For a cone-shaped cup or hopper filled point-down, as % of the height
Enter your values to see the result.
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How to use the Cone Volume Calculator
Choose radius or diameter and whether you know the vertical height or the slant height.
Enter the two measurements and pick the unit.
Read the volume in cubic units, liters and gallons.
Check the surface areas and slant height in the results and the working in the steps.
Optionally enter a fill depth to see how much a cone-shaped cup or hopper holds when partly full.
Cone formulas
Find
Formula
Example: r = 3, h = 4
Volume
V = (1/3)πr²h
12π ≈ 37.70
Slant height
s = √(r² + h²)
√25 = 5
Height from slant height
h = √(s² − r²)
√(25 − 9) = 4
Base area
πr²
9π ≈ 28.27
Lateral area
πrs
15π ≈ 47.12
Total surface area
πr(r + s)
24π ≈ 75.40
Apex angle
2 × arctan(r ÷ h)
73.74°
Partly filled cones
When a cone stands point down (a funnel, a paper cup, a grain hopper), the liquid inside is a smaller copy of the whole cone. Every length is scaled by the depth ratio, so the volume scales by its cube. That is why a cone looks half full long before it really is.
Depth (share of height)
Share of full volume
25%
1.56%
50%
12.5%
75%
42.19%
79.4%
50%
90%
72.9%
If the cone stands point up (like a pile), the empty space at the top is the small cone instead: filled share = 1 − (1 − depth)³.
Tips
Height means vertical height. If you measured along the sloping side, choose "Slant height" so the calculator finds the true height first.
For a truncated cone (a bucket or lampshade, called a frustum), subtract the volume of the missing tip cone from the full cone.
Compare shapes with the cylinder and sphere volume calculators: a cone, a hemisphere and a cylinder with the same radius r and height r have volumes in the ratio 1 : 2 : 3.
Frequently asked questions
What is the formula for the volume of a cone?
V = (1/3)πr²h, where r is the radius of the base and h is the perpendicular height. A cone with r = 3 cm and h = 4 cm holds (1/3) × π × 9 × 4 ≈ 37.70 cm³.
Why is a cone one third of a cylinder?
A cone and a cylinder with the same base and height satisfy V_cone = (1/3) × V_cylinder. You can show it by filling the cone with water three times to fill the cylinder, or with calculus by adding up thin circular slices whose radius shrinks linearly to zero.
How do I find the slant height of a cone?
The radius, height and slant height form a right triangle, so s = √(r² + h²). For r = 3 and h = 4, s = √25 = 5.
What is the surface area of a cone?
The lateral (side) area is πrs and the base is πr², so the total is πr(r + s). For r = 3 and s = 5 that is 15π + 9π = 24π ≈ 75.40 square units.
Can I calculate the volume from the slant height?
Yes. First find the height with h = √(s² − r²), then use V = (1/3)πr²h. The slant height must be longer than the radius.
How full is a cone-shaped cup when the liquid is halfway up?
Only one eighth full. The liquid forms a smaller cone that is similar to the whole cup, so its volume scales with the cube of the depth: (1/2)³ = 12.5%. At 80% of the depth a point-down cone is only 51.2% full.
How do I find the volume of a pile of sand or gravel?
A loose pile is roughly a cone. Measure the distance across the base and the height, then use V = (1/3)πr²h. A pile 3 m across and 1 m high holds about 2.36 m³.
Are all the tools on FreeToolCabin tested for quality and accuracy?
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