Perimeter and Area - Circumference and Area of a Circle

Concept 1: Definition of a Circle and its Center. A circle is a simple closed curve where every point on the boundary is at an equal distance from a fixed point inside called the center. Visually, you can imagine a compass where the needle stays fixed at the center while the pencil moves to draw the round boundary.

Concept 2: Radius and Diameter. The radius ($r$) is the distance from the center to any point on the circle's edge. The diameter ($d$) is a straight line that passes through the center and connects two points on the circle. Visually, the diameter is a straight line cutting the circle into two equal halves, and mathematically, it is twice the radius: $d = 2r$.

Concept 3: Circumference ($C$). The circumference is the distance around the edge of the circle, effectively its perimeter. If you imagine a circular wheel rolling one complete revolution, the distance it travels on the ground is equal to its circumference.

Concept 4: The Constant Pi ($\pi$). For every circle, the ratio of its circumference to its diameter is always constant. This constant is denoted by the Greek letter $\pi$. Since $\frac{C}{d} = \pi$, we derive the formula $C = \pi d$. For calculations, we usually use $\pi \approx \frac{22}{7}$ or $\pi \approx 3.14$.

Concept 5: Area of a Circle ($A$). The area is the measure of the region enclosed inside the circle's boundary. Visually, it is the flat surface area covered by the circle. The formula is derived as $A = \pi r^2$.

Concept 6: Relationship between Sectors and Area. If a circle is cut into several small sectors (like thin pizza slices) and rearranged, they form a shape resembling a rectangle. The height of this rectangle is the radius ($r$) and the base length is half the circumference ($\pi r$), leading to the area formula: $Area = \text{length} \times \text{breadth} = \pi r \times r = \pi r^2$.

Concept 7: Semi-circle Perimeter and Area. A semi-circle is half of a full circle, visually appearing like a protractor. Its area is $\frac{1}{2} \pi r^2$. Its perimeter consists of the curved arc ($\pi r$) plus the straight diameter ($2r$), giving the total perimeter as $\pi r + 2r$.