Question 1

What are the surface area formulas for common 3D shapes?

Accepted Answer

Cube (side s): SA = 6s². Rectangular prism: SA = 2(lw+lh+wh). Sphere: SA = 4πr². Cylinder: SA = 2πr² + 2πrh = 2πr(r+h). Cone: SA = πr² + πrl = πr(r+l) where l = slant height. Square pyramid: SA = b² + 2bl where b = base side, l = slant height.

Question 2

What is the difference between total and lateral surface area?

Accepted Answer

Total surface area includes all faces (top, bottom, sides). Lateral surface area includes only the side surfaces, excluding the bases. Example for a cylinder: lateral SA = 2πrh (just the curved side). Total SA = 2πrh + 2πr² (adds the two circular ends).

Question 3

How do I find the surface area of a cylinder?

Accepted Answer

SA = 2πr² + 2πrh = 2πr(r+h). The 2πr² covers the two circles (top and bottom). The 2πrh covers the lateral rectangle (if you unroll the cylinder, the side becomes a rectangle with width 2πr and height h). Example: r = 4, h = 10. SA = 2π(4)(4+10) = 8π×14 = 112π ≈ 351.9 sq units.

Question 4

How do I find the surface area of a cone?

Accepted Answer

Total SA = πr² + πrl, where r = base radius and l = slant height = √(r²+h²). Example: r = 3 cm, h = 4 cm. l = √(9+16) = 5. Base area = π×9 = 28.27 cm². Lateral area = π×3×5 = 47.12 cm². Total = 75.40 cm².

Question 5

Why does a sphere have the minimum surface area for a given volume?

Accepted Answer

Among all shapes with the same volume, a sphere has the smallest surface area. This is the isoperimetric inequality in 3D. It's why soap bubbles are spherical — surface tension minimizes surface area. For any shape with volume V: SA ≥ (36π)^(1/3) × V^(2/3), with equality only for a sphere.

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