Math & Science · Guide
Surface Area and Lateral Area Formulas
Lateral area is the sides only. Total surface area adds the ends. Here is every formula you are likely to need.
Want the answer for your own numbers? Use the free Area Calculator — it does every step below for you.
Lateral area vs total surface area
Lateral area is the area of the sides only — not the top or bottom. Think of the label on a can. Total surface area includes the bases too — the label plus the two lids.
Lateral area of a cone
Lateral area = π × r × l (l = slant height)
Total surface area = πrl + πr²
Slant height l = √(r² + h²)
Example: radius 3, height 4
Slant height: √(9 + 16) = 5.
Lateral area: π × 3 × 5 = 47.1. Base: π × 9 = 28.3.
Lateral area 47.1; total surface area 75.4
Use the slant height, not the vertical height. Mixing them up is the most common mistake.
Surface area of a prism
Surface area = 2 × base area + perimeter of base × length
Example: triangular prism with a 3-4-5 triangle, length 10
Base area: ½ × 3 × 4 = 6, so two bases = 12.
Perimeter: 3 + 4 + 5 = 12. Sides: 12 × 10 = 120.
Surface area = 132 square units
For a rectangular box: SA = 2(lw + lh + wh).
Other 3D shapes
| Shape | Lateral area | Total surface area |
|---|---|---|
| Cylinder | 2πrh | 2πrh + 2πr² |
| Cone | πrl | πrl + πr² |
| Sphere | — | 4πr² |
| Cube | 4s² | 6s² |
| Square pyramid | 2 × base edge × slant height | lateral + base edge² |
More worked examples
Cylinder: radius 3, height 10
Lateral area: 2 × π × 3 × 10 = 188.5.
Two bases: 2 × π × 3² = 56.5.
Lateral area 188.5; total surface area 245.0
Square pyramid: base edge 6, slant height 5
Lateral area: 2 × 6 × 5 = 60. Base: 6² = 36.
Lateral area 60; total surface area 96
Sphere: radius 5
4 × π × 5² = 4 × π × 25.
Surface area 314.2
Area formulas for 2D shapes
| Shape | Area |
|---|---|
| Rectangle | length × width |
| Triangle | ½ × base × height |
| Circle | πr² |
| Semicircle | ½πr² |
| Trapezoid | ½(a + b) × h |
| Parallelogram | base × height |
| Rhombus | ½ × d1 × d2 |
| Ellipse | π × a × b |
Area of a polygon formula
For a regular polygon (all sides equal):
Area = ½ × perimeter × apothem
The apothem is the distance from the centre to the middle of a side. A regular hexagon with 6 cm sides has an apothem of about 5.196 cm, so its area is ½ × 36 × 5.196 = 93.5 cm². For irregular polygons, split them into triangles and add the areas.
Cross-sectional area
The cross-sectional area is the area you see when you slice straight through an object. For a pipe, wire or beam, it is the area of the end shape. A round pipe with an inner diameter of 50 mm has a cross-sectional area of π × 25² = 1,963 mm². It is used for flow rates, electrical resistance and beam strength.
Find 2D areas instantly in the area calculator, and 3D surface areas alongside volume in the volume calculator.
Where you will actually use this
- Painting a room. The walls are the lateral area of a box. A 12 ft × 10 ft room with an 8 ft ceiling has 2 × (12 + 10) × 8 = 352 ft² of wall, before you subtract doors and windows. A gallon of paint typically covers 350–400 ft² per coat, so two coats need about 2 gallons.
- Wrapping and packaging. The total surface area of a box tells you the least amount of material needed to cover it.
- Labels and insulation. A can label or a pipe wrap covers the lateral area of a cylinder — no ends.
Common mistakes
- Using the vertical height where the formula needs the slant height (cones and pyramids).
- Using the diameter where the formula needs the radius.
- Forgetting to square the units: area is in cm² or ft², not cm or ft.
- Adding the bases when the question asks for lateral area only.
Frequently asked questions
Lateral area = πrl, where r is the base radius and l is the slant height. A cone with radius 3 and height 4 has a slant height of 5 and a lateral area of about 47.1.
Add twice the base area to the base perimeter times the prism's length. A triangular prism with a 3-4-5 base and length 10 has 12 + 120 = 132 square units.
Total surface area = 2πrh + 2πr² — the curved side plus two circular ends. For a radius of 3 and height of 10, that is 188.5 + 56.5 = 245.0 square units.
Lateral area covers only the sides of a solid. Total surface area adds the base or bases. For a cylinder, the lateral area is the label; the total surface area is the label plus both lids.