Calculate the total surface area of 3D shapes — sphere, cube, cylinder, cone, rectangular prism and pyramid — with lateral and base area breakdown.
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Surface Area of a sphere = 4πr², where r is the radius. For a sphere with radius 5: SA = 4 × 3.14159 × 25 = 314.16 square units. The surface area formula can also be written as πd² (where d is diameter). A sphere has no edges or flat faces — its entire surface curves uniformly. The surface area of Earth is approximately 510 million km².
Total Surface Area of a cylinder = 2πr² + 2πrh = 2πr(r + h). The formula has two parts: the two circular ends (2πr²) and the curved lateral surface (2πrh). For a cylinder with radius 3 and height 10: SA = 2π(3)(3 + 10) = 2 × 3.14159 × 3 × 13 = 244.9 sq units. Open-top containers use SA = πr² + 2πrh (only one circular end).
Surface area is the total area of all exterior faces of a 3D object — measured in square units (sq ft, sq m). Volume is the amount of 3D space enclosed by the object — measured in cubic units (cu ft, cu m). A hollow box and a solid box can have identical surface areas but very different volumes. Surface area determines material needed for coating; volume determines capacity or weight.
Total Surface Area of a cone = πr² + πrl = πr(r + l), where r is the base radius and l is the slant height (l = √(r² + h²)). The base is a circle (πr²) and the lateral surface unfolds into a flat sector. For a cone with radius 4 and height 6: slant height = √(16 + 36) = √52 = 7.21. SA = π × 4 × (4 + 7.21) = 140.9 sq units.
Surface Area = 2(lw + lh + wh), where l = length, w = width, h = height. Calculate the area of each pair of opposite faces and double it. For a box 5 × 3 × 4: SA = 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 2(47) = 94 sq units. For a cube (all sides equal): SA = 6s².
Lateral surface area (LSA) is the area of all surfaces excluding the top and bottom bases. For a cylinder: LSA = 2πrh (the curved side only). For a rectangular prism: LSA = 2h(l + w) (the four vertical walls only). LSA is useful for problems involving paint on walls (excludes floor and ceiling), labels on cans (excludes top and bottom lids), or wrapping the sides of a shape.
Surface area calculations appear in painting and coating (how much paint to buy), packaging design (material to make a box or wrapper), heat transfer (larger surface area = faster heat exchange), biology (lung alveoli, intestinal villi maximise surface area for absorption), chemistry (catalyst surface area affects reaction rate), and construction (roofing, siding, insulation coverage).
Total Surface Area = Base Area + Lateral Area = s² + 2sl, where s is the base side length and l is the slant height (l = √((s/2)² + h²), where h is the height). For a pyramid with base 6 and height 8: slant height = √(9 + 64) = √73 = 8.544. LSA = 2 × 6 × 8.544 = 102.5. Total SA = 36 + 102.5 = 138.5 sq units.
Of all 3D shapes, a sphere encloses the maximum volume for a given surface area — equivalently, it has the minimum surface area for a given volume. This is proven by the isoperimetric inequality. It explains why soap bubbles are spherical (minimising surface tension energy), why water droplets form spheres, and why cells in nature are often roughly spherical to minimise membrane material.
Calculate wall area: perimeter × height. Subtract doors (~20 sq ft each) and windows (~15 sq ft each). Ceiling area: length × width. Add walls + ceiling for total area. One gallon covers 350–400 sq ft for one coat. For two coats, divide total area by 175–200 to get gallons needed. For a 12 × 15 ft room with 9 ft ceilings: walls = (54 ft perimeter × 9 ft) − 2 doors − 2 windows ≈ 446 sq ft; ceiling = 180 sq ft; total ≈ 626 sq ft; two coats ≈ 4 gallons.
