Calculate total surface area, lateral surface area, and base area for cubes, rectangular prisms, spheres, cylinders, cones, triangular prisms, and square pyramids. Supports metric (cm/m) and imperial (in/ft) units.
Surface area is the total area of the outer surface of a three-dimensional shape. It measures how much material would be needed to wrap the shape completely — paint to cover a wall, sheet metal to fabricate a tank, or fabric to sew a cover. This free surface area calculator handles seven common 3D shapes with full support for both metric (centimetres, metres) and imperial (inches, feet) units, as used in the UK and USA respectively.
Two distinct measures are commonly used in geometry:
In real-world applications: when painting a cylindrical storage tank, you often only need the lateral area (the sides). When ordering material for a tin can, you need the total surface area including both lids.
A cube has 6 identical square faces.
Total SA = 6s² | Lateral SA = 4s² | Base Area = s²
| Side (s) | Total SA | Lateral SA |
|---|---|---|
| 5 cm | 150 cm² | 100 cm² |
| 10 in | 600 in² | 400 in² |
| 1 ft | 6 ft² | 4 ft² |
A box with length l, width w, and height h.
Total SA = 2(lw + lh + wh) | Lateral SA = 2h(l + w)
Example: A room 5m × 4m × 2.5m has wall area (lateral SA) = 2 × 2.5 × (5+4) = 45 m². Add the floor and ceiling for total SA: 2(5×4 + 5×2.5 + 4×2.5) = 2(20 + 12.5 + 10) = 85 m².
A sphere has only one continuous surface — no base, no top.
Total SA = 4πr²
Example: A ball with radius 10 cm has SA = 4 × π × 100 = 1,256.6 cm².
A cylinder has two circular bases and a curved lateral surface.
Total SA = 2πr(r + h) | Lateral SA = 2πrh | Each Base = πr²
A standard tin can with radius 4 cm and height 12 cm has lateral SA = 2 × π × 4 × 12 = 301.6 cm² and total SA = 2 × π × 4 × (4 + 12) = 402.1 cm².
A cone has a circular base and a slanted surface. The slant height l = √(r² + h²).
Total SA = πr(r + l) | Lateral SA = πrl | Base = πr²
A triangular prism with triangle sides a, b, c, triangle height h_t, and prism length L:
Total SA = (a + b + c)L + 2 × (½ × base × h_t)
This shape appears in tent designs, ramp structures, and triangular bar stock in manufacturing.
A square pyramid with base side b and slant height l = √((b/2)² + h²):
Total SA = b² + 2bl | Lateral SA = 2bl
A net is the 2D pattern you get when you unfold all faces of a 3D shape flat. The total area of the net equals the total surface area of the shape. Students in UK primary schools and US elementary schools often use nets to understand how surface area works. For example, the net of a cube is six connected squares. The net of a cylinder is two circles plus a rectangle whose width equals the circumference of the circles (2πr).
In the US, paint coverage is typically listed in square feet per gallon (around 350–400 sq ft per gallon). In the UK, coverage is listed in square metres per litre (approximately 10–12 m² per litre). To calculate how much paint you need:
A US living room 15 ft × 12 ft × 9 ft tall has wall area = 2 × 9 × (15+12) = 486 ft². At 400 sq ft/gallon with two coats, you need about 2.43 US gallons. The equivalent UK room at 4.5m × 3.7m × 2.7m has wall area = 44.6 m². At 11 m²/litre with two coats, you need about 8.1 litres.
The surface area of a rectangular box tells you the minimum wrapping paper needed. In practice, add 15–20% for overlaps and folds. For a box 30 cm × 20 cm × 10 cm, total SA = 2(30×20 + 30×10 + 20×10) = 2(600 + 300 + 200) = 2,200 cm² = 0.22 m².
Whether insulating a loft, a cylindrical water tank, or pipe lagging, the lateral surface area tells you how much insulation material to buy. A cylindrical hot water cylinder 45 cm diameter, 1.05 m tall has lateral SA = 2π × 22.5 × 105 = 14,843 cm² ≈ 1.48 m².
A standard 330 ml aluminium drink can has a diameter of approximately 66 mm (radius 33 mm) and height of about 115 mm. Its total surface area = 2 × π × 33 × (33 + 115) = approximately 30,659 mm² = 306.6 cm². Multiply by billions of cans produced annually and small material savings in surface area translate to massive cost reductions for manufacturers.
| Metric | Imperial Equivalent |
|---|---|
| 1 cm² | 0.155 in² |
| 1 m² | 10.764 ft² = 1,550 in² |
| 1 in² | 6.452 cm² |
| 1 ft² | 929.0 cm² = 0.0929 m² |
The surface area of a sphere is 4πr², where r is the radius. For example, a sphere with radius 7 cm has SA = 4 × π × 49 = 615.75 cm². Unlike most shapes, a sphere has no distinct base or lateral surface — the entire surface is curved and continuous.
Total surface area includes all faces of a shape — sides, top, and bottom. Lateral surface area includes only the sides, excluding the base(s). For a cylinder, lateral SA = 2πrh (just the tube). Total SA = 2πr(r+h) (tube plus both circular ends). The distinction matters in practical applications like painting only the sides of a tank or calculating the label area on a bottle.
For the walls only (lateral surface area), use: 2 × height × (length + width). For total SA including floor and ceiling: 2(length × width + length × height + width × height). A room 5m × 4m × 2.5m has wall area = 2 × 2.5 × 9 = 45 m² and total SA = 85 m².
Calculate wall area in square feet (perimeter × height, minus doors and windows), divide by paint coverage (typically 350–400 sq ft per US gallon), and multiply by number of coats. A 14 ft × 12 ft room with 9 ft ceilings has about 468 sq ft of wall area. With two coats at 400 sq ft/gallon, you need approximately 2.3 US gallons.
A net is a two-dimensional pattern that can be folded to form a three-dimensional shape. Every face of the 3D shape becomes a flat polygon in the net. The total area of all polygons in a net equals the total surface area of the shape. Nets are widely used in UK primary school maths and US elementary school geometry lessons to help students visualise 3D shapes.
The slant height of a cone is the distance along the slanted surface from the base edge to the apex. Using the Pythagorean theorem: l = √(r² + h²), where r is the base radius and h is the vertical height. This slant height is used in the lateral surface area formula: LSA = πrl.
Surface area is always measured in square units. In the metric system (UK, Europe): cm², m², mm². In the imperial system (US): in², ft², yd². Common conversions: 1 m² = 10.764 ft², 1 ft² = 929 cm². Always ensure all input dimensions use the same unit before calculating.
A typical 400g food tin is approximately 75 mm diameter and 113 mm tall. Radius = 37.5 mm. Total SA = 2π × 37.5 × (37.5 + 113) = 2π × 37.5 × 150.5 ≈ 35,441 mm² ≈ 354.4 cm². The label typically covers the lateral area only: 2π × 37.5 × 113 ≈ 26,624 mm² ≈ 266.2 cm².
Results are estimates based on standard geometric formulas. Real surfaces may vary due to thickness, imperfections, and measurement tolerances. Verify critical measurements before purchasing materials for construction or manufacturing.