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Surface Area of a Sphere
Geometry • Grades 7, 8, 9, 10
Geometry
Surface Area of a Sphere
Calculate the total outer surface area of a 3D sphere using A = 4πr².
The Formula
A = 4πr²
Used for: Finding the total outer surface area of a 3D sphere
ATotal surface area (square units)
rRadius of the sphere
dDiameter (2r)
⚡Live Interactive Quick Solver
Instant CalculationSurface Area (A):314.16 units²
Exact form: 100π• Step: A = 4 × π × 5² = 314.16
A = 4\pi r^2 = \pi d^2
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🎯 Key Concepts & Rules
Essential principles for calculating with this formula:
🌐
A = 4πr²Surface area is exactly 4 times the area of its great circle (πr²).
📏
Diameter Form: A = πd²Since d = 2r, (2r)² = 4r², giving A = πd².
🔢
Doubling radius quadruples areaBecause r is squared, 2r yields 4× the surface area.
📝 Worked Examples & Step-by-Step Solutions
Example 1: Radius = 5 cm
Problem:
Find total surface area of a sphere with radius r = 5 cm.1
Formula
A = 4 × π × r²2
Substitute
A = 4 × π × (5)²3
Calculate r²
A = 4 × π × 25 = 100π4
Decimal Form
A ≈ 100 × 3.14159 ≈ 314.16 cm²Answer: A = 100π cm² ≈ 314.16 cm²
💡 Pro Tips & Common Mistakes
Archimedes' Theorem
The surface area of a sphere equals the lateral surface area of an enclosing cylinder (2πr × 2r = 4πr²).
Hemisphere Surface Area
Total surface area of a solid hemisphere = 3πr² (2πr² curved + πr² base circle).
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