Surface Area Calculator
Find the total and lateral surface area of 9 common solids, with each face, the volume and every step shown.
A = 2πr² + 2πrh
Covering a cylinder with radius r = 4 cm and height h = 10 cm takes 351.8584 cm² of material.
Surface area breakdown
Each part of the surface, with its formula and your numbers.
Open at one end, like a cup or tank: 2πrh + πr² = 301.5929 cm².
Your calculation
The same steps the calculator runs, with your numbers.
Total surface area in other units
Converted with exact factors: 1 in = 2.54 cm, 1 ft = 12 in, 1 yd = 3 ft.
| Unit | Surface area |
|---|
Values are rounded to 4 decimal places. Real objects have seams, rims and overlaps, so add a margin when buying material.
What Is Surface Area?
Surface area is the total area of every face or curved surface on the outside of a solid. It answers the question of how much material covers the object: paint on a wall, cardboard in a box, sheet metal in a tank or the label on a can. It is measured in square units such as cm², m², in² or ft².
Surface area splits into two parts. The lateral surface area is the area of the sides only, such as the curved side of a cylinder or the four triangles of a pyramid. The base area is the area of the flat ends. The total surface area adds them together.
Surface area is different from volume. Volume measures the space inside in cubic units, while surface area measures the outside skin in square units. Two objects can hold the same volume and still need very different amounts of material.
How to Use the Surface Area Calculator
- Pick the shape. Choose from cube, box (rectangular prism), sphere, hemisphere, cylinder, cone, square pyramid, capsule and frustum.
- Choose one unit. Enter every length in the same unit, such as centimeters or feet. The result is in that unit squared, and the table converts it to the other units.
- Enter the measurements. Use the radius, not the diameter, for round shapes. For a cone or pyramid, enter the vertical height. The calculator works out the slant height.
- Read the result. The large number is the total surface area. The breakdown lists each face or part, then come the volume, the steps and the unit table.
Results update as you type. The quick examples include a drink can, a shipping box, the walls of a room, a sphere, a 3, 4, 5 cone and a square pyramid.
Surface Area Formulas for Every Shape
In these formulas, r is the radius, h the height, s the slant height and a the edge length. Each example result was computed with the calculator.
| Shape | Total surface area | Lateral area | Example |
|---|---|---|---|
| Cube | 6a² | 4a² | a = 3: 54 |
| Box (rectangular prism) | 2(lw + lh + wh) | 2h(l + w) | 40, 30, 20: 5,200 |
| Sphere | 4πr² | 4πr² | r = 12: 1,809.5574 |
| Hemisphere | 3πr² | 2πr² | r = 5: 235.6194 |
| Cylinder | 2πr² + 2πrh | 2πrh | r = 4, h = 10: 351.8584 |
| Cone | πr² + πrs | πrs | r = 3, h = 4: 75.3982 |
| Square pyramid | a² + 2as | 2as | a = 6, h = 4: 96 |
| Capsule | 2πr(2r + h) | 2πrh | r = 1, h = 4: 37.6991 |
| Frustum | π(r₁ + r₂)s + πr₁² + πr₂² | π(r₁ + r₂)s | 4, 2, h = 3: 130.7949 |
Slant heights come from the Pythagorean theorem: s = √(r² + h²) for a cone, s = √(h² + (a ÷ 2)²) for a square pyramid and s = √((r₁ − r₂)² + h²) for a frustum.
Worked Examples: Cube and Box
A box has three pairs of matching rectangles. Find one face from each pair, add them and double the result.
Worked Examples: Round Shapes
Keep the answer as a multiple of π until the last step, then multiply by π ≈ 3.14159265 to avoid rounding errors.
Pyramid, Capsule and Frustum
Square pyramid
A square pyramid has a square base and four matching triangles. Each triangle has base a and height equal to the slant height s, measured up the middle of the face. For a = 6 m and a vertical height of 4 m, s = √(4² + 3²) = 5 m. The four triangles give 2 × 6 × 5 = 60 m², the base adds 6² = 36 m² and the total is 96 m².
Capsule
A capsule is a cylinder with a hemisphere on each end. The two ends form one full sphere, so the area is 4πr² + 2πrh = 2πr(2r + h). With r = 1 and a cylinder length of 4, the area is 2π × 1 × 6 = 12π ≈ 37.6991 square units.
Conical frustum
A frustum is a cone with the top cut off parallel to the base, the shape of a bucket or lampshade. For r₁ = 4, r₂ = 2 and h = 3, the slant height is √(2² + 3²) = √13 ≈ 3.6056. The side is π × 6 × 3.6056 ≈ 67.9630, the two circles add π(16 + 4) ≈ 62.8319 and the total is about 130.7949 square units.
How to Calculate Surface Area Manually
- Sketch the net. Imagine unfolding the solid flat. A box becomes six rectangles, a cylinder becomes two circles and a rectangle, and a pyramid becomes a square and four triangles.
- Find each flat area. Use length × width for rectangles, ½ × base × height for triangles and πr² for circles. The curved side of a cylinder is a rectangle 2πr wide and h tall.
- Use the slant height on sloped faces. Triangles on a pyramid and the side of a cone use the slant height, found with the Pythagorean theorem.
- Add every part and write square units. Leave out any face that is open, such as the top of a box or a cup.
Area Unit Conversions
| From | Equals |
|---|---|
| 1 cm² | 100 mm² |
| 1 m² | 10,000 cm² |
| 1 in² | 6.4516 cm² |
| 1 ft² | 144 in² = 0.09290304 m² |
| 1 yd² | 9 ft² = 0.83612736 m² |
| 1 m² | ≈ 10.7639 ft² |
Area factors are the square of length factors. Because 1 m = 100 cm, 1 m² = 100 × 100 = 10,000 cm², not 100 cm².
