Geometry and Measurement - Perimeter and Area of 2D Shapes including Circles

Perimeter is the total linear distance around the outside boundary of a 2D shape. For any polygon, it is the sum of all its side lengths. Visually, if you were to place a piece of string along the edge of a shape and then straighten it out, the length of that string is the perimeter.

Area is the measure of the total surface space contained within the boundaries of a 2D shape, expressed in square units like $cm^2$ or $m^2$. You can visualize area as the number of $1 \times 1$ unit squares required to perfectly tile the inside of the shape.

A parallelogram is a quadrilateral with two pairs of parallel sides. Its area is calculated by multiplying the base by the perpendicular height. Visually, if you cut a right-angled triangle from one side of a parallelogram and shift it to the other side, it forms a rectangle, which explains why the area formula $A = b \times h$ is used.

A triangle's area is always exactly half the area of a parallelogram with the same base and height. When calculating the area, the height ($h$) must be the perpendicular distance from the base to the opposite vertex. Visually, any triangle can be seen as half of a rectangle or parallelogram that encloses it.

A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides (labeled $a$ and $b$). The area is the product of the average of the parallel sides and the perpendicular height. Imagine the trapezium being transformed into a rectangle by 'leveling out' the slanted sides to the average width.

The circle is defined by its radius ($r$), the distance from the center to the edge, and its diameter ($d$), which is the distance across the circle through the center ($d = 2r$). The perimeter of a circle is specifically called the circumference. The constant $\pi$ (approximately $3.14159$ or $\frac{22}{7}$) represents the fixed ratio between any circle's circumference and its diameter.

The area of a circle represents the space inside the curved boundary. Visually, if a circle is sliced into many thin sectors (like pizza slices) and rearranged, they form a shape resembling a rectangle with a width of $\pi r$ and a height of $r$, leading to the formula $A = \pi r^2$.

Composite shapes are complex figures made by combining two or more simple 2D shapes (such as a rectangle joined to a semi-circle). To find the total area, calculate the areas of the individual parts and add them. To find the perimeter, sum only the exterior lengths that form the outer boundary.