Math Detours · Grade 6 · Sharpen · 7

Why elephants have thick legs

Area and volume are easy to calculate and strange to scale. Double every length and the skin grows four times while the insides grow eight. That one fact shapes every animal on Earth.

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Tonight's idea

Three area facts. A parallelogram is a rectangle that has been pushed sideways: slice a triangle off one end and stick it on the other, and you have a rectangle with the same base and height. So area \(= \text{base} \times \text{height}\). A triangle is half a parallelogram (put two copies together), so area \(= \tfrac12 \times \text{base} \times \text{height}\). Cut a trapezoid along a diagonal and you get two triangles with bases \(a\) and \(b\) and the same height \(h\), so area \(= \tfrac12 (a + b) \times h\).

slide the end triangle acrosstwo triangles make a parallelogramabhtwo triangles, same height h

Volume of a box is length × width × height, and that still works with fractions. A box \(2\tfrac12\) by \(1\tfrac12\) by 4 holds \(15\) cubic units: fill it with half-unit cubes, \(5 \times 3 \times 8 = 120\) small cubes, each \(\tfrac18\) of a unit, and \(120 \div 8 = 15\).

Take any shape and double every length. Its area becomes \(2 \times 2 = 4\) times bigger. Its volume becomes \(2 \times 2 \times 2 = 8\) times bigger. Scale by 3 and area goes up 9 times, volume 27 times. Area and volume do not grow together.

Why that matters: an animal's weight is volume, but the strength of its legs is the cross-section area of its bones. Make an animal bigger and weight wins. That is why elephants have thick legs, and why a storybook giant would snap his shins.

Watch one

A cube with 1 cm edges has surface area \(6 \times 1 = 6\) cm² and volume 1 cm³. Double the edge to 2 cm: surface \(6 \times 4 = 24\) cm² (4 times), volume 8 cm³ (8 times). Triple it: surface 54 cm² (9 times), volume 27 cm³ (27 times).

Surface area of a 2 × 3 × 4 box from its net: three pairs of faces, \(2(2 \times 3 + 3 \times 4 + 2 \times 4) = 2(6 + 12 + 8) = 52\).

Your turn

  1. Find the areas: a triangle with base 9 and height 4; a parallelogram with base 7 and height 3; a trapezoid with parallel sides 5 and 9 and height 4.
  2. Volume of a box \(\tfrac34\) by \(\tfrac23\) by 6 units. Then: how many \(\tfrac1{12}\)-unit cubes would fill it? (Careful: count how many fit along each edge. Along the \(\tfrac34\) edge, \(\tfrac34 \div \tfrac1{12} = 9\) cubes fit.)
  3. Draw a net for a 1 × 2 × 3 box and find its surface area.
  4. A photo is enlarged so that every length is 3 times bigger. How many times more ink does it use? A model car is built at \(\tfrac1{10}\) scale. How many model cars' worth of paint covers the real car?
  5. Imagine a mouse scaled up 100 times in every direction. How many times heavier is it? How many times bigger is the cross-section of its leg bones? So each square centimetre of bone now carries how many times the load? Explain why the scaled-up mouse cannot stand.
  6. A round cake 20 cm across and a round cake 30 cm across have the same height. How many times more cake is the big one?

Stretch: pizza and statues

A pizzeria sells one 12-inch pizza or two 8-inch pizzas for the same price. Which is more pizza? (No \(\pi\) needed: circle areas scale like the square of the diameter, so compare \(12^2\) with \(2 \times 8^2\).)

A sculptor makes a statue 3 times as tall as her model, same shape. How many times more bronze does it need? How many times more gold leaf to cover it? Why does a small ice cube melt faster than a big one of the same shape, even though both are at the same temperature? (Think about surface compared with volume.)

Answers

  1. 18; 21; \(\tfrac12(5 + 9) \times 4 = 28\).
  2. \(\tfrac34 \times \tfrac23 \times 6 = 3\) cubic units. Along the edges: \(9\), \(8\) and \(72\) twelfth-cubes, so \(9 \times 8 \times 72 = 5184\) tiny cubes. Check: each has volume \(\tfrac1{1728}\), and \(5184 \div 1728 = 3\).
  3. Faces: two 1 × 2, two 2 × 3, two 1 × 3. Surface area \(2(2 + 6 + 3) = 22\).
  4. Ink is area: \(3^2 = 9\) times. Paint is area: the real car has \(10^2 = 100\) times the surface of the model.
  5. Weight scales with volume: \(100^3 = 1{,}000{,}000\) times heavier. Bone cross-section scales with area: \(100^2 = 10{,}000\) times. So each square centimetre carries \(1{,}000{,}000 \div 10{,}000 = 100\) times the load. Bone is bone; it cannot take 100 times its usual stress. The legs would have to be enormously thicker in proportion, which is exactly what elephants have.
  6. Area ratio \((30/20)^2 = 2.25\), so \(2\tfrac14\) times as much cake.

Stretch. \(12^2 = 144\) against \(2 \times 64 = 128\). One 12-inch pizza is more, by about 12%. Statue: bronze is volume, \(27\) times; gold leaf is area, \(9\) times. Ice: melting happens at the surface, and a small cube has more surface per unit of volume (a cube of side \(s\) has ratio \(6s^2 / s^3 = 6/s\), which grows as \(s\) shrinks). The same reason babies and small animals lose heat fast and must eat constantly.

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