Learn: Circles

A quick, calm intro — then you choose when to climb.

The idea

A circle has two main measurements — the rim and the inside. Two formulas, one job each.

The two formulas:

1
Circumference — distance around the rim: $C = 2\pi r$ — a length, in cm or m
2
Area — space inside: $A = \pi r^2$ — a surface, in cm² or m²
3
Read the question — rim or inside? Each formula answers exactly one of them

Quick memory tips

$\pi \approx \frac{22}{7}$ — use it when the radius plays nicely (7, 70, 14…)
Square the radius FIRST: $\pi r^2$ means $\pi \times (r \times r)$
Units tell the truth: cm = rim, cm² = inside

Circumferenceunit: cm

The distance around the rim.

C = 2\pi r

Areaunit: cm²

The space inside the circle.

A = \pi r^{2}

Radius vs Diameter

The radius is half the diameter — don't mix them up.

r = \frac{d}{2} \quad \text{or} \quad d = 2r

Example: diameter = 14 → radius = 14 ÷ 2 = 7 (not 14!)

Blue = around the rim (circumference) · Green = inside the circle (area) · Amber = radius vs diameter warning

Worked example

Every step explained — why we do it, and what happens.

1
$r = 7 \text{ cm}$
Why: The radius is the distance from the centre to the edge — half of the diameter.
What happens: A circle with radius 7.
2
$C = 2\pi r = 2 \times \frac{22}{7} \times 7 = 44 \text{ cm}$
Why: Circumference is the distance around the rim: 2πr. The 7 cancels the 7 in 22/7 nicely.
What happens: 44 cm around.
3
$A = \pi r^2 = \frac{22}{7} \times 7^2 = 154 \text{ cm}^2$
Why: Area is the space inside: πr². Square the radius FIRST (7² = 49), then multiply.
What happens: 154 cm² inside.
4
$C = 44 \text{ cm}, \qquad A = 154 \text{ cm}^2$
Same radius, two different answers: 44 (around, cm) vs 154 (inside, cm²). The formula decides.

Real-world example

A circular picnic table has a radius of 70 cm. A cloth must cover the whole tabletop, and a ribbon must go around the rim. How much cloth, and how much ribbon?

1
$r = 70 \text{ cm}$
Why: The radius of the table is 70 cm — the same r in both formulas.
What happens: Radius 70.
2
$A = \frac{22}{7} \times 70^2 = \frac{22}{7} \times 4900 = 15400 \text{ cm}^2$
Why: The cloth covers the top — that is the area: πr². 70² = 4900.
What happens: 15400 cm² of cloth.
3
$C = 2 \times \frac{22}{7} \times 70 = 440 \text{ cm}$
Why: The ribbon wraps the rim — that is the circumference: 2πr.
What happens: 440 cm of ribbon.
4
$\text{Trap: using } A = 2\pi r = 2 \times \frac{22}{7} \times 70 = 440$
Why: The glitch: swapping the formulas. That is the ribbon formula, not the cloth — 440 is the rim length in cm, not a surface at all. Cloth needs area: πr², in cm².
What happens: 440 cm is a length (ribbon); the cloth needs 15400 cm².
5
$A = 15400 \text{ cm}^2, \quad C = 440 \text{ cm}$
Why: The check: same radius, rim and inside both computed — each formula did its own job.
What happens: ✓ cloth 15400 cm², ribbon 440 cm.

Common glitch

Two classic glitches:

1.Swapping circumference and area

Wrong:Area = $2\pi r$ — using the rim formula for the inside

Right:Rim: $C = 2\pi r$ . Inside: $A = \pi r^2$ .

2.Using the diameter as the radius

Wrong:diameter = 14, so $r = 14$

Right:The radius is half the diameter: $r = 14 \div 2 = 7$ .

Area and Perimeter Calculator — Calculate a circle’s circumference and area.

Next up: Pythagorean Theorem →

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Common glitch

Two classic glitches:

1.Swapping circumference and area

Wrong:Area = $2\pi r$ — using the rim formula for the inside

Right:Rim: $C = 2\pi r$ . Inside: $A = \pi r^2$ .

2.Using the diameter as the radius

Wrong:diameter = 14, so $r = 14$

Right:The radius is half the diameter: $r = 14 \div 2 = 7$ .

Learn: Circles

The idea

The two formulas:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Circles

The idea

The two formulas:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator