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.
📖 Area Study Guide
1. What Is a Dimension?
• 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. Metric Units of Length
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
Key conversions: 1 m = 100 cm, 1 cm = 10 mm
3. US Customary Units of Length
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
Key conversions: 1 ft = 12 in, 1 yd = 3 ft
4. Converting Between Length Units
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
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
5. Units of Area
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.
6. Converting Area Units — The Critical Insight
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!)
The rule: if 1 unit = k sub-units, then 1 unit² = k² sub-units²
Common conversions:
• 1 cm² = 100 mm² → 1 mm² = 0.01 cm²
• 1 m² = 10,000 cm² → 1 cm² = 0.0001 m²
• 1 ft² = 144 in² → 1 in² = 1/144 ft²
• 1 yd² = 9 ft² → 1 ft² = 1/9 yd²
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.
7. Ratio of Areas — Same Dimensions & Units
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
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
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.
How do you calculate the area of a rectangle and a square?
The area of a rectangle is calculated as width × height (A = w × h). A square is a special type of rectangle where all sides are equal (w = h = s). Thus, the area of a square is side × side, or side squared (A = s²).
How do you calculate the area of a right-angled triangle?
The area of a right-angled triangle is calculated by multiplying its two perpendicular legs and dividing by 2 (A = ab / 2). This is because a right-angled triangle is exactly half of a rectangle with the same width and height.
How do you calculate the area of a general triangle?
The area of any general triangle is calculated as half the base times the height (A = (1/2) × b × h or A = bh/2), where the height is the perpendicular distance from the base to the opposite vertex.
How do you calculate the area of a circle?
The area of a circle is calculated as π times the radius squared (A = πr²). Since π (pi) is an irrational number, the area of a circle with a rational radius will always be an irrational number.
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). It is used to find a missing side length when the other two are known.
What is the area ratio of inscribed circles and squares?
For a square inscribed in a circle, the area ratio of the square to the circle is always exactly 2/π ≈ 0.637. For a circle inscribed in a square, the area ratio of the circle to the square is always exactly π/4 ≈ 0.785. These ratios are constant regardless of the shapes' actual sizes.
Can the area of a shape be an irrational number?
Yes, if the side lengths are irrational numbers (such as √2), the resulting area can be either rational or irrational. You can learn more about these classifications in our Number Sets - Real & Complex topic.
What are the main types of special triangles?
The three main types are: isosceles (two equal sides and two equal base angles), equilateral (all sides and angles equal — each angle is 60°), and the 30-60-90 right triangle (fixed side ratios of 1 : √3 : 2).
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.