Slant Height Of Cone Calculator - Diagonal Side From Radius and Height
Use this slant height of cone calculator to enter the base radius or diameter and height, and read the slant side, apex angle, base area, and lateral area.
Slant Height Of Cone Calculator
Results
What Is the Slant Height of a Cone?
Slant height of cone is the diagonal distance from the apex of a right circular cone down to a point on the rim of the circular base. It is the hypotenuse of the right triangle that lives inside the cone, with the base radius and the perpendicular height as the two legs, and it is the single value you need to compute the cone's lateral surface area, full apex angle, and flat side-wall pattern.
- • Geometry homework and exam problems: Solve textbook problems that give a base radius and a perpendicular height and ask for the side of a right circular cone.
- • Lampshades, traffic cones, party hats, and waffle cones: Estimate the side length of a real cone-shaped object before cutting paper, foil, fabric, or sheet metal for the side wall.
- • Lateral and total surface area: Plug the slant side into the lateral area formula pi * r * l to size the curved wall of a cone-shaped part.
The result is a length, not an area or a volume. Enter the base radius (or the diameter) and the perpendicular height in the same unit, and the page returns the diagonal in that unit.
The page assumes a right circular cone, where the apex sits directly above the center of the circular base. Oblique cones, frustums, and pyramids use different side-length formulas, so the values on this page will not match those shapes.
If the same cone also needs its curved side area, the Lateral Area Of Cone Calculator plugs the slant side and the base radius into the pi * r * l formula and returns the wall area and the total surface area in one panel.
How the Slant Height of Cone Calculator Works
The calculator takes the right triangle that lives inside the cone, where the radius, the vertical height, and the slant side are the three legs, and applies the Pythagorean theorem to solve for the diagonal. The base measurements, the apex angles, and the lateral area are then derived from that single diagonal.
- r: Base radius of the cone. Enter diameter instead and the calculator divides it by 2.
- h: Perpendicular height from the apex straight down to the base plane.
- l: Diagonal slant side from the apex to a point on the 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. The same diagonal drives the half apex angle alpha = atan(r / h), the full apex angle 2 alpha, and the lateral area pi * r * l.
Example: r = 3 cm, h = 4 cm (3-4-5 right triangle)
Base radius r = 3 cm, perpendicular 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. Lateral area = pi * 3 * 5 ≈ 47.12 cm^2.
Slant side = 5.0000 cm. Half apex angle ≈ 36.8699 degrees. Full apex angle ≈ 73.7398 degrees. Lateral area ≈ 47.1239 cm^2.
The slant side is one third longer than the perpendicular height, which is the same 3-4-5 ratio that shows up in any right triangle with legs 3 and 4. Use 5 cm as the side length of the cone wall when you cut a flat wedge pattern.
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 the perpendicular height is the unknown instead of the slant side, the Height Of Cone Calculator takes the same radius and the slant side and reports the perpendicular height of the same right circular cone.
Key Concepts Behind the Slant Height of a Cone
Four ideas decide whether the diagonal on this page matches the cone you are actually 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.
Slant side versus perpendicular height
The perpendicular height h is the straight vertical line from the apex to the base plane. The slant side 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 slant side. Half apex angle alpha satisfies tan(alpha) = r / h, so alpha = atan(r / h) and the full apex angle is 2 alpha.
Why the slant side drives the lateral area
The lateral surface area of the cone is A = pi * r * l, so a small change in the slant side scales the side-wall area in direct proportion.
Choosing the right height is the easiest place to go wrong. The perpendicular 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 slant side sits just above the perpendicular height and the half apex angle is small. When the cone is short and wide, the slant side sits well above the perpendicular 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 the larger radius, the smaller top radius, and the height to return the curved surface of the frustum.
How to Use This Slant Height of Cone Calculator
Run the calculator in five short steps with the radius and the height in the same length unit, then read the diagonal and the supporting values in the result panel.
- 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 and height: Type the radius or diameter value, then the perpendicular height from the apex to the base plane. Use the same length unit for both.
- 3 Pick the length unit: Select cm, m, in, or ft so the diagonal and the supporting values match the units you used for the inputs.
- 4 Read the slant side and supporting values: The primary result is the slant side l, with the half apex angle, the full apex angle, the base area, and the lateral area shown underneath for auditing.
Suppose a paper-craft cone has a base radius of 6 cm and a perpendicular height of 10 cm. Select Radius, enter 6 and 10 in cm, and read slant side ≈ 11.6619 cm, half apex angle ≈ 30.9638 degrees, full apex angle ≈ 61.9275 degrees, base area ≈ 113.0973 cm^2, and lateral area ≈ 219.9115 cm^2.
When you also need the inside capacity of the same cone, the Cone Volume Calculator applies (1/3) pi r^2 h with the same radius and perpendicular height and reports the volume alongside the slant side on this page.
Benefits of This Slant Height of Cone 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 work.
- • Slant side 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 length units: Enter cm, m, in, or ft and read the slant side in the same unit, which avoids manual unit conversion for craft, school, fabrication, and engineering tasks.
The calculator works for any right circular cone as long as you have a base dimension and the perpendicular height. The page does not handle cones given by the side alone, oblique cones, frustums, or cones with the apex cut off.
For other solids such as cubes, cylinders, square pyramids, and spheres, the Surface Area Calculator keeps each shape-specific surface area formula in a single place.
Factors That Affect Your Slant Height of Cone Result
The Pythagorean formula is short, but a few measurement choices decide whether the diagonal fits the cone you are actually cutting.
Height type
The perpendicular height goes straight from the apex to the base plane. A side-wall measurement used as height will inflate the diagonal and the apex angles.
Unit consistency
Mixing centimeters for the radius with inches for the height distorts the diagonal and the angles. Keep both inputs in the same unit before pressing Calculate.
Radius versus diameter
Doubling the radius by accident is the most common error. If the value feels too large, switch the input mode to Diameter and re-enter it.
Slope of the cone
A short and wide cone (large r compared to h) gives a slant side well above the height and a half apex angle close to 90 degrees. A tall and narrow cone (small r) gives a slant side close to the height and a small half apex angle.
Rounding when the value is reused
Rounding the slant side before recomputing the lateral area or the angle can shift the last decimal place. Keep full precision for downstream calculations.
- • This calculator is for a right circular cone. An oblique cone has a different side length and a different lateral area formula, and the Pythagorean relation does not apply.
- • The result panel shows rounded values, so the last digit can differ from a hand calculation that uses a truncated value of pi or rounds the slant side first.
Real material takeoffs usually need waste, seams, or overlap on top of the geometric length. Add your own allowance for the physical part on top of the formula result.
According to Wikipedia Cone, in a right circular cone the slant height is the distance from the apex to any point on the edge of the circular base, and it is the hypotenuse of the right triangle that also contains the base radius and the perpendicular height
When the cone is cut by a parallel plane to form a frustum, the Truncated Cone Volume Calculator reads the larger base radius, the smaller top radius, and the perpendicular height to return the truncated volume in the same length unit.
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 base radius, the perpendicular height, and the side of the cone.
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: How do you 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: How do you 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: Is the slant height of a cone longer than the perpendicular height?
A: Yes, for a right circular cone the slant side is always longer than the perpendicular 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: 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.