A cone combines a circular base with a single vertex. The most useful distinction is between perpendicular height h, which controls volume, and slant height ℓ, which controls the curved surface area of a right cone.
What the cone measurements mean
For a right circular cone, the apex is directly above the center of the circular base. The base radius is r, the perpendicular height is h, and the slant height is ℓ. Those three lengths form a right triangle in a vertical cross-section.
Core cone formulas
The base area is the area of a circle, B = πr². A right cone then follows from four closely related formulas:
V = ⅓πr²hℓ = √(r² + h²)L = πrℓA = πrℓ + πr² = πr(ℓ+r)Why the units change
| Quantity | Formula type | Unit |
|---|---|---|
| r, h, ℓ | length | cm, m, in, ft |
| πr², πrℓ | length × length | cm², m², in², ft² |
| ⅓πr²h | area × length | cm³, m³, in³, ft³ |
Worked example: r = 5, h = 12
Suppose a right circular cone has radius 5 units and perpendicular height 12 units. The cross-section is a 5–12–13 right triangle, so the slant height is exactly 13.
The result is therefore about 314.159 cubic units of volume and 282.743 square units of closed total surface area.
How volume responds to radius and height
The formula V = ⅓πr²h tells you exactly how the dimensions influence volume. Holding height fixed, doubling the radius multiplies volume by four because radius is squared. Holding radius fixed, doubling height only doubles volume.
Volume ×4, when h stays fixed.
Volume ×2, when r stays fixed.
Volume ×9, when h stays fixed.
Volume ×8 because r² contributes ×4 and h contributes ×2.
Right cone vs oblique cone
An oblique cone is tilted rather than centered over its base. Its volume is still ⅓Bh, where h is the perpendicular height. However, radius and perpendicular height alone do not determine a unique lateral surface area or a single slant height for an oblique cone.
| Shape | Volume | Surface-area data |
|---|---|---|
| Right cone | ⅓πr²h | r and h determine ℓ, L, A |
| Oblique cone | ⅓πr²h | additional geometry is required |
| Frustum | ⅓πh(R²+Rr+r²) | R, r, h determine slant and areas |
Frustum formulas
A frustum is what remains after a cone is cut by a plane parallel to its base. Let R be the larger radius, r the smaller radius, and h the perpendicular height.
V = ⅓πh(R² + Rr + r²)ℓ = √((R−r)² + h²)L = π(R+r)ℓA = L + πR² + πr²Open vs closed cone surface area
Total surface area includes the circular base. If the base is open—such as an open-top container—or the object is an ice-cream cone, the exposed area may be only the lateral area L = πrℓ.
| Surface choice | Formula | Includes |
|---|---|---|
| Lateral only | πrℓ | curved side |
| Total closed cone | πrℓ + πr² | curved side + base |
| Closed frustum | π(R+r)ℓ + πR² + πr² | side + both circular bases |
Using the calculator accurately
Choose the shape first. For a right cone, enter the base radius and perpendicular height. For a frustum, enter both radii and the height. Keep all lengths in compatible units; the calculator preserves those units conceptually, so a centimeter input produces square centimeters for area and cubic centimeters for volume.
Radius and height must be greater than zero for a non-degenerate solid.
For a frustum, the top radius cannot exceed the base radius in the model.
Use perpendicular height for volume—not the slant height.
For an oblique cone, only volume is determined by R and h here.
FAQ
For a circular cone, V = ⅓πr²h, where r is the base radius and h is the perpendicular height.
Height is perpendicular to the base. Slant height runs along the side from the vertex to the edge of the base.
A cone and a cylinder with the same base area and perpendicular height have volumes in the ratio 1:3, so the cone's volume is one-third of the matching cylinder's.
Not in general. They determine the volume, but additional information is needed for the oblique lateral geometry.
V = ⅓πh(R² + Rr + r²), where R is the larger radius, r is the smaller radius, and h is the perpendicular height.
Yes. For a closed right cone, total area is πrℓ + πr². Lateral area alone excludes the circular base.