Area is a fundamental concept in geometry that defines the amount of two-dimensional space a shape occupies. Simply put, it’s the measurement of a surface.
Imagine you want to paint a shape. The area is essentially how much paint you would need to cover that shape completely.
Visual representation of area as the amount of paint needed to cover shapes.
Area of Basic Shapes: Formulas for Simplicity
For common shapes like rectangles, squares, and circles, we have specific formulas that make calculating the area straightforward.
Calculating the Area of a Rectangle
Let’s take a rectangle as an example. Suppose we have a rectangle with a width of 5 units and a height of 3 units. How do we find its area?
The formula to calculate the area of a rectangle is:
Area = width × height
Where:
- width (w) is the length of the shorter side.
- height (h) is the length of the longer side.
In our example, the width (w) is 5, and the height (h) is 3. Plugging these values into the formula:
Area = 5 × 3 = 15 square units
Therefore, the area of this rectangle is 15 square units.
To delve deeper into calculating areas of various shapes, you can explore our detailed guide on Area of Plane Shapes.
Finding Area by Counting Squares
Another intuitive way to understand area is by using a grid of squares. We can overlay a shape on a grid and count the squares it covers.
If we place our rectangle on a grid, we can see that it perfectly covers 15 squares. This confirms our calculation: the rectangle has an area of 15 square units.
For instance, if each square in the grid represents 1 meter on each side, then the area of the rectangle is 15 m² (15 square meters).
Square Meter vs. Meter Squared: Understanding the Terminology
In the metric system, the standard unit for area is the square meter. A square meter is the area of a square with sides that are each 1 meter long.
1 square meter
It’s crucial to use the correct terminology. We say “square meters” when referring to area, not “meters squared“. Although “meters squared” mathematically represents m², in the context of area measurement, “square meters” is the standard and preferred phrasing.
Visual comparison of 3 square meters as area versus 3 meters squared as linear measurement.
Besides square meters, other units like “square millimeters” (mm²), “square centimeters” (cm²), and “square kilometers” (km²) are also used to measure area, depending on the scale. You can learn more about metric area units at Metric Area.
Approximating Area: Dealing with Irregular Shapes
Sometimes, shapes aren’t perfectly aligned with a grid, or they are irregular. In such cases, we can find an approximate area by counting squares with a simple rule:
- If a square is more than half covered by the shape, count it as 1.
- If a square is less than half covered, count it as 0.
Let’s see this in action with a pentagon:
Approximating area of a pentagon by counting squares, applying the half-square rule.
Using this method, we can approximate the area of this pentagon to be approximately 17 square units.
Another approach for approximation is to visually combine parts of squares that seem to add up to roughly a whole square.
For example, in this circle:
Approximating area of a circle by visually combining partial squares to estimate whole squares.
Areas marked “4” and “8” appear to combine to form approximately one whole square each. By adding up the whole and approximated squares, we estimate the circle’s area to be approximately 14 square units.
The Precision of Formulas: Best for Accurate Area Calculation
While counting squares provides a good visual understanding and approximation, using a formula is generally the most accurate method for calculating area, especially when dealing with shapes that have defined mathematical properties.
Example: Calculating the Area of a Circle with a Formula
Consider a circle with a radius of 2.1 meters. To find its area, we use the formula:
Area = π × r²
Where:
- π (pi) is a mathematical constant approximately equal to 3.1416 (you can learn more about pi at number pi).
- r is the radius of the circle.
Calculating the precise area of a circle using the formula and radius.
In our example, the radius (r) is 2.1 meters. Plugging this into the formula:
Area = 3.1416… × (2.1m)²
= 3.1416… × (2.1m × 2.1m)
= 13.854… m²
Rounding to two decimal places, the circle has an area of 13.85 square meters.
Area of Complex Shapes: Divide and Conquer
For more complex or irregular shapes, a useful strategy is to break them down into simpler shapes whose areas we know how to calculate.
Example: Finding the Area of an Irregular Shape
Let’s calculate the area of this L-shaped region:
An L-shaped region for area calculation, demonstrating complex shape area determination.
We can divide this shape into two simpler parts: a square (Part A) and a triangle (Part B).
Dividing the L-shaped region into a square (A) and a triangle (B) for easier area calculation.
Part A (Square):
The side of the square is 20m. The area of a square is side × side or side².
Area of A = a² = 20m × 20m = 400m²
Part B (Triangle):
Viewing the triangle sideways, we can consider the base as 20m and the height as 14m. The area of a triangle is ½ × base × height.
Area of B = ½ × b × h = ½ × 20m × 14m = 140m²
Total Area:
To find the total area of the L-shape, we add the areas of Part A and Part B.
Total Area = Area of A + Area of B
Area = 400m² + 140m² = 540m²
Area Calculation by Triangulation
Another powerful technique for finding the area of complex shapes is to divide them into triangles.
Dividing an irregular polygon into triangles for area calculation by triangulation.
By dividing any polygon into triangles, we can calculate the area of each triangle and sum them up to get the total area of the polygon. To do this, measure the base (b) and height (h) of each triangle:
Measuring base and height of triangles within a polygon to calculate individual triangle areas.
Then calculate the area of each triangle using the formula: Area = ½ × b × h and add them together.
For further exploration, you can also use tools that help calculate area by drawing polygons, such as the Area of a Polygon by Drawing Tool.
Animated image demonstrating an online tool for calculating the area of irregular polygons.
Advanced Area Calculation: Using Coordinates
When the coordinates of each corner point of a polygon are known, more advanced methods like the Area of Irregular Polygons method can be used for precise area calculation.
Area is a crucial concept with applications in various fields, from calculating the space needed for gardening (Garden Area Activity) to more complex applications in engineering and design. Understanding “What Is Area” and how to calculate it is a fundamental skill in geometry and beyond.
Explore further: Area of Plane Shapes | Geometry Index
