Geometry

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Surface Area Calculator

Surface area and volume of sphere, cone, cube, cylinder, and more. Live solve with Logic Trace.

Surface Area Calculator

Pick a shape, enter dimensions, and read surface area, volume, and lateral area. Results update live.

OR
Derived

Enter radius or diameter. The other updates in real time. The derived field uses a dashed mint border.

Results

Surface Area

314.1593 mยฒ

Volume

523.5988 mยณ

Logic trace

Schematic

dr

Surface Area Calculator: Formula Reference & Tips

Enter dimensions for 10 solidsโ€”surface area, volume, and lateral area update live. Logic Trace shows each formula step. Trusted by students and educators for geometry, packaging, and material estimates.

Quick Workflow

Choose Shape

10 solids: sphere, cone, cube, cylinder, rectangular tank, capsule, spherical cap, frustum, ellipsoid, square pyramid.

Input & Units

Enter dimensions. Spherical cap: any two of r, R, h. Units: m, ft, in, and more.

Logic Trace

See the formula applied step-by-step. Copy to paste the derivation for homework or reports.

Schematic

SVG diagram highlights the dimension you're editing. Copy Results for surface area, volume, lateral area.

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

  • 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).
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.

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 = ฯ€dยฒ and V = ฯ€dยณ/6.
A=4ฯ€r2A = 4\pi r^2
V=43ฯ€r3V = \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 2ฯ€r).
A=2ฯ€r2+2ฯ€rhA = 2\pi r^2 + 2\pi rh
V=ฯ€r2hV = \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 โˆš(rยฒ + hยฒ).
A=ฯ€r2+ฯ€rr2+h2A = \pi r^2 + \pi r\sqrt{r^2+h^2}
V=13ฯ€r2hV = \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=6a2A = 6a^2
V=a3V = 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)A = 2(lw + lh + wh)
V=lwhV = 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=2ฯ€r(2r+h)A = 2\pi r(2r + h)
V=ฯ€r2h+43ฯ€r3V = \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=a2+2aa24+h2A = a^2 + 2a\sqrt{\frac{a^2}{4}+h^2}
V=13a2hV = \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)โ€”provide any two to compute the third.
A=2ฯ€RhA = 2\pi R h
V=ฯ€h2(3Rโˆ’h)3V = \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 rโ‚, bottom radius rโ‚‚, and height h:
A=ฯ€(r12+r22+(r1+r2)(r1โˆ’r2)2+h2)A = \pi(r_1^2 + r_2^2 + (r_1+r_2)\sqrt{(r_1-r_2)^2+h^2})
V=13ฯ€h(r12+r1r2+r22)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=43ฯ€abcV = \frac{4}{3}\pi abc

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 2ฯ€rh; cones have ฯ€rโˆš(rยฒ+hยฒ). 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=4ฯ€r2A = 4\pi r^2
Cylinder (closed):
A=2ฯ€r2+2ฯ€rhA = 2\pi r^2 + 2\pi rh
Cone:
A=ฯ€r2+ฯ€rr2+h2A = \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 onlyโ€”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=43ฯ€abcV = \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. mยฒ, ftยฒ, inยณ).
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Mathematical Reference Note

Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.

Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.

Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.

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