Calculate the missing top radius, bottom radius, height, or angle of a conical frustum using three known values and unit selections.
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Conical Frustum Angle Formula
The calculator uses the right-triangle relationship between the height of the frustum and the difference between the top and bottom radii.
theta = arctan((R_t - R_b) / H)
R_t = R_b + H*tan(theta)
R_b = R_t - H*tan(theta)
H = (R_t - R_b) / tan(theta)
- theta = frustum angle, measured from the vertical height direction
- R_t = top radius
- R_b = bottom radius
- H = vertical height of the conical frustum
- tan = tangent function
- arctan = inverse tangent function
Enter any three values to solve for the fourth. If the angle is missing, the calculator finds the angle from the radius difference divided by height. If a radius is missing, it rearranges the tangent formula to solve for that radius. If height is missing, it divides the radius difference by the tangent of the angle.
Length inputs can use inches, feet, centimeters, or meters. The calculator converts all length values to a common unit before calculating, then converts the result back to the selected output unit. The angle can be entered or returned in degrees or radians.
Common Frustum Angle Interpretations
The angle from this calculator is based on the change in radius over the vertical height. Positive and negative results show which radius is larger.
| Result | Meaning | Shape Description |
|---|---|---|
| theta > 0 | Top radius is larger than bottom radius | Frustum widens upward |
| theta = 0 | Top and bottom radii are equal | Cylinder, no taper |
| theta < 0 | Top radius is smaller than bottom radius | Frustum narrows upward |
Angle Unit Reference
| Angle Unit | Conversion | Example |
|---|---|---|
| Degrees | degrees = radians × 180 / pi | 0.5236 rad ≈ 30° |
| Radians | radians = degrees × pi / 180 | 45° ≈ 0.7854 rad |
Example Calculations
Example 1: Find the frustum angle
Suppose the top radius is 8 in, the bottom radius is 5 in, and the height is 12 in.
theta = arctan((8 - 5) / 12)
theta = arctan(0.25) = 14.0362°
The frustum angle is about 14.0362°.
Example 2: Find the top radius
Suppose the bottom radius is 10 cm, the height is 20 cm, and the angle is 5°.
R_t = 10 + 20*tan(5°)
R_t = 10 + 1.7498 = 11.7498 cm
The top radius is about 11.7498 cm.
FAQ
What angle does this conical frustum calculator measure?
It measures the taper angle using the vertical height and the difference between the top and bottom radii. In other words, it is the angle between the side taper and the vertical height direction, based on tan(theta) = radius difference / height.
Why can the angle be negative?
The angle is negative when the top radius is smaller than the bottom radius. This happens because the formula uses top radius minus bottom radius. A negative result does not mean the shape is invalid. It only shows the direction of the taper.
Can the top radius and bottom radius use different units?
Yes. You can enter the radii and height using different supported length units. The calculator converts them to a common base unit before applying the formula, then returns the missing value in the unit selected for that field.
