Area
In geometry, area measures the amount of space enclosed within a two-dimensional figure. In simple terms, it tells us how much surface is covered by a shape on a plane.
Area is one of the most important concepts in geometry. It applies to two-dimensional figures, such as squares, rectangles, triangles, circles, and polygons.
Mathematically, area is a scalar quantity represented by a non-negative real number that indicates the size of a region in the plane.
The standard unit of area in the International System of Units (SI) is the square meter (m2). Smaller and larger units, such as square centimeters (cm2) and square kilometers (km2), are used depending on the size of the surface being measured.
Other units are commonly used in specific fields. For example, land areas are often measured in hectares (ha), where one hectare corresponds to 10,000 m2.
Note. The study of area dates back thousands of years. Ancient Greek mathematicians such as Euclid and Archimedes developed some of the first systematic methods for calculating the areas of geometric figures. Today, area plays a central role not only in mathematics but also in architecture, engineering, geography, agriculture, cartography, and urban planning.
Area in Physics
In physics, area is treated as a derived physical quantity because it is obtained from the fundamental quantity of length.
The SI unit of area is the square meter (m2), defined as the area of a square whose sides are each 1 meter long.
From the perspective of dimensional analysis, area has the dimension of length squared:
$$ [A] = L^2 $$
This reflects the fact that area is calculated by multiplying two lengths together.
For example, the area of a rectangle is found by multiplying its base by its height:
$$ A = \text{base} \times \text{height} $$
If a rectangle has a base of 10 meters and a height of 5 meters, then:
$$ A = (10 \, \text{m}) \times (5 \, \text{m}) = 50 \, \text{m}^2 $$
The result shows both the numerical calculation and the dimensional calculation:
$$ 10 \times 5 = 50 \qquad \text{and} \qquad m \times m = m^2 $$
Geometrically, an area of 50 m2 means that fifty unit squares, each measuring 1 meter by 1 meter, can fit inside the rectangle.
How to Calculate Area
The formula for area depends on the shape being considered.
- Area of a Triangle
Multiply the base by the corresponding height and divide the result by two. $$ A = \frac{\text{base} \times \text{height}}{2} $$ - Area of a Square
Square the length of one side. $$ A = s^2 $$ - Area of a Rectangle and a Parallelogram
Multiply the base by the corresponding height. $$ A = \text{base} \times \text{height} $$ - Area of a Rhombus
Multiply the diagonals and divide by two. $$ A = \frac{d_1 d_2}{2} $$ - Area of a Trapezoid
Add the lengths of the two parallel sides, multiply by the height, and divide by two. $$ A = \frac{(B+b)h}{2} $$ - Area of a Circle
Multiply the square of the radius by the constant pi. $$ A = \pi r^2 $$ - Area of a Regular Polygon
Multiply the perimeter by the apothem and divide by two. $$ A = \frac{Pa}{2} $$ - Area of an Irregular Figure
The area of an irregular figure can often be found by dividing it into simpler shapes whose areas are easier to calculate. In more advanced mathematics, integration can also be used.
Multiples and Submultiples of the Square Meter
The square meter (m2) is the standard metric unit of area. It represents the area of a square with sides measuring 1 meter.
The most commonly used multiples and submultiples are:
| mm2 | square millimeter |
| cm2 | square centimeter |
| dm2 | square decimeter |
| m2 | square meter |
| dam2 | square decameter |
| hm2 | square hectometer |
| km2 | square kilometer |
Each step upward in the table increases the area by a factor of 100, while each step downward decreases it by the same factor.
For example:
$$ 1 \, m^2 = 100 \, dm^2 $$
and
$$ 1 \, m^2 = 0.01 \, dam^2 $$
Converting Area Units
Because area is measured in square units, conversion factors must also be squared.
Some common conversions are:
- $$ 1 \, cm^2 = 10^{-4} \, m^2 $$
- $$ 1 \, mm^2 = 10^{-6} \, m^2 $$
- $$ 1 \, km^2 = 10^6 \, m^2 $$
For example, since:
$$ 1 \, cm = 10^{-2} \, m $$
then:
$$ 1 \, cm^2 = (10^{-2} \, m)^2 = 10^{-4} \, m^2 $$
The same principle applies to all area conversions: whenever a length conversion factor is used, it must be squared.
Suppose you want to convert an area of 100 m² into square decimeters. Since 1 meter equals 10 decimeters, the conversion factor is 102. $$ 100 \, m^2 = 100 \times 10^2 \, dm^2 = 10,000 \, dm^2 $$ To convert the same area into square centimeters, use the square of 100. $$ 100 \, m^2 = 100 \times (10^2)^2 \, cm^2 = 100 \times 10^4 \, cm^2 = 10^6 \, cm^2 $$ To convert it into square millimeters, use the square of 1000. $$ 100 \, m^2 = 100 \times (10^3)^2 \, mm^2 = 100 \times 10^6 \, mm^2 = 10^8 \, mm^2 $$ The same method applies to all area-unit conversions.
Equivalence Classes of Area
Two geometric figures belong to the same equivalence class with respect to area if they have exactly the same area.
From a mathematical perspective, area defines an equivalence relation among geometric figures.
Any two polygons with equal areas belong to the same equivalence class and are commonly known as equivalent polygons.
This concept is not limited to polygons and can be extended to other types of plane figures.
