Turn the circle into numbers — circumference, arc length, sector area, and the polygons that approach it.
So far the circle has been all shape — a center, a distance, and the symmetry that comes with them. Now we make it number. The distance around the rim is the circumference, a tidy π times the diameter — the very same π for every circle, because all circles are the same shape scaled up or down. Take only part of the way around and you get an arc of proportional length; take the matching slice of the inside and you get a sector. Along the way the regular polygons step out of the circle — triangle, square, hexagon — straight pieces closing in on the curve. By the end, shape has become quantity, and we are ready for the next strand, where numbers and intervals describe position itself.
Wrap a string snugly around any circular jar, mark where it meets, then unroll the string and lay it flat. Its length is the circumference — the distance all the way around. Do this for jars of every size and a startling thing happens: the circumference is always the same multiple of the diameter. That multiple has a name, the Greek letter π (pi), and its value is π ≈ 3.14159…, on and on forever without repeating.
So for every circle,
C = πd = 2πr
The ratio comes out the same for every circle precisely because all circles are similar (Stage 17): blow a circle up by a factor and both its circumference and its diameter scale by that same factor, so their ratio never budges. That fixed ratio is π.
A circle has radius r = 7. Then C = 2π·7 = 14π. As a decimal that is 14 × 3.14159… ≈ 43.98; with the school approximation π ≈ 22⁄7 it is a clean 14 × 22⁄7 = 44.
π is a true constant, not a "rounding." 3.14 and 22⁄7 are just handy approximations of it. Leaving an answer as 14π is exact; turning it into 43.98 is the approximation.
Mark the rim into n equal arcs and join neighbouring points with straight chords. The figure you get is a regular n-gon — all sides equal, all angles equal — and every one of its vertices sits on the circle, so we say it is inscribed in that circle. With n = 3 you get an equilateral triangle; n = 4 a square; n = 6 a regular hexagon.
Because the n arcs are equal, the n central angles from O to each side are equal too, and they share the full turn evenly:
each central slice = 360°n
The headline is the last line: the more sides, the closer the polygon hugs the circle. A regular 100-gon is, to the eye, a circle. Hold that thought — it is exactly how a circle's circumference and area get pinned down: as the limit of polygons we already know how to measure.
A regular polygon carries two circles of its own. Its circumscribed circle passes through every vertex; the distance from the center to a vertex is the circumradius R. Its inscribed circle touches the midpoint of every side; the distance from the center to a side's midpoint is the apothem a. The apothem is the perpendicular from the center to a side — drop it and you cut one of the n equal slices into two matching right triangles.
Each slice is an isosceles triangle (two sides are radii R). The apothem splits it into a right triangle whose angle at O is half the central angle — 180/n — with the half-side opposite it and the apothem adjacent. Reading off the right triangle:
| quantity | formula | why |
|---|---|---|
| side | s = 2R·sin(180/n) | twice the half-side, opp. 180/n |
| apothem | a = R·cos(180/n) | adjacent to 180/n |
| perimeter | P = n·s | n equal sides |
| area | A = ½·P·a | n triangles, each ½·s·a |
The area rule is worth a second look: the polygon is n thin triangles meeting at O, each with base s and height a, so each has area ½·s·a, and all n together give ½·(n·s)·a = ½·P·a — base-times-height in disguise.
Side s = 2·10·sin30° = 2·10·½ = 10 — a hexagon's side equals its circumradius! Apothem a = 10·cos30° = 10·√32 = 5√3 ≈ 8.66. Perimeter P = 6·10 = 60. Area A = ½·60·8.66 ≈ 259.8 (exactly 150√3). Notice: P = 60 is already close to the circle's 2πR = 20π ≈ 62.83, and the area 259.8 is reaching toward πR² = 100π ≈ 314.16.
And there is the bridge. As n grows, the side s shrinks but there are more of them: the perimeter P → 2πR and the area A → πR². The polygons we can measure with right triangles close in on the circle we are trying to measure — which is exactly where C = 2πr and the circle's area come from.
An arc is just part of the trip around the rim. If a central angle measures n°, that arc takes up the fraction n360 of the full turn — so it is that same fraction of the whole circumference:
arc length ℓ = n360 · 2πr = n·π·r180
A 90° arc in a circle of radius 8: the fraction is 90⁄360 = ¼, so ℓ = ¼ · 2π · 8 = ¼ · 16π = 4π ≈ 12.57.
Arc length (in units) is not arc measure (in degrees). Two arcs can both measure 90°, yet the one in a bigger circle is longer — its 90° wraps around more rim. Measure tells you the share of the turn; length multiplies that share by the actual size, 2πr.
A sector is the pizza slice between two radii and the arc joining their tips. If those radii open a central angle of n°, the sector is the very same fraction n360 of the whole disk, so:
sector area A = n360 · πr²
There is a second face to the same formula. The sector is, in spirit, a triangle whose base is the arc and whose height is the radius. Writing the arc length as ℓ:
A = ½ · ℓ · r
A = ½·ℓ·r is just ½·base·height wearing a curved coat — the arc ℓ is the "base," the radius r is the "height." It is the polygon area rule ½·P·a taken to the limit: as the slices shrink, the apothem a becomes the radius r, the perimeter piece becomes the arc ℓ.
