Area and perimeter formulas Knowledge-Based Study

Study notes

1. What is Perimeter?

   The perimeter of a two-dimensional shape is the total distance around its outer boundary. To find the perimeter of any polygon, you simply add the lengths of all its exterior sides.

   Key Formulas

   • Square: Since all four sides are equal in length, $P = 4s$, where s is the side length.
   • Rectangle: Opposing sides are equal in length, giving $P = 2l + 2w = 2(l + w)$, where l represents length and w represents width.
   • Regular Polygon: For any regular polygon with n sides of equal length s, $P = n \cdot s$.

   Perimeter is a one-dimensional measurement, so it is expressed in linear units such as centimeters (cm), meters (m), or inches (in).

   What is Area?

   The area of a figure measures the amount of two-dimensional space enclosed within its boundary. It is always reported in square units, such as square centimeters ($\text{cm}^2$) or square meters ($\text{m}^2$).

   Rectangles and Squares

   • Rectangle: $A = l \cdot w$
   • Square: $A = s^2$

   Parallelograms and Triangles

   • Parallelogram: $A = b \cdot h$, where b is the base and h is the perpendicular height.
   • Triangle: A triangle covers exactly half the area of a parallelogram with the identical base and height: $A = \frac{1}{2} b \cdot h$.

   Note: The height must always be perpendicular (at a right angle) to the chosen base, not along the slanted side.

   Circumference and Circular Area

   Circles do not have straight edges, so their boundary length is referred to as the circumference ($C$). Every circle has a radius ($r$), which is the distance from the center to any point on the edge, and a diameter ($d$), which is twice the radius ($d = 2r$).

   Circumference Formula

   The ratio of a circle's circumference to its diameter is the constant $\pi$ (approximately 3.14159):

   • $C = 2\pi r$ or $C = \pi d$

   Area Formula

   The area enclosed by a circle depends on the square of its radius:

   • $A = \pi r^2$

   Example: A circle with radius $r = 5\text{ cm}$ has a circumference of $C = 2\pi(5) = 10\pi\approx 31.42\text{ cm}$ and an area of $A = \pi(5^2) = 25\pi\approx 78.54\text{ cm}^2$.

   Working with Composite Shapes

   A composite figure is a shape made up of two or more basic geometric figures, such as rectangles, triangles, or semicircles.

   Step-by-Step Strategy

   1. Decompose the figure: Divide the complex shape into simpler components whose formulas you know.
   2. Calculate component values: Determine the missing dimensions using known side lengths and properties.
   3. Combine the results: For area, sum the areas of the individual regions (or subtract non-shaded cutouts). For perimeter, sum only the segments that form the outer boundary.

   Caution: When finding the perimeter of a composite shape, do not include internal shared boundary lines that lie inside the figure.