Slant Height Calculator - Radius, Height, and Angle
Use this slant height calculator to enter the cone's base radius or diameter and height and read the slant side, apex angle, and lateral area in any unit.
Slant Height Calculator
Results
What Is the Slant Height Calculator?
A slant height calculator finds the diagonal distance from the apex of a right circular cone to a point on the rim of the base. It is useful for cone-shaped parts such as funnels, lampshades, traffic cones, conical roofs, paper-craft cones, and waffle cones, and the value drives the cone's lateral area, total surface area, and apex angles.
- • Geometry homework: Confirm the diagonal side of a right circular cone in problems that give you the radius and the height.
- • Sheet metal and paper craft: Plan the diagonal length of a cone's side wall before cutting foil, paper, fabric, or sheet metal.
- • Funnels, lampshades, and hats: Estimate the side length of a party hat, lampshade, traffic cone, or waffle cone from the base and the height.
- • Cone angle checks: Read the half apex angle and the full apex angle to compare a measured cone to a target spread or taper.
The result is a length, not an area or a volume. Enter the base radius (or diameter) and the vertical height in the same unit, and the calculator returns the diagonal in that unit along with the angles and the side-wall area.
The page assumes a right circular cone, where the apex sits directly above the center of the circular base. Oblique cones, frustums, and regular pyramids need different inputs and the values on this page will not match.
If the same cone also needs its curved side area, the Lateral Area Of Cone Calculator uses this diagonal together with the radius in the pi * r * l formula.
How the Calculator Works
The calculator takes the right triangle that lives inside the cone, where the radius, the vertical height, and the diagonal are the three legs of the triangle, and applies the Pythagorean theorem to solve for the diagonal.
- r: Radius of the circular base. Enter diameter instead and the calculator divides it by 2.
- h: Perpendicular height from the apex straight down to the base plane.
- l: Diagonal from the apex to a point on the base rim; the hypotenuse of the right triangle.
The Pythagorean relation l = sqrt(r^2 + h^2) is the only step you need once the base radius and the height are in hand. Internally, the calculator keeps the full double-precision value and rounds the displayed numbers to two decimals.
This page focuses on the radius-and-height case taught in geometry classes, so the input panel only accepts a base dimension and a perpendicular height.
Example: r = 3 cm, h = 4 cm (3-4-5 right triangle)
Radius r = 3 cm, height h = 4 cm.
l = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 cm. Half apex angle = atan(3 / 4) ~ 36.87 degrees. Full apex angle ~ 73.74 degrees.
Diagonal = 5.00 cm. Half apex angle ~ 36.87 degrees. Full apex angle ~ 73.74 degrees. Base circumference ~ 18.85 cm. Lateral surface area ~ 47.12 cm^2.
The side is exactly one third longer than the height, which is the same ratio that shows up in the 3-4-5 right triangle. Use 5 cm as the diagonal of the side wall when you cut a flat pattern for this cone.
According to Wolfram MathWorld, the slant height s of a right circular cone with base radius r and vertical height h is s = sqrt(r^2 + h^2), and that same diagonal appears in the lateral surface area A = pi * r * s.
When you also need the inside capacity of the same cone, the Cone Volume Calculator returns the (1/3) pi r^2 h volume in the same units, alongside this diagonal.
Key Concepts Explained
These four ideas decide whether the formula matches the cone you are measuring.
Right circular cone
A cone whose apex sits directly above the center of a circular base. The Pythagorean relation l = sqrt(r^2 + h^2) only applies to this symmetric shape.
Diagonal vs height
The height h is the straight line from the apex to the base plane. The diagonal l is the longer line from the apex to a point on the rim, so l is always greater than h.
Half apex angle
The angle between the cone's central axis and the side. In the right triangle, half apex angle alpha satisfies tan(alpha) = r / h, so alpha = atan(r / h).
Diagonal and lateral area
The lateral surface area of the cone is A = pi * r * l, so a small change in l scales the side-wall area in direct proportion. Knowing l is the shortcut to that area.
Choosing the right height is the easiest place to go wrong. The vertical height is the straight line from the apex to the base plane, not a measurement taken along the outside of the cone.
When the cone is very tall and narrow, the diagonal sits just above the height and the half apex angle is small. When the cone is short and wide, the diagonal sits well above the height and the half apex angle is large.
If the cone is cut by a parallel plane to form a truncated cone, the Frustum Cone Area uses a related side length from R, r, and H for the curved surface of the frustum.
How to Use This Calculator
Use the calculator with matching length units, and double-check whether you have the radius, diameter, or diagonal in hand before you start.
- 1 Pick the input mode: Choose Radius if you measured from the center of the base to its edge, or Diameter if you measured across the full base.
- 2 Enter the base dimension: Type the radius or diameter value in the same unit you will use for the height.
- 3 Enter the height: Use the perpendicular height from the apex to the base plane, not the side.
- 4 Pick the unit: Select cm, m, in, or ft so the diagonal is shown in the same length unit as the inputs.
- 5 Read the diagonal: The primary result is the side length, with the half apex angle, full apex angle, base circumference, base area, and lateral area shown underneath for auditing.
