Surface Area | Sphere, Cylinder & Cone

Surface Area Calculator: Sphere, Cylinder, Cone & More

How to calculate surface area of sphere, cylinder, cone, and 7 more solids. Free calculator with volume and lateral area. Logic Trace, formulas, dimension-highlighted schematics. Trusted by students and educators. No sign-up.

What This Surface Area Calculator Does

Purpose & Use Cases

This surface area calculator computes surface area, volume, and lateral area for 10 common 3D solids. Use it to calculate surface area of a sphere, cylinder surface area formula, cone surface area, or any supported shape. Choose meters, feet, or inches. Ideal for geometry homework, packaging, material estimates, and learning how to find surface area. Lateral area (sides only) is shown for cylinders, cones, frustums, and pyramids.

Shapes:

Sphere, cone, cube, cylinder, rectangular tank, capsule, spherical cap, conical frustum, ellipsoid, square pyramid.
Outputs:

Surface area (A), volume (V), lateral area (L when applicable).

Surface Area and Volume Formulas

Sphere

The surface area of a sphere is the total area of its curved surface; the volume is the space it encloses. Use radius (r) or diameter (d); since d = 2r, you can write A = πd² and V = πd³/6.

A = 4\pi r^2

V = \frac{4}{3}\pi r^3

Cylinder

The surface area of a closed cylinder can be calculated by summing the total areas of its base and lateral surface: two circular bases plus the curved side (unwrapped as a rectangle of height h and width 2πr).

A = 2\pi r^2 + 2\pi rh

V = \pi r^2 h

Cone

The surface area of a right circular cone combines the circular base area with the lateral (slant) surface. The slant height is √(r² + h²).

A = \pi r^2 + \pi r\sqrt{r^2+h^2}

V = \frac{1}{3}\pi r^2 h

Cube, Rectangular Tank, Capsule & More

Cube

A cube has six congruent square faces; surface area is six times the area of one face. With edge length a:

A = 6a^2

V = a^3

Rectangular Tank

A rectangular prism has three pairs of opposite rectangular faces; surface area is the sum of all six faces.

A = 2(lw + lh + wh)

V = lwh

Capsule

A capsule (cylinder with hemispherical caps on both ends) combines the lateral surface of the cylinder with the surface area of two hemispheres. Here h is the cylinder height only.

A = 2\pi r(2r + h)

V = \pi r^2 h + \frac{4}{3}\pi r^3

Square Pyramid

A square pyramid has one square base and four triangular lateral faces meeting at the apex. With base edge a and height h, the slant height appears in the lateral area.

A = a^2 + 2a\sqrt{\frac{a^2}{4}+h^2}

V = \frac{1}{3}a^2 h

Spherical Cap, Frustum & Ellipsoid

Spherical Cap

A spherical cap is the portion of a sphere cut off by a plane; its surface area is the curved dome portion (excluding the circular base). Use base radius (r), ball radius (R), and height (h), provide any two to compute the third.

A = 2\pi R h

V = \frac{\pi h^2(3R-h)}{3}

Conical Frustum

A frustum is a cone with its tip cut off by a plane parallel to the base. Surface area includes both circular bases and the lateral (slant) surface. With top radius r₁, bottom radius r₂, and height h:

A = \pi(r_1^2 + r_2^2 + (r_1+r_2)\sqrt{(r_1-r_2)^2+h^2})

V = \frac{1}{3}\pi h(r_1^2 + r_1 r_2 + r_2^2)

Ellipsoid

An ellipsoid extends an ellipse to three dimensions with semi-axes a, b, c. Volume is exact; surface area has no closed form in elementary functions, so we use Knud Thomsen's approximation.

V = \frac{4}{3}\pi abc

How to Use This Calculator

Select a shape from the dropdown, sphere, cone, cube, cylinder, rectangular tank, capsule, spherical cap, conical frustum, ellipsoid, or square pyramid. Enter the required dimensions in the input fields; for a cylinder that means radius and height, while a spherical cap accepts any two of base radius (r), ball radius (R), and height (h) and solves for the third. Choose your preferred length unit (meters, feet, inches, centimeters, and more) and all outputs display in that unit squared for area or cubed for volume. The results panel shows total surface area, volume, and lateral area when applicable, lateral area represents the sides only, excluding bases. The SVG schematic highlights whichever dimension you are currently editing, making it easy to verify inputs visually. Open the Logic Trace to see the formula applied step by step with your substituted values, and use Copy Results to export surface area, volume, and lateral area for reports or homework.

Lateral Area and When to Use Each Formula

Lateral area is the surface area of the sides only, excluding base(s). Cylinders have lateral area 2πrh; cones have πr√(r²+h²). Spheres and cubes do not have a separate lateral area. Use the surface area formula for a sphere when wrapping a ball or sizing a spherical tank; the cylinder formula for pipes and cans; the cone formula for traffic cones or party hats; the rectangular tank formula for boxes and rooms.

Surface Area Calculator FAQ

How do I calculate surface area of a sphere, cylinder, or cone?

Sphere:

A = 4\pi r^2

Cylinder (closed):

A = 2\pi r^2 + 2\pi rh

Cone:

A = \pi r^2 + \pi r\sqrt{r^2+h^2}

Enter radius and height; results show surface area, volume, and lateral area (when applicable) in your chosen unit.

What is lateral area and when is it shown?

Lateral area is the surface area of the sides only, excluding base(s). Cylinders, cones, frustums, and pyramids have distinct lateral area. Spheres and cubes do not; their total surface area is all-encompassing.

Why is ellipsoid surface area approximate?

Ellipsoid surface area has no closed form in elementary functions. We use Knud Thomsen's approximation for accuracy. Volume is exact:

V = \frac{4}{3}\pi abc

What units are supported?

Meters, feet, inches, and 8 other length units. All outputs (surface area, volume, lateral area) display in your selected unit squared or cubed (e.g. m², ft², in³).

Surface Area Calculator