Area and circumference of a circle — one relationship connects both formulas.
The Mnemonic
Two Formulas Built From the Same Radius
A circle has two core measurements: Area, the space inside it (A = πr²), and Circumference, the distance around it (C = 2πr). Both formulas are built entirely from the radius r — the distance from the center to any point on the circle.
The diameter (d = 2r) is the full width of the circle through the center — twice the radius. Circumference can also be written as C = πd, since 2r and d are the same value.
Radius r goes center to edge. Diameter d = 2r goes edge to edge through the center
💡 Memory Trick
"Area and circumference of a circle — one rhyme covers both." A = πr² (area uses radius SQUARED). C = 2πr (circumference uses radius, not squared). The squared term is always the giveaway for area — squaring naturally produces a measurement of surface/space, while circumference is just a length, so it stays linear (no exponent).
Why It Works
Where π Actually Comes From
π (pi) is defined as the ratio of any circle's circumference to its diameter — literally C/d = π for every circle that has ever existed, regardless of size. Rearranging that definition directly gives C = πd, and since d = 2r, that becomes C = 2πr.
The area formula is less immediately obvious but can be shown by slicing a circle into many thin pie-shaped wedges and rearranging them into an approximate rectangle: one side approximates the radius, and the other approximates half the circumference (πr) — multiplying those gives Area ≈ r × πr = πr². As the wedges get thinner, this approximation becomes exact.
Using It In A Proof
Converting Between the Four Related Values
Circle problems frequently give you one of radius, diameter, area, or circumference and ask for a different one — the skill is picking the right formula and correctly handling the square root or squaring involved.
1
Identify what's given and what's needed
If given the diameter, convert to radius first (r = d/2) before using either formula, since both are built around r.
2
Going from area to radius
Since A = πr², solving for r requires dividing by π FIRST, then taking the square root: r = √(A/π).
3
Going from circumference to radius
Since C = 2πr, solving for r is simple division, no square root needed: r = C/(2π).
Full Worked Example
Finding Area From a Given Circumference
Given: A circle has a circumference of 44 inches. Find: its area (using π ≈ 3.14).
1
Solve for the radius first
C = 2πr, so 44 = 2π r, so r = 44 / (2π) ≈ 44 / 6.28 ≈ 7.0.
2
Plug the radius into the area formula
A = πr² ≈ 3.14 × (7.0)² = 3.14 × 49.
3
Calculate
A ≈ 153.9 square inches.
Notice radius had to be found FIRST — you can't jump directly from circumference to area without passing through r, since the two formulas use r differently (squared vs. not squared).
🎯 Quick Worked Example
A circle has a diameter of 20 cm. Find its area (using π ≈ 3.14).
1
Convert diameter to radius. r = d/2 = 20/2 = 10 cm.
2
Apply the area formula. A = πr² ≈ 3.14 × 10² = 3.14 × 100.
3
Calculate. A ≈ 314 square cm.
📌 Exam Application
A very common exam trap is plugging the diameter directly into A = πr² instead of converting to radius first — since the formula squares whatever value you plug in, using diameter instead of radius produces an answer 4 times too large (because (2r)² = 4r², not r²).
⚠️ Most Common Circle Formulas Mistakes
Trap 1 — Plugging in diameter where radius is required: Both formulas are defined in terms of r specifically — always confirm whether a given value is the radius or the diameter, and convert (divide by 2) before using either formula if needed.
Trap 2 — Square-rooting incorrectly when solving for r from area: r = √(A/π) requires dividing by π BEFORE taking the square root — taking the square root of A alone, or dividing by π after square-rooting, both produce a wrong answer.
✓ Quick Self-Test
1) State the area formula for a circle. 2) State the circumference formula for a circle. 3) A circle has radius 6 — find its area and circumference. 4) A circle has an area of 78.5 (using π ≈ 3.14) — find its radius. 5) Why can't you convert directly from circumference to area without finding the radius first?