Lateral Area Of Cone Calculator - Slant Height and Radius

A lateral area of cone calculator finds the curved side area of a right circular cone, leaving out the base circle. It is useful for geometry homework, sheet metal, paper craft, conical tanks, funnels, traffic cones, and any other cone-shaped surface where the side wall is the part that needs to be covered or quoted. The tool returns the lateral area, slant height, base area, and total surface area in one pass.

• • Geometry homework: Confirm the curved side area in problems that separate the lateral area from the base.
• • Sheet metal and paper craft: Estimate how much material is needed for the side of a funnel, lampshade, planter, or party hat.
• • Conical tanks and hoppers: Plan coatings, paint, or liner for the curved side of a cone-shaped tank, hopper, or roof.
• • Everyday cone objects: Work out the side area of waffle cones, megaphones, traffic cones, and other real cone shapes.

The result is an area, not a length or a volume. Keep the radius and the height in the same unit, and the calculator returns the area in the matching square unit, such as square centimeters, square meters, square inches, or square feet.

The page assumes a right circular cone, where the apex sits directly over the center of the circular base. If the shape is an oblique cone or a truncated cone, the formula and the supporting values are different.

If the same cone also needs its inside capacity, the Cone Volume Calculator gives the matching (1/3) pi r^2 h volume in the same units.

The calculator adds the side of the cone as a single curved surface, then derives the slant height from the right triangle inside the cone.

• 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: Slant height, the diagonal from the apex to a point on the base rim.
• A_lateral: Lateral area, equal to pi times the radius times the slant height.

The lateral area formula pi * r * l is the only step you need once the slant height is known. Internally, the calculator keeps the full double-precision value, then rounds the displayed outputs to two decimals so the numbers are easy to read.

This page focuses on the standard radius-and-height case taught in geometry classes, so the input panel only accepts a base dimension and a perpendicular height. If a drawing gives you the slant height directly, the same formula A = pi * r * l applies and you can finish the calculation by hand; the slant height result shown here is for auditing the radius-and-height path, not for use as an input.

According to Wolfram MathWorld, the lateral surface area of a right circular cone is A = pi * r * s, where s is the slant height, and s = sqrt(r^2 + h^2) for a cone with base radius r and vertical height h.

When the cone is cut by a parallel plane to form a truncated cone, the Frustum Cone Area computes the slant side, base, top, and total area from R, r, and H in one pass.

These four ideas decide whether the formula matches the cone you are measuring.

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 the slant side. A common mistake is to use a measurement taken along the outside of the cone, which is actually the slant height.

When the side wall and the base are exposed, use the total surface area. When the base sits on a table, sits in soil, or is open, the lateral area is the value you actually need.

For other solids such as cubes, cylinders, square pyramids, and spheres, the Surface Area Calculator keeps each shape-specific formula in a single place.

Use the calculator with matching length units, and double-check whether you have the radius, diameter, or slant height in hand before you start.

1. 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. 2 Enter the base dimension: Type the radius or diameter value in the same unit you will use for the height.
3. 3 Enter the height: Use the perpendicular height from the apex to the base plane, not the slant side.
4. 4 Pick the unit: Select cm, m, in, or ft so the area output is shown in the matching square unit.
5. 5 Read the lateral area: The primary result is the curved side area, with slant height, base area, and total area shown underneath for auditing.

Suppose a funnel has a base radius of 6 cm and a height of 10 cm. Enter Radius, dimension 6, height 10, unit cm. The calculator returns slant height ~ 11.66 cm, lateral area ~ 219.91 cm^2, base area ~ 113.10 cm^2, and total area ~ 333.01 cm^2. If you are quoting foil for the side wall only, use 219.91 cm^2; if you are quoting foil for the entire outside, add the base area.

After reading square centimeters or square inches, the Area Converter turns the finished area into another square unit if your quote needs a different system.

Splitting the answer into lateral area, base area, slant height, and total area makes the result easier to use in real decisions.

• • Side-only area in one tap: Get the curved side area without doing the pi * r * l step by hand, and skip the base circle you may not need.
• • Audit-friendly values: Read slant height, base area, and total area in the same panel, so each formula step is visible to a teacher or a reviewer.
• • 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 matching square unit for the area outputs, which avoids manual unit conversion.
• • Material estimates: Use the lateral area to size sheet metal, paper, fabric, paint, or liner for the cone's side wall.

The calculator works for any right circular cone as long as you have a base dimension (radius or diameter) and the perpendicular height. The slant height is derived inside the page from those two inputs, so the page does not handle cones given by slant height alone. It also assumes the apex is over the center of the base, so it does not handle oblique cones, frustums, or cones whose apex has been cut off.

If you need volume, slant height from an angle, or surface area for a different 3D shape, 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 handles the same radius-style inputs for a truncated cone volume.

The formula is short, but a few measurement choices decide whether the final area fits the cone you are actually covering.

• • This calculator is for a right circular cone. An oblique cone, where the apex is not centered over the base, has 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 slant height first.

Real material takeoffs usually need waste, seams, or overlap on top of the geometric area. 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 or sits on a floor that hides the base, use the lateral area as the covering area. If the base is exposed, add the base area to the lateral area to get the total outside surface.

According to OpenStax Prealgebra 2e, the total surface area of a right circular cone is the sum of the base area pi * r^2 and the lateral area pi * r * s, where s is the slant height, and the same decomposition is what this page shows in the result panel so the side-only and full-surface readings stay auditable side by side.

For a six-sided pyramid with a regular hexagon base, the Hexagonal Pyramid Surface Area Calculator separates total, lateral, and base area the same way this cone page does.