Areas Related to Circles • Topic 1 of 3

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 = $\pi \cdot r^2$
MeasurementPhysical MeaningFormulaPrimary Units
CircumferenceOuter boundary line length$2 \cdot \pi \cdot r$cm, m, km (linear units)
AreaInside flat space surface$\pi \cdot r^2$sq. cm, sq. m (square units)

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DIAGRAM 1: BOUNDARY LINE VS INSIDE SURFACE

           Circumference (Perimeter)             Area (Enclosed Space)
                . - - ~ ~ - - .                     . - - ~ ~ - - .
            .                   .               . * * * * * * * * * .
          /                       \            / * * * * * * * * * * \
         /                         \          / * * * * * * * * * * * \
        ;             O-------------;        ; * * * * * * O * * * * * ;
         \               Radius (r)/          \ * * * * * * * * * * * /
          \                       /            \ * * * * * * * * * * /
            .                   .               . * * * * * * * * * .
                . - - _ _ - - .                     . - - _ _ - - .

DIAGRAM 2: LINEAR RELATIONSHIPS
        
        <----------------------- Diameter (d) ----------------------->
        . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
        |                             |                             |
        <-------- Radius (r) --------><-------- Radius (r) -------->
Solved Examples
1
Worked Example
A circular garden has a radius of 14 meters. Find the total distance a gardener walks to complete one full lap around its outer fence, and calculate the total surface area of the grass inside. (Take $\pi = 22/7$).
Solution
  1. Step 1: Identify the given dimensions.
  2. Radius ($r$) = 14 meters.
  3. Step 2: Calculate the perimeter (circumference) of the circular garden.
  4. Circumference = $2 \cdot \pi \cdot r = 2 \cdot (22 / 7) \cdot 14$
  5. Circumference = $2 \cdot 22 \cdot 2 = 88$ meters.
  6. Step 3: Calculate the surface area of the grass field.
  7. Area = $\pi \cdot r^2 = (22 / 7) \cdot 14 \cdot 14$
  8. Area = $22 \cdot 2 \cdot 14 = 44 \cdot 14 = 616$ square meters.
  9. Answer: Circumference is 88 meters and Area is 616 square meters.
2
Worked Example
The total outer boundary length of a circular pond is found to be 176 meters. Calculate the exact radius of the pond and find its total area. (Take $\pi = 22/7$).
Solution
  1. Step 1: Set up the circumference equation to isolate the unknown radius ($r$).
  2. Circumference = $2 \cdot \pi \cdot r = 176$
  3. $$2 \cdot (22 / 7) \cdot r = 176$$
  4. $$(44 / 7) \cdot r = 176$$
  5. Step 2: Solve for $r$.
  6. $$r = 176 \cdot (7 / 44)$$
  7. Since $176 / 44 = 4$, we get $r = 4 \cdot 7 = 28$ meters.
  8. Step 3: Substitute this calculated radius into the area formula.
  9. Area = $\pi \cdot r^2 = (22 / 7) \cdot 28 \cdot 28$
  10. Area = $22 \cdot 4 \cdot 28 = 88 \cdot 28 = 2464$ square meters.
  11. Answer: Radius is 28 meters and Area is 2464 square meters.
3
Worked Example
A large bicycle wheel has a diameter of 70 cm. How many full, complete revolutions must the wheel make to travel a total road distance of 1.1 kilometers? (Take $\pi = 22/7$).
Solution
  1. Step 1: Convert all given values to consistent linear units.
  2. Diameter = 70 cm, which means Radius ($r$) = 35 cm.
  3. Total target distance = 1.1 km = $1.1 \cdot 1000$ meters = 1100 meters = 110000 cm.
  4. Step 2: Find the distance covered in exactly one single full wheel rotation.
  5. Distance in 1 revolution = Circumference of the wheel = $2 \cdot \pi \cdot r$
  6. Distance = $2 \cdot (22 / 7) \cdot 35 = 2 \cdot 22 \cdot 5 = 220$ cm.
  7. Step 3: Calculate the total number of required rotations.
  8. Number of Revolutions = Total Target Distance / Distance in 1 Revolution
  9. Number of Revolutions = $110000 / 220 = 11000 / 22 = 500$ revolutions.
  10. Answer: 500 revolutions.
  11. --

Key Points

  • The perimeter of any circle is also called its circumference, calculated as $2\pi r$.
  • The area measures the inside flat space using the formula $\pi r^2$.
  • The constant Pi ($\pi$) is an irrational ratio value roughly equal to $22/7$ or $3.14$.
  • When a circular object rolls forward on the ground, the linear distance it travels in exactly one full spin equals its circumference.
  • If you double the radius of a circle, its perimeter doubles, but its area increases by four times ($2^2$).
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The circumference of a circle is:
Explanation: $2\pi r$.
Q2.The area of a circle is:
Explanation: $\pi r^2$.
Q3.With $r=7$ and $\pi=\tfrac{22}{7}$, the area is:
Explanation: $\tfrac{22}{7}\cdot49=154$.
Q4.With $r=7$, the circumference is:
Explanation: $2\cdot\tfrac{22}{7}\cdot7=44$.
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