A sphere with radius 3 has a volume of 113.10 — and a surface area of 113.10. The same number, twice. Students who notice it usually ask the same question: is that a coincidence? It is not. It is the one radius where two different formulas cross, and understanding why they cross is the fastest way to stop mixing the formulas up.
Volume is r³. Surface area is r². That is the whole trick.
Volume fills three dimensions, so its formulas always carry a cube: a cube's volume is s³, a sphere's is (4/3)πr³. Surface area covers two dimensions, so its formulas always carry a square: a cube's surface area is 6s², a sphere's is 4πr². If your sphere volume answer has r² in it, you found an area, not a volume — check the power before you check the arithmetic.
The two sphere formulas share the 4, the π and the r, so memory swaps them. The fix is to say the power out loud before writing anything: "volume is r cubed."
Why r = 3 is the only radius where they match
Divide surface area by volume and almost everything cancels: SA ÷ V = 4πr² ÷ (4/3)πr³ = 3/r. At r = 3 the ratio is exactly 1, so the numbers are equal: V = (4/3)π × 27 = 36π ≈ 113.10 and SA = 4π × 9 = 36π ≈ 113.10. At r = 2 the ratio is 1.5 — surface area 50.27 against volume 33.51. At r = 5 it is 0.6 — volume 523.60 against surface area 314.16. The numbers cross once, at r = 3, and then volume pulls away for good, because r³ outgrows r².
And the units never match: volume is 113.10 cubic units, surface area is 113.10 square units. Equal digits, different quantities. A volume answer written in cm² is wrong even when the number is right.
The five shapes, one idea each
Cube — V = s³. Side 4 gives 4³ = 64, and surface area 6 × 16 = 96. The space diagonal is 4√3 ≈ 6.93.
Rectangular prism — V = l × w × h. A 3 × 4 × 5 box gives 60, with surface area 2(12 + 15 + 20) = 94. A 5 × 5 × 10 box gives 250 for both — volume and surface area genuinely equal there, at 250 each.
Cylinder — V = πr²h. Base area times height: r = 2, h = 5 gives π × 4 × 5 = 20π ≈ 62.83, with surface area 28π ≈ 87.96. Doubling the radius quadruples the volume; doubling the height only doubles it.
Sphere — V = (4/3)πr³. The odd one out: no flat base, so the formula is the one students mix up. r = 3 gives 36π ≈ 113.10 — twice.
Cone — V = (1/3)πr²h. One third of the cylinder with the same base and height. r = 3, h = 4 gives 12π ≈ 37.70, with slant height √(9 + 16) = 5 and surface area 24π ≈ 75.40. Drop the 1/3 and you get the cylinder's 113.10 — three times too big.
The three mistakes that cost the most marks
Using the diameter as the radius. A sphere with diameter 6 has radius 3, so its volume is 113.10. Using 6 as the radius gives (4/3)π × 216 = 904.78 — eight times too big, because 2³ = 8. On a cylinder the same mistake costs four times: r = 4 instead of r = 2 with h = 5 gives 251.33 instead of 62.83.
Forgetting the 1/3 in the cone. A cylinder with r = 3 and h = 4 has volume 113.10; the cone with the same base and height has (1/3) × 113.10 = 37.70. The cone is a third of the cylinder, always.
Counting only the curved side as surface area. A cylinder's surface area is lateral area plus two bases: 2πrh + 2πr². For r = 2, h = 5 the lateral area alone is 62.83 — the same digits as the volume, another trap — while the full surface area is 87.96. An open-top tank has only one base, so read the question before counting bases.
A box that holds 9 litres
A shipping box measures 30 × 20 × 15 cm. Its volume is 30 × 20 × 15 = 9,000 cm³, and since 1 litre = 1,000 cm³, that is 9 litres. Its surface area is 2(600 + 450 + 300) = 2,700 cm² of cardboard. Volume tells you what fits inside; surface area tells you how much material the box is made of.
Want the steps shown instead of just the answers? A free volume calculator works through all five shapes — cube, prism, cylinder, sphere and cone — with volume and surface area shown step by step. The guide to sphere volume walks through the 3/r rule and the full radius table, and the guide to the cylinder volume formula covers V = πr²h step by step with five worked examples. Check your own numbers on a free volume calculator and the r² vs r³ confusion stops working on you.
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