Geometry · 3D Surface Area
Surface Area
Calculator
Calculate total, lateral, and base surface areas for 11 three-dimensional shapes — with component breakdown bars, SVG diagrams, unit conversion, and step-by-step formulas.
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Total SA
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Lateral SA
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Shape
🔮 Sphere
Total SA = 4πr²\nNo lateral/base distinction (all curved)
SA in cm²
Diagram
Total Surface Area
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units²
Lateral Surface Area
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units²
Base Area (×count)
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units²
Total SA
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Lateral SA
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Base Area
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LSA % of Total
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🧩 Surface Area Component Breakdown
📋 Step-by-Step Derivation
Surface Area Formulas for 3D Shapes
Surface area is the total area of all outer faces of a 3D solid. It's measured in square units. Total surface area (TSA) includes bases; lateral surface area (LSA) includes only the sides.
All Surface Area Formulas
Sphere: SA = 4πr² (all curved, no base)
Cube: SA = 6a² LSA=4a², Base=a²×2
Rect. Box: SA = 2(lw+lh+wh) LSA=2h(l+w)
Cylinder: LSA = 2πrh SA=2πr(r+h)
Cone: LSA = πrl SA=πr(r+l) l=slant
Pyramid: LSA = 2al SA=a²+2al l=slant
Capsule: SA = 2πr(2r+h)
Frustum: LSA = π(r₁+r₂)l SA=πl(r₁+r₂)+πr₁²+πr₂²
Sph. Cap: SA = 2πRh + πa² a=base radius
Hemisphere: SA = 3πr² (curved=2πr², base=πr²)
Tri. Prism: SA = perimeter×L + 2×tri. area
What is a conical frustum and how is its surface area calculated?
A frustum is what you get when you cut the top off a cone with a cut parallel to the base. It has two circular bases (radii r₁ and r₂) and a slanted lateral surface. The slant height is l = √(h² + (r₂−r₁)²). Lateral SA = π(r₁+r₂)×l. Total SA = πl(r₁+r₂) + πr₁² + πr₂². Frustums appear in real life as buckets, tapered cups, lampshades, and architectural columns.
What is a spherical cap?
A spherical cap is the region of a sphere that lies above (or below) a given plane. If the sphere has radius R and the cap has height h, then the base circle has radius a = √(2Rh − h²). Curved surface area = 2πRh. Total SA (including base) = 2πRh + πa² = π(2Rh + a²). When h = R, you have a hemisphere: curved SA = 2πR², base = πR², total = 3πR².
Why does a sphere have SA = 4πr²?
Archimedes proved this elegantly: the surface area of a sphere equals the lateral surface area of the cylinder that just contains it (same radius and height 2r). That cylinder's LSA = 2πr×2r = 4πr². The calculus proof: integrate circular strips from −r to +r: ∫ 2π·r·dx = 2πr × 2r = 4πr². Another way: the sphere has exactly 4 times the area of its great circle (πr²): 4 × πr² = 4πr².
What is the ratio of surface area to volume?
The SA-to-volume ratio (SA/V) is critical in biology, chemistry, and materials science. For a sphere: SA/V = 4πr²/((4/3)πr³) = 3/r. Smaller radius → higher SA/V ratio. This is why small cells (like bacteria) have much higher SA/V than large cells, allowing faster nutrient exchange. Catalysts are ground into fine powders to maximize SA/V. Nanoparticles have enormous SA/V ratios — why they're so reactive. The sphere minimizes SA for a given volume (isoperimetric inequality).