Surface Area vs Volume
Scaling a solid by a factor k multiplies its surface area by k² and its volume by k³. Doubling the radius of a sphere gives 4 times the surface area and 8 times the volume.
The surface area to volume ratio therefore falls as objects grow. A cube with 1 cm edges has 6 cm² of surface for 1 cm³ of volume, a ratio of 6. A cube with 10 cm edges has 600 cm² for 1,000 cm³, a ratio of 0.6.
The calculator shows this ratio as a tile. It is expressed per unit of length, so it changes when you switch units.
Where Surface Area Is Used
Painting a room
The walls of a room form the lateral area of a box. A room 12 ft by 10 ft with 8 ft walls has 2 × (12 × 8 + 10 × 8) = 352 ft² of wall, or 472 ft² with the ceiling. Subtract doors and windows, then divide by the coverage printed on the paint can.
Boxes, wrapping and packaging
A 40 × 30 × 20 cm box needs 5,200 cm² of cardboard before flaps and overlaps. An open box without a lid needs 4,000 cm².
Labels and tanks
A can 6.6 cm wide and 12.2 cm tall has a curved side of about 252.9610 cm², the area of a label that wraps all the way around. For a tank, the surface area sets how much metal or coating is needed.
Common Mistakes
- Using the diameter as the radius. A can 6.6 cm across has r = 3.3 cm. Using 6.6 makes the end areas 4 times too large.
- Using the vertical height on sloped faces. The side of a cone with r = 3 and h = 4 uses s = 5, not 4.
- Mixing units. Convert every length to one unit first. 2 m and 50 cm should be entered as 2 and 0.5 meters.
- Converting area with length factors. 1 ft² is 144 in², not 12 in².
- Counting faces that are not there. An open box, a cup or a pipe has fewer faces than the closed solid. Use the breakdown to remove them.
- Writing cubic units. Surface area is always in square units. Cubic units belong to volume.
Related Calculations
The Volume Calculator finds the space inside the same shapes, and the Area Calculator covers flat shapes such as rectangles, triangles and trapezoids. The Circle Calculator finds the radius, circumference and area of the round faces.
Slant heights come from the Pythagorean Theorem Calculator, and the Square Footage Calculator handles floor and wall areas for home projects.
Frequently Asked Questions
How do you calculate the surface area of a cylinder?
The total surface area of a cylinder is 2πr² + 2πrh, the two circular ends plus the curved side. The curved side unrolls into a rectangle whose width is the circumference 2πr and whose height is h. For radius 4 cm and height 10 cm, the ends give 2π × 4² = 100.5310 cm² and the side gives 2π × 4 × 10 = 251.3274 cm², for a total of 112π ≈ 351.8584 cm². If you know the diameter, halve it first. A cylinder open at one end, such as a cup, leaves out one end: 2πrh + πr².
What is the formula for the surface area of a sphere?
The surface area of a sphere is 4πr², exactly four times the area of a circle with the same radius. For a radius of 12 cm, the area is 4π × 12² = 4π × 144 = 576π ≈ 1,809.5574 cm². A sphere with radius 1 has a surface area of about 12.5664 square units. Because the radius is squared, doubling the radius multiplies the surface area by 4. A hemisphere has a curved area of 2πr², and adding its flat circular base gives 3πr².
How do you find the surface area of a rectangular prism?
A rectangular prism, or box, has six rectangular faces in three matching pairs, so its surface area is 2(lw + lh + wh). For a box 40 cm long, 30 cm wide and 20 cm high, the top and bottom give 2 × 1,200 = 2,400 cm², the front and back give 2 × 800 = 1,600 cm², and the two ends give 2 × 600 = 1,200 cm². The total is 5,200 cm². A cube is the special case where every edge is equal, so its area is 6a²: a cube with 3 ft edges has 54 ft².
What is the difference between lateral and total surface area?
Lateral surface area counts only the sides of a solid and leaves out the bases. Total surface area adds the bases. For a cylinder with radius 4 cm and height 10 cm, the lateral area is 2πrh = 251.3274 cm² and the total is 351.8584 cm². For a cone, the lateral area is πrs, where s is the slant height. The walls of a room are the lateral area of a box: a room 12 ft by 10 ft with 8 ft walls has 352 ft² of wall, and 592 ft² including the floor and ceiling.
How do you find the surface area of a cone?
The total surface area of a cone is πr² + πrs, the circular base plus the curved side, where s is the slant height. If you know the vertical height h, find the slant height with the Pythagorean theorem: s = √(r² + h²). For r = 3 cm and h = 4 cm, s = √(9 + 16) = 5 cm. The side is π × 3 × 5 = 15π ≈ 47.1239 cm², the base is π × 3² = 9π ≈ 28.2743 cm², and the total is 24π ≈ 75.3982 cm². Using h in place of s is the most common mistake.
Sources & References
- OpenStax. Prealgebra 2e, 9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solids, cylinders, spheres and cones). openstax.org
- OpenStax. Calculus Volume 2, 2.4 Arc Length of a Curve and Surface Area (lateral area of a cone and a frustum from the slant height). openstax.org
- NIST. Guide to the SI (Special Publication 811), Appendix B.9 Factors for Units Listed by Kind of Quantity (area conversion factors). nist.gov
Results are for educational purposes. Every worked number on this page was recomputed with the same formulas the calculator uses. Methodology reviewed: 2026.