Compute the total outer area of a 3D shape - cube, sphere, cylinder, cone or rectangular prism. Used to estimate paint required, wrapping material, heat transfer area and packaging.
Surface area (Rectangular prism)
148
Result is in (input units)²
कुछ सामग्री अब भी अंग्रेज़ी में दिखती है।
Surface area is the total outer skin of a 3D shape - what you'd cover if you wrapped it in paper, painted it, or coated it with anything. Distinct from volume (which measures inside space), surface area governs heat exchange, paint quantity, wrapping material, packaging, and any process happening at the boundary of an object. For Indian homeowners, surface area calculations help estimate paint needed for interior and exterior walls, packaging dimensions for parcels, heating/cooling surface area, and material wrapping. For students, it's a Class 8-10 mensuration chapter staple and a frequent JEE/NEET problem.
This calculator handles five common 3D shapes - cube, rectangular prism, cylinder, cone and sphere - with formulas displayed for verification. The cylinder and cone formulas distinguish between lateral surface area (only the curved side, used for label-printing on cans) and total surface area (curved side plus top and bottom faces). For composite shapes like a hemisphere on a cylinder (typical water tank or temple dome), compute each shape's relevant surface separately and sum. The slant height for cones uses Pythagoras (ℓ = √(r² + h²)) - a useful tie-in for school geometry.
Pick the shape.
Enter the dimensions.
Read the surface area.
Surface area sums the area of every face. For curved shapes (sphere, cylinder, cone) the formulas are derived by 'unrolling' the curved surface and computing its flat area, plus any flat faces.
Cube: 6s² Rect prism: 2(lw + lh + wh) Cylinder: 2πr(r + h) Cone: πr(r + ℓ), ℓ = √(r² + h²) Sphere: 4πr²
Standard surface area formulas: Cube = 6 × side² (six square faces). Rectangular prism (cuboid) = 2(lw + lh + wh) - three pairs of rectangular faces. Cylinder total = 2πr(r + h) - combining 2πrh lateral and 2πr² for two circular ends. Cylinder lateral only (open cans, labels) = 2πrh. Cone total = πr(r + ℓ) where slant ℓ = √(r² + h²). Cone lateral only = πrℓ. Sphere = 4πr² - same as 4× area of a circle of the same radius.
Why do curved surface formulas have these specific forms? Cylinder lateral surface 'unrolls' into a rectangle: 2πr (circumference) × h (height) = 2πrh. Cone lateral surface unrolls into a sector of a circle with radius equal to slant height ℓ. Sphere surface (4πr²) is exactly the derivative of sphere volume (4/3)πr³ with respect to r - a beautiful calculus relationship.
Composite shapes: real-world objects often combine basic shapes. A water bottle is cylinder + hemisphere top - compute cylinder lateral area + half sphere surface, possibly excluding the connecting circle. A pencil is cylinder + cone tip. A house with sloped roof is rectangular prism walls + roof (triangular prisms or pyramid). Always identify which faces are visible/relevant and sum only those. For a closed box, sum all faces; for an open container, exclude the top.
Lateral surface = perimeter × height = 16 × 3 = 48 m² = 516 sq ft. Subtract one door 21 sq ft + one window 16 sq ft = 479 sq ft net. At 140 sq ft/L coverage for 2 coats: 479 × 2 / 140 = 6.84L. Round up to 7L premium emulsion.
Cylinder lateral: 2π × 0.6 × 1.2 = 4.52 m². Hemisphere: 2π × 0.6² = 2.26 m². Bottom (flat): π × 0.6² = 1.13 m². Total external surface: 7.91 m² ≈ 85 sq ft. Useful for anti-rust coating estimation.
Surface area = 2(50×30 + 50×20 + 30×20) = 2(1500 + 1000 + 600) = 2 × 3100 = 6200 cm² = 0.62 m². Used for cardboard, wrapping paper or printing area calculation.
Surface area scales as length², volume as length³. As objects grow, surface-to-volume ratio shrinks - why elephants have small ears relative to body, and why ice cubes melt faster than icebergs.
Find total wall surface area, subtract doors and windows, divide by paint coverage (typically 350 sq ft/litre for one coat). Add 10-15% wastage.
When you 'unroll' a cone's lateral surface, it becomes a sector of a circle with radius equal to the slant - not the cone's height. Slant = √(r² + h²) by Pythagoras.
Surface = 4πr², Volume = (4/3)πr³. Notice surface area is exactly the derivative of volume with respect to r - geometric interpretation of integration.
Sphere - that's why bubbles, water drops and many cells are spherical. Minimises surface energy.
Long thin shapes. That's why heat sinks have fins and lungs have alveoli - maximises exchange surface in compact volume.
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