# Surface Area Calculator

Calculate the surface area of 3D shapes: cube, rectangular prism, sphere, cylinder, and cone. Get total, lateral, and base areas. Free online calculator.

## What this calculates

Calculate the total surface area, lateral surface area, and base area of common 3D shapes. Select a shape, enter its dimensions, and get instant results.

## Inputs

- **Shape** — options: Cube, Rectangular Prism, Sphere, Cylinder, Cone — Select the 3D shape.
- **Length / Side** — min 0 — Side length (cube), length (rect prism), or not used for sphere/cylinder/cone.
- **Width** — min 0 — Width of the rectangular prism.
- **Height** — min 0 — Height of the prism, cylinder, or cone.
- **Radius** — min 0 — Radius of the sphere, cylinder, or cone.
- **Slant Height** — min 0 — Slant height of the cone (if 0, calculated from radius and height).

## Outputs

- **Total Surface Area** — The total surface area of the shape.
- **Lateral Surface Area** — The surface area excluding the base(s).
- **Base Area** — The area of one base.

## Details

Surface area is the total area of the outer surfaces of a three-dimensional shape. It is measured in square units and is essential in applications like painting, packaging, heat transfer, and material estimation.

The formulas for common 3D shapes are: Cube: 6s² (where s is the side length). Rectangular prism: 2(lw + lh + wh). Sphere: 4πr². Cylinder: 2πr² + 2πrh (two circular bases plus the lateral rectangle). Cone: πr² + πrl (one circular base plus the lateral surface), where l is the slant height.

Lateral surface area excludes the bases and represents only the side surfaces. For a cylinder, it is the rectangle you would get if you unrolled the side. For a cone, it is the sector of a circle formed by the slant surface. Understanding the distinction between total and lateral surface area is important in applications where the base is not exposed, such as the sides of a paint can.

## Frequently Asked Questions

**Q: What is surface area?**

A: Surface area is the total area of all the outer faces of a 3D object. It tells you how much material is needed to cover the object completely. It is measured in square units (cm², m², in², etc.).

**Q: What is the difference between lateral and total surface area?**

A: Lateral surface area includes only the side surfaces, excluding any bases. Total surface area includes everything: sides plus bases. For example, a cylinder's lateral area is 2πrh (the curved side), while the total is 2πrh + 2πr² (adding both circular ends).

**Q: How do I find the slant height of a cone?**

A: The slant height l of a cone can be found using the Pythagorean theorem: l = √(r² + h²), where r is the base radius and h is the vertical height. This calculator automatically computes the slant height if you leave it at 0.

**Q: Why is the surface area of a sphere 4πr²?**

A: The surface area of a sphere equals exactly 4 times the area of its great circle (πr²). This result, first proven by Archimedes, can be derived through calculus or by noting that the area of a sphere equals the lateral area of its circumscribing cylinder.

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Source: https://vastcalc.com/calculators/math/surface-area
Category: Math
Last updated: 2026-04-21