Calculate the lateral, base, and total surface area of a cone from radius and height or slant height. Includes apex angle, unrolled sector angle, presets, and reference table.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a single point called the apex. Its surface area has two components: the lateral (side) surface — the curved part you can see when looking at an ice cream cone — and the circular base.
The lateral surface area of a cone is πrl, where r is the base radius and l is the slant height. The slant height is the straight-line distance from the edge of the base to the apex, found via the Pythagorean theorem: l = √(r² + h²), where h is the perpendicular height. The base area is simply πr². Adding them gives the total surface area: SA = πr(l + r).
Cones appear everywhere in daily life and engineering. Traffic cones, party hats, ice cream cones, funnels, and volcanic cinder cones are all conical shapes. In manufacturing, knowing the lateral surface area tells you how much material you need to make the cone — when unrolled, the lateral surface forms a circular sector with radius l and arc length 2πr.
An interesting geometric property is the apex angle — the full opening angle at the tip — which equals 2·arctan(r/h). It ranges from nearly 0° (a needle-thin spike) to nearly 180° (a nearly flat disk). The sector angle when the cone is unrolled is (r/l) × 360°, which determines how much of a full circle the flat pattern occupies.
This calculator supports three input modes: radius + height, radius + slant height, or diameter + height. You can toggle whether the base is included. Presets for party hats, traffic cones, and more let you explore instantly, while breakdown bars and a reference table make comparison easy.
Cone problems often require several linked measurements, not just a single area formula. If you know radius and height, you first need the slant height before you can find the lateral surface; if you know diameter instead of radius, you need to convert that as well. This calculator handles those steps automatically and shows the full breakdown into lateral area, base area, total surface area, volume, and cone angles in one place.
That makes it useful for both classroom geometry and practical design work. Anyone making party hats, funnels, traffic cones, or sheet-metal patterns can compare lateral-only versus closed-base totals, while students can see how the right-triangle relationship between radius, height, and slant height drives every result.
Slant Height: l = √(r² + h²) Lateral SA: πrl Base Area: πr² Total SA: πr(l + r) Volume: (1/3)πr²h Apex Angle: 2·arctan(r/h) Sector Angle (unrolled): (r/l) × 360° Base Circumference: 2πr
Result: Slant Height = 13 cm, Lateral SA ≈ 204.20 cm², Total SA ≈ 282.74 cm²
Choose Radius + Height mode, enter val1 = 5 and val2 = 12, and keep the base included. The calculator finds the slant height from √(5² + 12²) = 13 cm, then uses πrl for the lateral area: π × 5 × 13 = 65π ≈ 204.20 cm². Adding the base area π × 5² ≈ 78.54 cm² gives a total surface area of about 282.74 cm². The same setup also reports the cone volume and the unrolled sector angle.
The curved surface of a right circular cone is easier to understand when you imagine cutting it from apex to base and flattening it. Once unrolled, that curved side becomes a sector of a circle. The sector radius is the cone's slant height, and the arc length of that sector equals the circumference of the base, $2pi r$. That is why the lateral surface area formula is $pi r l$: it is the area of a sector whose size depends on both the base radius and the slant height.
This perspective also explains why steep, narrow cones can have a surprisingly large lateral area. Even if the base radius is small, a long slant height stretches the sector outward and increases the area of the material needed to wrap the cone.
Most mistakes in cone geometry come from mixing up height and slant height. The perpendicular height runs straight from the base center to the apex, while the slant height lies along the outside surface. If a problem gives radius and height, you must first use the Pythagorean theorem to find $l = sqrt{r^2 + h^2}$. If a problem gives diameter and height, halve the diameter before substituting into any surface-area formula.
This calculator makes those conversions explicit by supporting radius + height, radius + slant height, and diameter + height modes. That is especially useful for worksheet problems where the cone might be described in different ways depending on whether the focus is geometry, manufacturing, or drafting.
Cone surface area is not just a classroom topic. Costume makers use it to size party hats, food packaging teams use it for cone sleeves and dispensers, and metal fabricators use it when rolling flat stock into hoppers or transition pieces. In those cases, the lateral area often matters more than the total because the circular base may be open.
The extra outputs in this calculator help with those applications. Apex angle describes how sharp the cone is, while the sector angle tells you how large a wedge to cut from a flat sheet before rolling it into shape. Together, those values make the tool more useful than a formula-only answer.
Total SA = πr(l + r), where r is the base radius and l is the slant height. The lateral part is πrl and the base is πr².
Use the Pythagorean theorem: l = √(r² + h²), where r is the radius and h is the height. The radius, height, and slant height form a right triangle.
Height (h) is the perpendicular distance from base to apex, measured inside the cone. Slant height (l) is the distance along the surface from the base edge to the apex. l is always greater than h.
Cut a sector from a circle of radius = slant height (l). The sector angle should be (r/l) × 360°. Roll it into a cone shape and attach the straight edges.
The apex angle is the full opening angle at the tip: 2·arctan(r/h). A narrow cone (like a needle) has a small apex angle; a flat cone approaches 180°.
A cone has 1/3 the volume of a cylinder with the same base and height. Its lateral surface area is also smaller since the slant height replaces the full height: πrl vs. 2πrh.