Perimeter and area of a circle

What is the perimeter and area of a circle? The perimeter of a circle is the total distance around its outer boundary. In geometry, this specific boundary length is called the circumference. The area of a circle represents the total amount of flat space enclosed inside that boundary. Imagine a circular running track: if you run all the way around the outer white line, you have covered the circumference. If you need to cover the grass field inside the track with fresh turf, you are calculating the area.

To measure these values, mathematicians use a special constant called Pi (written as the Greek symbol $\pi$). Pi represents a fixed ratio: the circumference of any circle divided by its diameter. No matter how small a coin or how massive a ferris wheel is, this ratio is always the same! For calculations, we approximate $\pi$ as 22/7 or 3.14.

• Radius (r): The straight-line distance from the exact center of the circle to any point on its outer edge.
• Diameter (d): The maximum straight distance across a circle, passing through the center. It is always equal to twice the radius ($d = 2r$).

Formulas for calculations:

• Circumference of a circle = $2 \cdot \pi \cdot r$
• Area of a circle = $\pi \cdot r^2$

Deriving area from circumference

Why is the area $\pi r^2$? Slice a circle into many thin sectors and rearrange them alternately to form a near-rectangle. The rectangle's length is half the circumference, $\frac{1}{2}(2\pi r)=\pi r$, and its breadth is the radius $r$. Area $=\text{length}\times\text{breadth}=\pi r\times r=\pi r^2$. The thinner the slices, the closer the shape gets to a true rectangle, which is why the formula is exact.

Distance covered by a rolling wheel

When a wheel makes one complete turn, every point on its rim returns to its start, and the wheel advances forward by exactly one circumference. So:

$$\text{Distance} = (\text{Number of revolutions}) \times (\text{Circumference})$$

Rearranged, $\text{Revolutions} = \dfrac{\text{Total distance}}{2\pi r}$. Always convert speed (km/h) to distance first, and keep every length in the same unit before dividing.

Area of a ring (annulus)

A circular path, washer, or pipe cross-section is the region between two concentric circles. If the outer radius is $R$ and the inner radius is $r$, the shaded ring area is the big circle minus the small one:

$$\text{Ring area}=\pi R^2-\pi r^2=\pi\left(R^2-r^2\right)=\pi(R+r)(R-r).$$

For a path of uniform width $w$ around a circle of radius $r$, the outer radius is $R=r+w$.

Two circles combined into one

If two circles of radii $r_1$ and $r_2$ are melted/recast into a single circle of radius $R$, the property that is conserved decides the equation. If their areas add up: $\pi R^2=\pi r_1^2+\pi r_2^2\Rightarrow R^2=r_1^2+r_2^2$. If their circumferences add up: $2\pi R=2\pi r_1+2\pi r_2\Rightarrow R=r_1+r_2$.

Common mistakes to avoid

• Radius vs diameter: a question often gives the diameter. Halve it first. A 70 cm wheel has $r=35$ cm, not 70.
• Choosing $\pi$: use $\pi=\frac{22}{7}$ when the radius is a multiple of 7 (the 7 cancels neatly); use $\pi=3.14$ otherwise, or when the problem says so. Never mix both in one calculation.
• Units: circumference is a length (cm, m), area is square units ($\text{cm}^2$, $\text{m}^2$). Convert km to m and m to cm before computing revolutions.
• Scaling: doubling $r$ doubles the circumference but multiplies the area by $4$, because area depends on $r^2$.