A full-grown horse weighs around 500 kg — more than most cars. Yet horses can swim. Meanwhile, a tiny iron nail sinks straight to the bottom. The key is not weight, but density: how much mass is packed into each unit of volume. If an object is less dense than water, it floats — no matter how large or heavy it is.
Key formula:
$$\rho = \frac{m}{V}$$
$\rho$ (rho) = density (g/cm³ or kg/m³)
$m$ = mass (g or kg)
$V$ = volume (cm³ or m³)
Rearranged: $m = \rho \times V$ · $V = m \div \rho$
Floating rule: If $\rho_\text{object} < \rho_\text{liquid}$ → floats. If $\rho_\text{object} > \rho_\text{liquid}$ → sinks.
Density of water $= 1$ g/cm³. Density of seawater $\approx 1.025$ g/cm³.
Math connection — Lesson 19 (Area and Perimeter): Volume of a box $= l \times w \times h$. For a cylinder it's $V = \pi r^2 h$. You already know how to calculate the base area — volume is just that area times the height. Density = mass ÷ volume uses the same arithmetic as speed = distance ÷ time.
| Material | Density (g/cm³) | Floats in water? |
|---|
| Cork | 0.24 | Yes ✓ |
| Wood (pine) | 0.55 | Yes ✓ |
| Ice | 0.92 | Yes ✓ |
| Horse | ≈ 0.83 | Yes ✓ (surprising!) |
| Water | 1.00 | — (reference) |
| Aluminium | 2.70 | No ✗ |
| Iron | 7.87 | No ✗ |
| Gold | 19.3 | No ✗ |
Worked Example
Worked Example — Does it float?
A block of pine wood has dimensions 20 cm × 10 cm × 5 cm and mass 550 g. Find its density and decide if it floats.
$$V = 20 \times 10 \times 5 = 1\,000 \text{ cm}^3$$
$$\rho = \frac{m}{V} = \frac{550}{1\,000} = 0.55 \text{ g/cm}^3$$
Since 0.55 < 1.00 (water), the wood floats. The fraction submerged $\approx 0.55/1.00 = 55\%$ — the rest sticks out above the surface.
Ships are hollow: A solid iron ball sinks because iron has $\rho = 7.87$ g/cm³. But a hollow steel ship has enormous air-filled space inside, so its average density (total mass ÷ total volume including air) drops well below 1 g/cm³ — and it floats. Shape changes the effective density.
Warm-Up — formula given
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Problem 1
A rock has mass 3 600 g and volume 1 200 cm³. Calculate its density. Does it sink in water?
$\rho = 3\,600 \div 1\,200$
divide → compare to 1 g/cm³ →
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Problem 2
Ice has density 0.92 g/cm³. An iceberg has volume 800 000 m³. What is its mass in kg?
Hint: 1 m³ = 1 000 000 cm³, or use density in kg/m³: ice = 920 kg/m³.
$m = \rho \times V = 920 \times 800\,000$
multiply → give in kg →
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Problem 3
An aquarium is 80 cm long, 35 cm wide, and 40 cm tall, filled to the top with water (density 1 g/cm³).
(a) Find the volume in cm³.
(b) Find the mass of the water in grams and in kg.
(c) 1 litre = 1 000 cm³. How many litres of water does it hold?
V = l × w × h → mass = V × ρ → convert to kg → litres →
Core Problems
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Problem 4
A horse has mass 480 kg and volume approximately 0.58 m³.
(a) Calculate the horse's density in kg/m³.
(b) Density of water = 1 000 kg/m³. Does the horse float?
(c) What fraction of the horse's body would be submerged? (Submerged fraction = $\rho_\text{horse} \div \rho_\text{water}$.)
density → compare → fraction submerged →
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Problem 5
A rectangular wooden block is 15 cm × 10 cm × 8 cm and has density 0.6 g/cm³.
(a) Find the volume and mass of the block.
(b) The block floats. How many cm of its height stick up above the water? (Height above water $= h \times (1 - \rho_\text{wood}/\rho_\text{water})$.)
V → m → height above water →
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Problem 6
An iceberg floats with $\frac{1}{8}$ of its volume above water. What is the density of ice? (Use $\rho_\text{ice} = \rho_\text{water} \times \text{submerged fraction}$.)
Submerged fraction $= 1 - \tfrac{1}{8} = \tfrac{7}{8}$
multiply by density of water →
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Problem 7 Challenge
A hollow metal sphere has an outer radius of 10 cm and wall thickness of 0.5 cm. The metal has density 8 g/cm³.
(a) Find the volume of the metal shell. ($V_\text{outer} = \frac{4}{3}\pi r^3$, inner radius = 9.5 cm.)
(b) Find the mass of the sphere.
(c) The sphere is placed in water. Does it float? (Total volume of sphere ≈ 4 189 cm³.)
outer volume → inner volume → shell volume → mass → average density = mass ÷ outer volume → compare to 1 →
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Problem 8 Open
Design a "density column": layer three liquids of different densities in a tall glass without mixing them. Research the densities of: honey, water, cooking oil, and rubbing alcohol. In which order would you pour them? What objects (coin, grape, bottle cap, raisin) would float at each boundary layer? Draw and label your column.
look up densities → order from densest → predict object positions → draw →
Show answers
Problem 1
$\rho = 3\,600 \div 1\,200 = 3$ g/cm³ — denser than water, sinks.
Problem 2
$m = 920 \times 800\,000 = 736\,000\,000$ kg $= 736\,000$ tonnes — just over 700 000 metric tons!
Problem 3
(a) $V = 80 \times 35 \times 40 = 112\,000$ cm³ · (b) Mass $= 112\,000$ g $= 112$ kg · (c) $112\,000 \div 1\,000 = 112$ litres
Problem 4
(a) $\rho = 480 \div 0.58 \approx 828$ kg/m³ · (b) Yes — less than 1 000 kg/m³ · (c) Fraction submerged $= 828 \div 1\,000 = 0.828 \approx 83\%$
Problem 5
(a) $V = 15 \times 10 \times 8 = 1\,200$ cm³ · $m = 0.6 \times 1\,200 = 720$ g · (b) Height above water $= 8 \times (1 - 0.6) = 8 \times 0.4 = 3.2$ cm above water.
Problem 6
$\rho_\text{ice} = 1.00 \times \tfrac{7}{8} = 0.875$ g/cm³ (actual ≈ 0.917 — close! The real iceberg fraction is more like $\tfrac{9}{10}$ underwater.)
Problem 7
$V_\text{outer} = \frac{4}{3}\pi (10)^3 \approx 4\,189$ cm³ · $V_\text{inner} = \frac{4}{3}\pi (9.5)^3 \approx 3\,591$ cm³ · $V_\text{shell} \approx 598$ cm³ · Mass $= 598 \times 8 \approx 4\,784$ g · Average density $= 4\,784 \div 4\,189 \approx 1.14$ g/cm³ — sinks (just barely).
Coming up next → s08: Light, Angles, and Shadows
How do you measure the height of a tree without climbing it? With a shadow, a ruler, and the similar triangles you already know from Lesson 15. You'll also learn why mirrors work — and why the sky is blue.