Skip to main content
All Calculators & ConvertersAll calculators

Surface Area Calculator

Calculate total surface area and, when relevant, lateral surface area for common 3D solids with formulas, step-by-step solutions, unit support, and labeled sketches.

Background

Students often remember the shape but forget whether to use total surface area, lateral area, or how to handle bases. This calculator helps by grouping solids into clear families, showing the formula before substitution, and using a labeled preview so dimensions like r, h, ℓ, and s are easier to identify.

Choose a solid

Tip: Start with the solid family first, then pick the exact solid tile below.

Solid preview

14 solids total

Click a tile to preview the selected solid with labeled dimensions.

Selected solid: Cube Formula: SA = 6s²

Enter the measurements for the selected solid below.

Solid inputs

Surface area is shown in squared units such as cm², m², in², or ft².

Use total surface area for the full outside of the solid. Use lateral area when you want only the side surface, excluding base(s), where supported. For a hemisphere, lateral means curved surface only, while total includes the circular base.

Display options

Chips prefill and calculate immediately.

Result

No results yet. Choose a solid, enter values, and click Calculate.

How to use this calculator

  • Choose a solid family, then click the exact solid card.
  • Review the live preview panel to confirm the solid and labeled dimensions.
  • Enter the required measurements.
  • Choose whether you want total surface area or lateral surface area only when available.
  • Select an input unit such as cm, m, in, or ft.
  • Click Calculate to get the result, formula, substitution, and labeled sketch.

How this calculator works

  • Prisms & boxes: use base area, perimeter of base, and height, often in the form SA = Ph + 2B.
  • Cylinders & cones: use radius, height, and sometimes slant height for curved surfaces.
  • Pyramids: use base area plus triangular face area or slant height formulas.
  • Round solids: use curved-surface formulas such as 4πr² for spheres and approximation formulas for ellipsoids.

Formula & Equations Used

Cube: TSA = 6s², LSA = 4s²

Rectangular Prism: TSA = 2(lw + lh + wh), LSA = 2h(l + w)

Triangular Prism: TSA = PL + 2B, LSA = PL

Regular Prism: TSA = Ph + 2B, LSA = Ph

Cylinder: TSA = 2πrh + 2πr², LSA = 2πrh

Cone: TSA = πrℓ + πr², LSA = πrℓ

Frustum of a Cone: TSA = π(R + r)ℓ + πR² + πr², LSA = π(R + r)ℓ

Square Pyramid: TSA = s² + 2sℓ, LSA = 2sℓ

Rectangular Pyramid: TSA = lw + lℓ₁ + wℓ₂, LSA = lℓ₁ + wℓ₂

Frustum of a Pyramid: TSA = 2(B + b)ℓ + B² + b², LSA = 2(B + b)ℓ

Sphere: TSA = 4πr²

Hemisphere: TSA = 3πr², Curved\ Surface = 2πr²

Capsule: TSA = 2πrL + 4πr²

Ellipsoid: TSA \(\approx\) 4π\(\left\)(\(\frac{(ab)^p + (ac)^p + (bc)^p}{3}\)\(\right\))^{1/p}, where p \(\approx\) 1.6075

What this calculator can show

  • Total surface area
  • Lateral surface area for supported solids
  • Base area or combined base area when helpful
  • Formula and substitution
  • Step-by-step solution
  • Result sketch with actual entered values

Example Problems & Step-by-Step Solutions

Example 1 — Cylinder

Find the total surface area of a cylinder with radius 4 and height 10.

  1. Use TSA = 2πrh + 2πr².
  2. Substitute: TSA = 2π(4)(10) + 2π(4²).
  3. Simplify: TSA = 80π + 32π = 112π.

Example 2 — Cone

Find the lateral surface area of a cone with radius 3 and slant height 5.

  1. Use LSA = πrℓ.
  2. Substitute: LSA = π(3)(5).
  3. Simplify: LSA = 15π.

Example 3 — Hemisphere

Find the total surface area of a hemisphere with radius 5.

  1. Curved area: 2πr² = 2π(25) = 50π.
  2. Base circle area: πr² = 25π.
  3. Total: 50π + 25π = 75π.

Example 4 — Rectangular Prism

Find the total surface area of a rectangular prism with length 8, width 5, and height 3.

  1. Use TSA = 2(lw + lh + wh).
  2. Substitute: TSA = 2(8·5 + 8·3 + 5·3).
  3. Simplify: TSA = 2(40 + 24 + 15) = 2(79) = 158.

Frequently Asked Questions

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

Total surface area includes the entire outside of the solid. Lateral surface area excludes the base or bases and keeps only the side surface when that interpretation is standard for the selected solid.

Q: What does lateral area mean for a hemisphere?

For a hemisphere, lateral area means the curved surface only, 2πr². Total surface area includes the flat circular base too, so it becomes 3πr².

Q: Why does the cone use slant height?

The curved surface of a cone depends on the slant height, not just the vertical height. If you enter radius and height, the calculator computes the slant height automatically.

Q: Why is ellipsoid surface area labeled as approximate?

Unlike a sphere, an ellipsoid does not have a simple elementary exact surface area formula. This calculator uses a standard approximation formula that is accurate for most student use cases.

Q: Can the result sketch show my actual values?

Yes. The live preview uses symbolic labels, and the result sketch can show the actual values you entered. You can also enable units inside result sketch labels.