Circumference and Area of a Circle

By the end of this lesson you will be able to:

• Name the parts of a circle and explain what π means
• Calculate circumference using
• Calculate area using
• Decide which formula a real-life problem needs

Questions About the School Water Tank

A cylindrical water tank has a circular base, diameter 2 m.

• Metal band around the rim: how long?
• Sheet metal for the floor: how much?

Two different questions, same circle.

Recall: What Is a Circle's Perimeter Called?

Last lesson: perimeter = total distance around a shape's boundary.

Quick recall: What would you call the perimeter of a circle?

The distance around a circle is called its circumference.
Circumference IS the perimeter of a circle — just a special name for a round shape.

Parts of a Circle: r, d, C

• Centre — fixed middle point
• Radius () — centre to edge; cm here
• Diameter () — edge to edge through centre;
• Circumference () — distance around

What Is π? A Ratio Every Circle Shares

Always the same ratio: π.

Use for multiples of 7; otherwise.

Bridge: π and d Give Us Circumference

School water tank:

• m = 200 cm

One step:

Estimate the band length. (Think first.)

Circumference Formula Derived From π Definition

• Diameter given →
• Radius given →
• Length only: answer in cm or m, never cm²

Probe: Catch the Circumference Error

Tank diameter = 200 cm. A student writes:

"C = π × r = 22/7 × 100 = 314 cm"

Right or wrong? Decide before turning to the next slide.

Pause and commit: write your prediction now.

Probe Explained: Diameter, Not Radius

Student used cm instead of cm:

Half the diameter → half the answer.

does not exist: use or .

Worked Example: Circumference from Radius

cm, cm:

cm, cm:

22/7 cancels: exact answers.

Real-Life Circumference: How Far a Wheel Rolls

Diameter = 70 cm. Distance per full turn?

One turn = 220 cm = 2.2 m.

Key idea: Circumference = distance per turn. Larger wheel → more ground.

Check-In: Find Three Circle Circumferences

Find the circumference of each circle. Show your working. Use .

(a) Diameter = 28 cm

(b) Radius = 14 m

(c) Diameter = 35 mm

(Answers: (a) 88 cm (b) 88 m (c) 110 mm)

Bridge: Circumference Is a Length, Not Area

Circumference answers "how far around?" — a length.

The tank floor question is different: how much surface does the base cover?

gives length, not area. A different formula is needed — derived by cutting the circle apart.

Where Does Come From?

1. Cut into 8 equal sectors
2. Interleave — point up, point down
3. Near-parallelogram forms:
   • Base ≈ (half circumference)
   • Height ≈

More sectors → straighter edges → exactly

The Area Formula:

• Input: radius — halve if diameter given
• Answer in square units (cm², m²)
• Using not → 4× too large
• π is a number: always substitute and compute

Worked Example: Area from Radius

Example 1: cm

Example 2: cm

Doubling quadruples the area.

Worked Example: Area When Diameter Is Given

cm:

If not halved first:

4× too large: halve the diameter first.

Check-In: Find Three Circle Areas

Find the area of each circle. Use . Show full working.

(a) cm

(b) m

(c) m

(a) 1 386 cm² · (b) 154 m² · (c) 38.5 m²

Bridge: Which Formula Does the Problem Need?

Before computing: length or covering?

The expected unit tells you which formula.

Decision Framework: Length or Covering?

1. Going around the edge? →

   • Fencing, ribbon, rolling distance

2. Covering the inside? →

   • Painting, grass, sheet metal

Tip: Answer in cm/m → circumference. Answer in cm²/m² → area.

Sorting Task: Classify Before Computing

C or A? No computing.

1. Pond — fencing the edge
2. Tabletop — painting it
3. Tin — paper label around the side
4. Lawn — spread fertiliser across
5. Wheel (d = 56 cm) — distance per turn

(Answers: 1-C, 2-A, 3-C, 4-A, 5-C)

Worked Example: Same Circle, Two Questions

Circular pond, m.

Fencing (circumference):

Water surface (area):

Same circle, two formulas, two units: 22 m of fence; 38.5 m² of surface.

Worked Example: The Tank Rim and Floor

Tank: cm, cm.

Rim band:

Floor:

Rim ≈ 6.3 m; floor ≈ 3.14 m². Same π — different units.

Worked: Area of a Composite Figure

Rectangle m × m with a semicircular end (diameter m, m).

Rectangle:

Semicircle:

Total:

Unscaffolded Challenge: Flower Bed Cost

Flower bed, diameter 14 m.

(a) Edging length around it.

(b) Soil area inside for planting.

(c) Edging UGX 2 000/m; soil prep UGX 5 000/m². Total?

(a) 44 m · (b) 154 m² · (c) UGX 858 000

Closer: One Ratio, Two Formulas, Two Units

— one ratio, every circle.

• Circumference: → cm, m
• Area: → cm², m²

Remember:

1. not
2. Halve before squaring
3. Unit → formula

Next: