Frustum Volume Calculator
Bucket volume, side, and slant from radii.
What it does
Buckets, planters, and lampshades are frustums: cones with the tip cut off. Two radii plus height fix volume, side area, and slant together. The same inputs price concrete for a footing and sheet for a shade.
The tool above runs the trio. With big radius 6 cm, small 4 cm, and height 10 cm, volume is 795.9 cm3 with a 320.38 cm2 side and 10.2 cm slant. With 8 over 5 and height 6, volume is 810.5 cm3. The imperial mirror at 2.4 over 1.6 with 3.9 gives 49.7 in3.
Ends set the family. Zero small radius makes a cone, matched radii make a cylinder, and the frustum sits between. Slant follows Pythagoras on the radius gap and height, which is why shallow tapers cut long gentle sides.
This page models solid right circular frustums. It does not handle wall thickness, elliptical tubs, or off center cuts. For sheet work, add seams and waste after.
How to use
- Pick metric (cm) or imperial (in).
- Enter the big end radius.
- Enter the small end radius below big.
- Enter the vertical height.
- Read volume first, then side.
- Read slant for layout work.
Formula
- R
- Big end radius in cm or in.
- r
- Small end radius below big.
- h
- Vertical height between ends.
Right circular frustum with flat parallel ends. At zero small radius the solid is a cone; at matched radii it is a cylinder. Hollow buckets need wall thickness math.
Worked example
Example: frustum 6 over 4 with 10 height
Defaults: metric big radius 6 cm with small 4 cm and 10 cm height. Imperial mirror is 2.4 over 1.6 with 3.9.
- Inputs (metric default): big 6 cm with small 4 cm and height 10 cm.
- Squares: 36 plus 24 plus 16 is 76.
- Volume: pi times 10 / 3 times 76 is 795.9 cm3.
- Slant: root of 4 plus 100 is 10.2 cm.
- Side: pi times 10 times 10.2 is 320.38 cm2. Tool shows the same.
Reference table
| Ends and height | Volume | Side |
|---|---|---|
| 6 over 4, h10 cm | 795.9 cm3 | 320.38 cm2 |
| 8 over 5, h6 cm | 810.5 cm3 | 273.97 cm2 |
| 5 over 3, h5 cm | 256.6 cm3 | 135.34 cm2 |
| 2.4 over 1.6, h3.9 in | 49.7 in3 | 50.03 in2 |
| 4 over 0, h9 cm | 150.8 cm3 | 123.76 cm2 |
Limits
- Right circular only. Elliptical excluded.
- Solid assumed. Hollow needs thickness.
- Parallel ends. Tilted cuts excluded.
- No seams or waste.
Common mistakes
- Using diameters as radii, 8x on volume.
- Allowing small above big.
- Mixing cm with inches.
- Reading slant as vertical height.
- Forgetting the divide by 3.
Try it yourself
Quick checks. Answers sit tucked under each question.
Big 8 with small 5 and height 6 cm?
Volume is pi times 6 / 3 times 129 is 810.5 cm3. Slant is root of 45 is 6.71 cm.
Big 5 with small 3 and height 5 cm?
Volume is pi times 5 / 3 times 49 is 256.6 cm3. Slant is root of 29 is 5.39 cm with side 135.34 cm2.
Questions
What is the frustum formula?
V = pi h / 3 (R squared plus R r plus r squared). Here 795.9 cm3.
How is slant figured?
Root of (R minus r) squared plus h squared. Here 10.2 cm.
How is side area figured?
Pi (R plus r) times slant. Here 320.38 cm2.
What happens with zero small radius?
At zero small radius the frustum is a cone pi R squared h / 3.
Can radii match?
No. Radii must differ with small below big. Equal radii give a cylinder.
What happens if I double the size?
Volume cubes with size while area squares. Double all and volume grows eightfold.