A 60° sector of radius 6: fraction 60⁄360 = ⅙, so A = ⅙ · π · 36 = 6π ≈ 18.85. Check it the other way: arc ℓ = ⅙·2π·6 = 2π, so ½·ℓ·r = ½·2π·6 = 6π ✓. The two routes agree.
Here is a delight to end on. Take a paper cone — an ice-cream cone, a party hat — and slit its curved side straight up from the rim to the tip. Lay it flat, and it opens into a sector of a circle. The sector's radius is the cone's slant height (the straight distance from tip to rim), and the sector's arc length is exactly the circumference of the cone's base — because that arc was the base rim before you unrolled it.
That single picture turns a three-dimensional surface into a flat sector you can measure with this lesson's tools — the lateral surface of the cone is exactly the sector's area, ½·ℓ·r with ℓ the base circumference and r the slant height.
And now the closing turn of all of plane geometry. We began Stage 18 with shape — a center and a distance — and we end it with number: circumference, arc length, sector area. To say precisely where a point sits — on a circle, on a line, anywhere — we need a clean language of numbers and intervals. That is exactly what the next strand, on sets and intervals, begins. Shape has become quantity; quantity is about to become the address of every point.
Six ideas turn the circle into number:
| quantity | formula | in words |
|---|---|---|
| circumference | C = πd = 2πr | distance all the way around |
| area of the disk | A = πr² | the whole inside |
| arc length | ℓ = (n/360)·2πr | the n/360 share of the rim |
| sector area | A = (n/360)·πr² = ½·ℓ·r | the n/360 share of the disk |
| regular n-gon side | s = 2R·sin(180/n) | apothem a = R·cos(180/n) |
| n-gon area | A = ½·P·a | n triangles → πR² as n grows |
Three threads run through all of it. First, π is the same for every circle, because all circles are similar. Second, a part of the circle is always the n/360 fraction of the whole — that one fraction gives both arc length and sector area. Third, the regular polygons close in on the curve: their straight-edge measurements (right triangles, ½·P·a) become the circle's own (2πr, πr²) as the number of sides grows without bound. Shape has become quantity.
Find the circumference of a circle with radius r = 7. Give an exact answer and a decimal.
C = 2πr = 2π·7 = 14π ≈ 43.98 (or a clean 44 using π ≈ 22⁄7).
A 90° arc sits in a circle of radius 8. How long is the arc?
The fraction is 90⁄360 = ¼, so ℓ = ¼·2π·8 = ¼·16π = 4π ≈ 12.57.
Find the area of a 60° sector of radius 6.
A = (60/360)·πr² = ⅙·π·36 = 6π ≈ 18.85. (Check: arc ℓ = ⅙·2π·6 = 2π, and ½·ℓ·r = ½·2π·6 = 6π ✓.)
A regular hexagon is inscribed in a circle of radius 10. Find its perimeter.
For a hexagon the side equals the circumradius: s = 2R·sin(180/6) = 2·10·sin30° = 2·10·½ = 10. So perimeter = 6·10 = 60.
What is the central angle of one slice of a regular pentagon?
360/n = 360/5 = 72°.
A sector has arc length 5 and radius 4. Find its area, and explain which formula you used.
Use A = ½·ℓ·r (no need for the angle): A = ½·5·4 = 10. It works because a sector is "½·base·height" with the arc as base and the radius as height.
Six questions to lock it in. Tap the answer you think is right.
This final Stage-18 lesson does the great conversion: it turns the circle from a shape into quantity. The single most powerful idea is proportional reasoning — a central angle of n° claims the fraction n/360 of the circle, and that one fraction governs both arc length (a share of the circumference 2πr) and sector area (a share of the disk πr²). Tie the two sector formulas together explicitly: (n/360)·πr² = ½·ℓ·r is "½·base·height" for a slice, which also makes the regular-polygon area rule ½·P·a feel like the same idea, viewed before the limit.
The misconception to hunt down is confusing arc length (units) with arc measure (degrees): two 90° arcs are equal in measure but unequal in length when the circles differ in size — measure is a share of the turn, length multiplies that share by the actual 2πr. Two more to watch: using d where r belongs (or vice versa) in C and in the area, and dropping the n/360 fraction so a sector gets computed as a whole disk. Because area scales like the square of length, sector area uses r², not r — a frequent slip. Encourage exact answers in terms of π (14π) and treat 3.14 or 22⁄7 as approximations applied only at the end.
This lesson supports 7.G.B.4 (know and use the area and circumference of a circle), G-C.B.5 (arc length is proportional to the radius; derive and use the sector-area formula), and reaches toward G-GMD / G-MG with the cone whose lateral surface unrolls into a sector. The regular-polygon section also previews how a circle's circumference and area are pinned down as limits of figures students can already measure.