Area: Dimensions & Units

Understand dimensions and units of area. Convert between metric (mm2, cm2, m2) and US customary (in2, ft2, yd2) area units, and practise ratio problems with composite shapes.

Dimensions & Units

Use the workspace below. Write equations like A = 30 to solve for the area.

Learning Topics

📖 Area Study Guide

1. What Is a Dimension?

A dimension tells us how many independent directions we need to measure a shape.
• A line is 1-dimensional (1D) — we only need one measurement: its length.
• A flat surface (like a rectangle or circle) is 2-dimensional (2D) — we need two measurements: width and height. This is why area is measured in square units.
• A solid object (like a box) is 3-dimensional (3D).

2. US Customary Units of Length

The US customary system is used primarily in the United States. Unlike the metric system, it does not follow powers of 10:

Units (small → large):
• in (inch): about the width of a thumb — 12 inches = 1 foot
• ft (foot): about the height of a standard ceiling tile — 1 ft = 12 in
• yd (yard): about one long step — 1 yd = 3 ft = 36 in
• mi (mile): about 20 minutes walking — 1 mi = 1,760 yd = 5,280 ft

US Customary Length Conversions

1\text{ mi} = 1{,}760\text{ yd} \\ 1\text{ yd} = 3\text{ ft} \\ 1\text{ ft} = 12\text{ in}

3. Metric Units of Length

The metric system is used in most countries worldwide. It is based entirely on powers of 10:

Units (small → large):
• mm (millimetre): about the width of a pencil tip
• cm (centimetre): about the width of a fingernail — 1 cm = 10 mm
• dm (decimetre): 1 dm = 10 cm = 100 mm
• m (metre): about the height of a door — 1 m = 100 cm = 1,000 mm
• km (kilometre): about 10 minutes walking — 1 km = 1,000 m

Metric Length Conversions

1\text{ m} = 10\text{ dm} \\ 1\text{ dm} = 10\text{ cm} \\ 1\text{ cm} = 10\text{ mm}

4. Converting Between Length Units

To convert a length, multiply when going from a larger unit to a smaller one, and divide when going from a smaller unit to a larger one:

US Customary:
• ft → in: multiply by 12 (e.g. 2 ft = 24 in)
• in → ft: divide by 12
• yd → ft: multiply by 3
• ft → yd: divide by 3

Metric:
• cm → mm: multiply by 10 (e.g. 5 cm = 50 mm)
• mm → cm: divide by 10 (e.g. 30 mm = 3 cm)
• m → cm: multiply by 100
• cm → m: divide by 100

Metric Prefixes

\begin{aligned} \text{tera}\text{ (T)} &= 10^{12} \\ \text{giga}\text{ (G)} &= 10^9 \\ \text{mega}\text{ (M)} &= 10^6 \\ \text{kilo}\text{ (k)} &= 10^3 \\ \text{centi}\text{ (c)} &= 10^{-2} \\ \text{milli}\text{ (m)} &= 10^{-3} \\ \text{micro}\text{ (}\mu\text{)} &= 10^{-6} \\ \text{nano}\text{ (n)} &= 10^{-9} \\ \text{pico}\text{ (p)} &= 10^{-12} \end{aligned}

Scale factors relative to the base unit

5. Converting Between US & Metric Units

Sometimes we need to convert measurements between the US customary and metric systems. Here are the most common length conversion factors:

• inch to cm: multiply by 2.54 (e.g., 2 in = 5.08 cm)
• foot to cm: multiply by 30.48
• mile to km: multiply by 1.609 (e.g., 10 mi ≈ 16.09 km)

To go the opposite way (metric to US customary), simply divide by these conversion factors instead of multiplying.

US & Metric Length Conversions

\begin{aligned} 1\text{ in} &= 2.54\text{ cm} \\ 1\text{ ft} &= 30.48\text{ cm} \\ 1\text{ mi} &\approx 1.609\text{ km} \end{aligned}

Converting between US Customary and Metric length units

6. Units of Area

Because area is length × length, the unit of area is the square of the length unit:

Metric area units: mm², cm², dm², m², km²
US area units: in², ft², yd², mi²

For example, 1 cm² means one square that is 1 cm wide and 1 cm tall. A rectangle 3 cm wide and 5 cm tall has an area of 15 cm².

7. Converting Area Units — The Critical Insight

⚠️ Area conversion is NOT the same as length conversion!

Because area = length × length, when you convert the length unit, you must apply the conversion factor twice (once for each dimension).

Example: 1 cm² to mm²
1 cm = 10 mm, so:
1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm²

1 cm² = 100 mm² (not 10!)

Think of it this way: a 1 cm × 1 cm square can be divided into a 10 × 10 grid of 1 mm × 1 mm squares — that is 100 tiny squares in total.

Area Unit Conversion Rule

\text{if}\ 1\ \text{unit} = k\ \text{sub-units} \\ \text{then}\ 1\ \text{unit}^2 = k^2\ \text{sub-units}^2

Metric Area Conversions

1\text{ m}^2 = 100\text{ dm}^2 \\ 1\text{ dm}^2 = 100\text{ cm}^2 \\ 1\text{ cm}^2 = 100\text{ mm}^2

US Customary Area Conversions

1\text{ mi}^2 = 3{,}097{,}600\text{ yd}^2 \\ 1\text{ yd}^2 = 9\text{ ft}^2 \\ 1\text{ ft}^2 = 144\text{ in}^2

8. Ratio of Areas — Same Dimensions & Units

A ratio compares two quantities of the same type and same unit. When both areas share the same unit, the ratio is a pure number — the units cancel out.

Example: Part A has an area of 12 cm². Part B has an area of 8 cm².
Ratio of A to B = 12 cm² / 8 cm² = 12/8 = 3/2
Written as: 3:2 (simplified).

Important: If the two areas are in different units, you must convert them to the same unit first, then take the ratio.

Enter your ratio answer using the a:b format (e.g. 3:2), fully simplified.

Mastering SealMath: Ratio & Unit Answers

For ratio problems: enter your answer in a:b format — for example, type 3:2 for a ratio of 3 to 2. Make sure the ratio is fully simplified (no common factors).

For area calculation problems: enter your numerical answer with A = value as usual. The unit is displayed in the question — you do not need to type it in the answer.

Frequently Asked Questions

Why does 1 cm² equal 100 mm² and not just 10 mm²?

Because area is two-dimensional (length × length), the conversion factor must be applied twice. Since 1 cm = 10 mm, we have 1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm². In general, if 1 unit = k sub-units, then 1 unit² = k² sub-units². The same logic applies to all other area unit conversions.

How is the area of a shape defined?

The area of a shape is defined by how many unit squares of 1 by 1 fit inside it. For example, if a rectangle can be divided exactly into 30 squares of 1 by 1, its area is 30.