The Cone Calculator converts base radius and perpendicular height into volume, slant height, lateral surface area and total surface area for a right circular cone. The apex sits directly above the center of the circular base. The page redraws a labelled figure with radius, height and slant marked to the entered values.
Enter radius and height, or supply slant height when a net problem gives the generators instead. Related pages cover the Cylinder Calculator for the non-tapering solid with the same base, the Conical Frustum Calculator when the tip is cut off, and the Pyramid Calculator for a polygonal base.
Calculate the volume of a cone
Volume of a right circular cone is one-third the volume of a cylinder with the same base radius and height. The Cone Calculator multiplies (1/3) * pi * r^2 by height. Radius and height must be positive. Diameter input halves before volume runs.
V = (1/3) * pi * r^2 * hThe one-third factor is the usual source of exam errors: students who forget it report three times the true volume. Volume still scales with r squared and with h. Doubling radius multiplies volume by four; doubling height multiplies volume by two.
For the reference case r = 3 and h = 4:
V = (1/3) * pi * 9 * 4 = 12 * pi approx 37.70Exact form 12 * pi pairs cleanly with the slant-height 5 triple below. When volume and radius are known, height recovers as 3 * V / (pi * r^2). When volume and height are known, radius recovers as the positive square root of 3 * V / (pi * h).
An oblique cone with the same base area and the same perpendicular height has the same volume, but the slant and surface formulas on this page assume the right circular case.
Calculate the slant height
Slant height is the distance from any point on the base circle to the apex measured along the cone's side. In a right circular cone it is the hypotenuse of the right triangle formed by radius and perpendicular height. The Cone Calculator applies the Pythagorean relation.
l = sqrt(r^2 + h^2)Slant height is required for lateral surface area. Perpendicular height alone is not enough for the curved skin. When a net gives slant and radius, height recovers as sqrt(l^2 - r^2) provided l is longer than r.
For r = 3 and h = 4:
l = sqrt(9 + 16) = sqrt(25) = 5The 3-4-5 triple is intentional: it matches the Pythagorean classroom fixture and keeps later surface numbers tidy. If a problem states generators of length 5 on a base of radius 3, height is 4 without another measurement.
Calculate lateral and total surface area
Lateral surface area is the curved sector that forms when the cone's side is unrolled. Total surface area adds the circular base. The Cone Calculator reports both. Lateral area uses pi * r * l with slant height l, not perpendicular height.
Lateral SA = pi * r * l
Total SA = pi * r * (r + l) = pi * r^2 + pi * r * lUsing h in place of l under-counts the wall whenever the cone is not degenerate. Base area is pi * r^2. Open cones that omit the base use lateral area alone; closed solids use total area.
For r = 3, h = 4, l = 5:
Lateral SA = pi * 3 * 5 = 15 * pi approx 47.12
Base area = pi * 9 = 9 * pi approx 28.27
Total SA = 24 * pi approx 75.40A conical frustum keeps two circular ends after the tip is removed; its lateral area uses both radii and a different slant. Link to the Conical Frustum Calculator for truncated cones such as buckets and lampshades.
Solve a cone with radius 3 and height 4
Take r = 3 and h = 4 as the reference cone. Slant height is 5. Volume and surface areas factor with pi. Diameter is 6. 1. Slant height
l = sqrt(3^2 + 4^2) = 52. Volume
V = (1/3) * pi * 9 * 4 = 12 * pi approx 37.703. Lateral surface area
Lateral SA = pi * 3 * 5 = 15 * pi approx 47.124. Total surface area
Total SA = pi * 3 * (3 + 5) = 24 * pi approx 75.40A cylinder with the same r = 3 and h = 4 has volume 36 * pi, three times the cone. The same cylinder's total surface area is 2 * pi * 3 * (3 + 4) = 42 * pi, which is not three times the cone's total area because the side shapes differ.
Unroll the lateral sector
Lateral surface area pi * r * l matches a sector with radius equal to the slant height l and arc length equal to the base circumference 2 * pi * r. For r = 3 and l = 5, arc length is 6 * pi and lateral SA is 15 * pi.
The sector angle in radians is (2 * pi * r) / l = 6 * pi / 5; in degrees that is 216 degrees. Nets cut that sector plus a separate disk of radius r when the base is included.
Starting from slant and radius recovers height: h = sqrt(l^2 - r^2). For l = 5 and r = 3, h = 4, which returns volume 12 * pi. Starting from volume and radius recovers height with h = 3 * V / (pi * r^2). Those rearrangements keep the Cone Calculator aligned with worksheet data that omit one of the three lengths r, h and l.
Unit note: radius, height and slant share one length unit. Volume uses cube units; lateral and total surface areas use square units. A frustum (truncated cone) needs two radii; move to the Conical Frustum Calculator when the tip is missing.
Frequently asked questions
What is cone volume?
Cone volume is V = (1/3) * pi * r^2 * h. For r = 3 and h = 4, V = 12 * pi approx 37.70.
How do I find slant height?
Slant height is l = sqrt(r^2 + h^2). For r = 3 and h = 4, l = 5.
What is lateral surface area of a cone?
Lateral surface area is pi * r * l using slant height. For r = 3 and l = 5, lateral SA = 15 * pi approx 47.12.
What is total surface area?
Total surface area is pi * r * (r + l), which adds the base. For the reference cone, total SA = 24 * pi approx 75.40.
Why is volume one-third of a cylinder?
A cone and a cylinder with the same base and height satisfy V_cone = (1/3) * V_cylinder. Forgetting the one-third is the most common error.
Can the Cone Calculator start from slant height?
Yes. With radius and slant known, height is sqrt(l^2 - r^2), then volume and surface areas follow.
What is a conical frustum?
A conical frustum is a cone with the tip cut off parallel to the base. Use the Conical Frustum Calculator for two radii and a height.
What units does the Cone Calculator use?
Radius, height and slant share one length unit. Volume uses cube units; surface areas use square units.
What is the reference cone volume?
For r = 3 and h = 4, volume is 12 * pi approx 37.70 with slant height 5.
Why must lateral area use slant height?
The unrolled side is a sector whose arc relates to the base circumference and whose radius is the generator length l, not the perpendicular height h.
Summary
Cone Calculator returns volume, slant height, lateral surface area and total surface area from base radius and height, with substituted arithmetic and a labelled figure. Worked numbers match r = 3 and h = 4: slant 5, V = 12 * pi and total SA = 24 * pi.
Use the Cylinder Calculator for the matching non-tapering solid and the Conical Frustum Calculator when the apex is removed.