Suppose a paper-craft cone has a base radius of 6 cm and a vertical height of 10 cm. Enter Radius, dimension 6, height 10, unit cm. The calculator returns diagonal ~ 11.66 cm, half apex angle ~ 30.96 degrees, full apex angle ~ 61.93 degrees, base circumference ~ 37.70 cm, base area ~ 113.10 cm^2, and lateral surface area ~ 219.91 cm^2. Use 11.66 cm as the side length of the wedge when you cut a flat pattern for the cone.
For a four-sided pyramid with a square base, the Square Pyramid Volume Calculator applies a similar diagonal-style formula that depends on a slant edge rather than this circular side.
Benefits of This Calculator
Returning the diagonal, the apex angles, the base measurements, and the side-wall area in one panel keeps the result easy to use in real decisions.
- • Side length in one tap: Get the diagonal from the apex to the rim without doing the sqrt(r^2 + h^2) step by hand.
- • Audit-friendly angles: Read the half apex angle and the full apex angle so a teacher or reviewer can confirm the spread of the cone.
- • Radius or diameter input: Switch between radius and diameter from a dropdown instead of dividing by 2 in your head or in a spreadsheet.
- • Metric and imperial units: Enter cm, m, in, or ft and read the diagonal in the same unit, which avoids manual unit conversion.
- • Material and pattern sizing: Use the diagonal with the base circumference to size a flat wedge pattern for paper, foil, fabric, or sheet metal.
The calculator works for any right circular cone as long as you have a base dimension and the perpendicular height. The diagonal is derived from those two inputs, so the page does not handle cones given by the side alone, oblique cones, frustums, or cones whose apex has been cut off.
If you need the volume, lateral area, or total surface area of the same cone, the related calculators in the math-conversion category cover those cases with their own formula and inputs.
When the cone is cut by a parallel plane to form a frustum, the Truncated Cone Volume Calculator reads the same radius-style inputs and reports a related side length for the frustum.
Factors That Affect Your Results
The formula is short, but a few measurement choices decide whether the diagonal fits the cone you are actually cutting.
Height type
Perpendicular height goes straight from the apex to the base plane. A side measurement used as height will inflate the diagonal and the apex angles.
Unit consistency
Mixing centimeters for the radius with inches for the height will distort the diagonal and the angles. Keep both inputs in the same unit before you press Calculate.
Radius vs diameter
Doubling the radius by accident is the most common error. If the value feels too large, switch the input mode from Radius to Diameter and re-enter it.
Slope of the cone
A short and wide cone (large r compared to h) gives a diagonal well above the height and a half apex angle close to 90 degrees. A tall and narrow cone (small r compared to h) gives a diagonal close to the height and a small half apex angle.
Rounding when reused
Rounding the diagonal before recomputing the lateral area or the angle can shift the last decimal place. Keep full precision for any downstream calculation.
- • This calculator is for a right circular cone. An oblique cone, where the apex is not centered over the base, has a different side and a different lateral area formula, and is not handled by this page.
- • The result panel shows rounded values, so the last digit can differ from a hand calculation that uses a truncated value of pi or that rounds the diagonal first.
Real material takeoffs usually need waste, seams, or overlap on top of the geometric length. Treat the calculator as a formula reference for the ideal solid and add your own allowance for the physical part.
If the shape is open at the top, use the diagonal as the side-wall length. If the cone is closed, add an allowance for the base and the cap when you cut a flat pattern.
According to OpenStax Prealgebra 2e, the slant height of a cone is the distance from the apex to any point on the edge of the circular base, and it forms the hypotenuse of the right triangle that also contains the radius and the vertical height
For other solids such as cubes, cylinders, square pyramids, and spheres, the Surface Area Calculator keeps each shape-specific formula in a single place.
Frequently Asked Questions
Q: What is the slant height of a cone?
A: It is the diagonal distance from the apex of a right circular cone straight down to a point on the rim of the circular base. The value is the hypotenuse of the right triangle formed by the radius, the vertical height, and the side of the cone.
Q: How do I find the slant height of a cone with radius and height?
A: Square the base radius, square the perpendicular height, add the two squares, and take the square root. In short, l = sqrt(r^2 + h^2). For r = 3 and h = 4, l = sqrt(9 + 16) = sqrt(25) = 5, which is the classic 3-4-5 right triangle.
Q: What is the formula for the slant height of a cone?
A: The slant height of a right circular cone is l = sqrt(r^2 + h^2), where r is the base radius and h is the perpendicular height. The same diagonal also appears in the lateral surface area A = pi * r * l, so it is the bridge between the radius-and-height inputs and the side-wall area.
Q: Is the slant height of a cone longer than the height?
A: Yes, for a right circular cone the side is always longer than the vertical height, because it is the hypotenuse of the right triangle formed with the radius and the height. The two values match only in the limiting case where the radius is zero.
Q: How do I find the slant height of a cone with the diameter?
A: Divide the diameter by 2 to get the radius, then use l = sqrt(r^2 + h^2). The calculator does this when you switch the 'Calculate Using' dropdown to Diameter, so you do not have to do the division by hand before pressing Calculate.
Q: What is the slant height of a cone used for?
A: It is the supporting value behind the cone's lateral surface area A = pi * r * l and the flat pattern of the side wall. It also drives the half apex angle alpha = atan(r / h) and the full apex angle 2 alpha, which describe how wide or how narrow the cone spreads from the apex.