10.1 Perimeter
Perimeter is the total length of the surrounding boundary of an enclosed region or shape.
Calculating Perimeter
- To find the perimeter of any polygon or composite shape, sum the lengths of all its outermost sides.
- Ensure all unit measurements are converted to the same unit (e.g., convert all lengths to centimeters or meters) before adding.
Estimating Perimeter
- Grid Method: Count the grid boundary segments or estimate side lengths using a square grid.
- Measuring Tape / Thread Method: For curved regions, use a string along the boundary and measure its length with a ruler.
10.2 Area of Triangles, Quadrilaterals, and Parallel Lines
Area Formulas
- Triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
- Parallelogram: $\text{Area} = \text{base} \times \text{height}$ (Height must be perpendicular to the base)
- Rhombus: $\text{Area} = \text{base} \times \text{height}$ OR $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ (where $d_1, d_2$ are lengths of the diagonals)
- Trapezium: $\text{Area} = \frac{1}{2} \times (a + b) \times h$ (where $a$ and $b$ are the parallel sides, and $h$ is the perpendicular distance between them)
- Kite: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ (where $d_1, d_2$ are the lengths of the two perpendicular diagonals)
Estimating Area
- 1 cm × 1 cm Grid Method:
- Count fully covered squares ($1 \text{ cm}^2$ each).
- Count half or more than half-covered squares as $1 \text{ cm}^2$.
- Ignore squares that are less than half covered.
10.3 Relationship between Perimeter and Area
Fixed Perimeter vs. Maximum Area
For a fixed perimeter, shapes with more equal sides enclose a larger area. For rectangles with a fixed perimeter:
- A square (where length = width) yields the maximum area.
- As the difference between length and width increases, the enclosed area decreases.
Fixed Area vs. Minimum Perimeter
For a fixed area, rectangles closer in dimension yield a smaller perimeter:
- A square yields the minimum perimeter for a given area.
- As the difference between length and width increases, the perimeter increases.