Circumference and area refresher

Before tackling sectors and segments, let's revisit the two fundamental measurements of a full circle. Both involve the same mysterious constant $\pi$ (read "pi"), which is the ratio of the circumference to the diameter of any circle. This ratio is the same for every circle , small or large , and is approximately $22/7 \approx 3.1428$ or, more accurately, $3.14159\ldots$.

Definitions and formulas

For a circle of radius $r$ (and hence diameter $d = 2r$):

• Circumference (perimeter, the total length around the circle): $C = 2\pi r = \pi d$.
• Area (the region enclosed): $A = \pi r^2$.

The first formula is the definition of $\pi$: $\pi = C/d$. The second can be derived by slicing the circle into thin wedges and rearranging them into an approximate rectangle of width $r$ and length $\pi r$ , giving area $\pi r^2$.

Which value of $\pi$ to use?

For arithmetic questions, the textbook often specifies. Common choices:

• $\pi = 22/7$: convenient when the radius (or some quantity) is divisible by $7$. Use it when the problem says "Use $\pi = 22/7$" or when the numbers obviously simplify with this value.
• $\pi = 3.14$: convenient when the radius is in decimals or when no specific instruction is given.
• $\pi$ as a symbol: sometimes the question asks for the answer "in terms of $\pi$" , leave $\pi$ unevaluated.

Always read the question carefully. Using $3.14$ instead of $22/7$ can be the difference between $0$ and full marks if the textbook specifies.

Some classic computations

Computation 1. Radius $7$ cm. Circumference and area.

$C = 2 \cdot 22/7 \cdot 7 = 44$ cm.

$A = 22/7 \cdot 49 = 22 \cdot 7 = 154$ cm$^2$.

Computation 2. Diameter $14$ cm. Same.

$r = 7$. Same answers as above.

Computation 3. Circumference $44$ m. Find the area.

$44 = 2 \cdot 22/7 \cdot r \Rightarrow r = 44 \cdot 7 / 44 = 7$ m. So $A = 22/7 \cdot 49 = 154$ m$^2$.

Computation 4. Two circles with radii $r_1$ and $r_2$. The ratio of their circumferences is $2\pi r_1 : 2\pi r_2 = r_1 : r_2$. The ratio of their areas is $\pi r_1^2 : \pi r_2^2 = r_1^2 : r_2^2$.

If $r_1 : r_2 = 2 : 3$, then circumferences are in $2:3$ but areas in $4:9$.

Two circles, sums and differences

When two circles are involved in a problem, you often need the radius of a third circle whose area or circumference equals the sum or difference. Use the fact that area scales with $r^2$ and circumference with $r$.

Problem. The areas of two circles are $S_1$ and $S_2$. A third circle has area $S_1 + S_2$. Find its radius in terms of $r_1$ and $r_2$.

$\pi R^2 = \pi r_1^2 + \pi r_2^2 \Rightarrow R = \sqrt{r_1^2 + r_2^2}$.

Problem. The circumferences of two circles add up to $C$. The radius of a third circle whose circumference is $C$ equals the sum of the two radii: $R = r_1 + r_2$. (Easy because circumference is linear in $r$.)

Ratio of areas as $\pi$ cancels

Whenever you compute a ratio of two areas (or two circumferences), $\pi$ cancels out. Use this fact to avoid computing $\pi$ explicitly when only a ratio is asked.

Worked examples

Example 1. A circle has radius $21$ cm. Find its circumference and area.

$C = 2 \cdot 22/7 \cdot 21 = 2 \cdot 22 \cdot 3 = 132$ cm.

$A = 22/7 \cdot 441 = 22 \cdot 63 = 1386$ cm$^2$.

Example 2. A circle's area is $616$ cm$^2$. Find its radius.

$\pi r^2 = 616 \Rightarrow r^2 = 616 \cdot 7/22 = 196 \Rightarrow r = 14$ cm.

Example 3. The circumference of a circle exceeds its diameter by $30$ cm. Find the radius.

$2\pi r - 2r = 30 \Rightarrow 2r(\pi - 1) = 30 \Rightarrow r = 15/(\pi - 1)$. With $\pi = 22/7$, $\pi - 1 = 15/7$, so $r = 15 \cdot 7/15 = 7$ cm.

Example 4. A wheel of diameter $42$ cm rolls $44$ metres. Find the number of revolutions.

Circumference $= \pi d = 22/7 \cdot 42 = 132$ cm. Distance $= 4400$ cm. Revolutions $= 4400/132 \approx 33.33$. (If the question insists on a whole number, the wheel completes $33$ full revolutions.)

Example 5. Two circles have radii in ratio $3:4$. Find the ratio of their areas.

$r_1^2 : r_2^2 = 9 : 16$.

Try it yourself

1. Find the circumference of a circle of radius $14$ cm.
2. Find the area of a circle of diameter $28$ cm.
3. A circle has circumference $66$ cm. Find its area.
4. Two circles of radii $5$ and $12$. Find the radius of a circle whose area equals the sum.
5. A wheel of radius $35$ cm rolls $22$ m. Find the number of revolutions.
6. The ratio of areas of two circles is $9 : 16$. Find the ratio of their circumferences.
7. A circle's area is $154$ cm$^2$. Find the radius and circumference.
8. The circumference of a circle exceeds its diameter by $11$ cm (use $\pi = 22/7$). Find $r$.
9. A circular flower bed has area $1386$ m$^2$. A path of width $3$ m runs around it. Find the area of the path (the annulus).
10. Two circles of radii $r_1, r_2$. Show that the area enclosed between the two concentric circles is $\pi(r_1^2 - r_2^2)$.
11. A circular pond of radius $10$ m has a path of width $2$ m around it. Find the path's area.
12. Find the diameter of a circle whose area equals the sum of the areas of two circles of diameters $10$ and $24$.

Pitfalls / Insight

A common confusion is between circumference (perimeter) and area. The first has units of length; the second has units of length squared. Always check units before submitting an answer.

Also, "use $\pi = 22/7$" is a specific instruction , use it whenever the textbook says so. Switching to $3.14$ in such problems leads to non-clean answers that the examiner spots immediately.

Finally, when adding circumferences or areas, add the linear or quadratic combinations: $C_3 = C_1 + C_2 \Rightarrow r_3 = r_1 + r_2$, and $A_3 = A_1 + A_2 \Rightarrow r_3 = \sqrt{r_1^2 + r_2^2}$. Don't mix these up.