Slant Height of a Cone Calculator

How to use this slant height of a cone calculator

What this calculator does

This calculator finds the slant height of a right circular cone from its base radius and perpendicular height. It also reports the base angle and the apex half-angle formed by the same right-triangle cross-section. The result is an exact geometric identity for an ideal right cone; it does not account for wall thickness, a rounded tip, a tilted axis, manufacturing tolerances, or an irregular base.

When to use it

Use the calculator when sizing a paper or sheet-metal cone, checking an ice-cream or traffic-cone dimension, preparing a geometry exercise, or obtaining the slant length needed for a later lateral-surface-area calculation. The underlying relationship is the same one described in the Wolfram MathWorld definition of slant height.

How to calculate

1. The calculator opens with a complete demonstration: radius 2.5 cm and height 15 cm. Its result and a validated example Excel workbook are available immediately.
2. Replace Radius (r) with the distance from the center of the circular base to its rim.
3. Replace Height (h) with the perpendicular distance from the base plane to the apex.
4. Choose the Length unit. Changing it converts both entered dimensions rather than only relabeling them.
5. Read Slant height (l), Base angle (α), and Apex angle (β), then use the details table to check the quantities together.
6. Select Download Excel to export the current inputs and results. Reset clears the demonstration and all calculated content; Excel export remains unavailable until a new complete valid radius and height are entered.

Input guide

Radius (r) is required, must be a finite positive number, and uses the selected length unit. Enter plain dot-decimal notation such as 2.5; standard comma grouping such as 1,250.5 is accepted, while decimal-comma notation such as 2,5 is rejected to prevent ambiguity. A larger radius increases the slant height and widens the apex half-angle. Do not enter the diameter by mistake; divide a measured diameter by two first.

Height (h) is required, positive, and measured perpendicular to the base. A realistic example is 15 cm. Increasing height increases slant height and makes the cone steeper, which raises the base angle and reduces the apex half-angle. Do not substitute the sloping side itself for the vertical height.

Length unit applies to radius, height, and slant height. Available choices are millimeters, centimeters, meters, inches, and feet. The conversion preserves the physical size, so switching 2.5 cm to millimeters produces 25 mm. The angles remain in degrees because they are independent of the chosen length unit.

Output guide

Slant height (l) is the distance along the cone's side from the base rim to the apex. It is shown in the selected unit and is always greater than either positive leg of the right triangle. Base angle (α) is measured between the base radius and the sloping side; taller, narrower cones have larger base angles. Apex angle (β) is the half-angle between the central axis and the sloping side, not the full tip angle. The Angle check pill confirms that α + β equals 90°. The formula substitution line shows the current radius and height placed into the square-root identity, and the interpretation note classifies the cone as narrow, moderate, or broad from its radius-to-height ratio. The details table restates every model value and its geometric meaning.

Worked example

With the demonstration values r = 2.5 cm and h = 15 cm, square both legs and add them: 2.5² + 15² = 6.25 + 225 = 231.25. Taking the square root gives l = √231.25 = 15.2069... cm, displayed as 15.21 cm. The base angle is arctan(15 ÷ 2.5) = 80.54°, and the apex half-angle is arctan(2.5 ÷ 15) = 9.46°. Their displayed sum is 90.00°.