Area of a Rectangle: Calculated as the product of its length and width.
$$\text{Area} = \text{length} \times \text{width}$$
Unit Squares: Area is measured by finding the number of non-overlapping unit squares (e.g., $1\text{ cm}^2$) that can pack a region.
Perimeter vs. Area: Perimeter (boundary length) is not a measure of area. Two regions can have the same perimeter but different areas, or a larger perimeter but smaller area.
Dissection: The process of cutting a geometric figure into pieces and rearranging them to form a different figure of equal area.
7.2 Triangles
Formulas & Geometric Properties
Area of a Triangle: Derived from half the area of an enclosing rectangle.
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
Median Property: In any triangle, the line joining a vertex to the midpoint of its opposite side (the median) divides the triangle into two triangles of equal area.
Triangles between Parallel Lines: Triangles with a common base and vertices lying on a line parallel to the base have equal areas (since they share the same base and height).
Minimum Perimeter: Among all triangles with a common base and vertices on a parallel line, the isosceles triangle (where the vertex lies on the perpendicular bisector of the base) has the minimum perimeter.
7.3 Polygons, Parallelograms, and Rhombuses
Area of Polygons & Parallelograms
Area of any Polygon: Can be evaluated by dividing (triangulating) the polygon into non-overlapping triangles and summing their areas.
Area of a Parallelogram: Formulated by dissecting a right triangle from one side and moving it to the other to form a rectangle.
$$\text{Area} = \text{base} \times \text{height}$$
Area of a Rhombus
Area of a Rhombus: Since diagonals of a rhombus are perpendicular bisectors of each other, it can be dissected into a rectangle of equal area.
$$\text{Area} = \frac{1}{2} \times \text{product of diagonals} = \frac{1}{2} \times d_1 \times d_2$$
7.4 Trapeziums
Area of a Trapezium
Area Formula: Derived by splitting the trapezium into a rectangle and two triangles, or by joining two identical copies to form a larger parallelogram.
$$\text{Area} = \frac{1}{2} \times \text{height} \times \text{sum of parallel sides} = \frac{1}{2} \times h \times (a + b)$$