The area of a circle of radius 4 is 50.265482 — and, exactly, 16π. Both are worth having, and the difference between them matters more than it looks: π is transcendental, so πr² can never be written as a fraction or as a finite decimal. Every decimal you have ever seen for a circle's area is a descr...
The exact area of a circle of radius 4 is 16π, not 50.2655. π is transcendental, so πr² can never be written as a fraction or a finite decimal — the decimal is a description of the answer. This gives the exact value first and the rounding beside it, works from the radius, diameter, circumference or area, and shows the sliced rearrangement that proves the formula rather than asking you to take it on trust.
Any one of the four. Whole numbers, decimals and fractions all work, and π can be written as pi.
RADIUS — centre to edge
AREA
50.265482
= 16π exactly
Rounded from exactly 16π. π is transcendental, so the exact area can never be written as a fraction or a finite decimal — use 16π where a question asks for an answer in terms of π.
| Radius | 4 | 4 |
| Diameter | 8 | 8 |
| Circumference | 8π | 25.132741 |
| Area | 16π | 50.265482 |
THE ANSWER PUT BACK
A = C × r ÷ 2 → 16π ✓
r = d ÷ 2 → 4 ✓
πr² numerically → 50.265482 ✓
DIAMETER
8
twice the radius
CIRCUMFERENCE
8π
2πr — a length, not an area
RADIUS SQUARED
16
the r² before the π is applied
IF THE RADIUS DOUBLED
201.0619
four times the area, not twice
WHAT IS HAPPENING
A = π × (4)² = 16π
The squaring comes first and the π afterwards. Because the radius is squared, the area grows with the square of the radius — doubling r multiplies the area by four, not by two.
C = 2π × 4 = 8π
One π and one r, where the area has one π and two rs. That single difference is why the two are so easily confused and why their units differ — one is a length, the other a length squared.
Why that arrangement proves the formula. Cutting and moving the slices changes nothing about how much material there is, so whatever the row weighs, the circle weighed. Half the slices point up and half point down, so the base is half the way round the circle — that is πr — and each slice is as tall as the radius. A shape of width πr and height r has area πr². The catch, and it is a real one, is that the top edge is scalloped rather than straight: the argument is exact only as the slices become infinitely thin, which is what the bar below measures.
Exact values throughout · π kept as π, never replaced by 3.14 or 22/7
Choose which measurement you have — radius, diameter, circumference or area — and type it in. Fractions such as 3/4 and decimals both work, and π can be written as pi.
Read the exact area first. It will carry a π, because it always does; the decimal underneath is a rounding of it and not the value.
Watch the slices unroll. Half point up and half point down, so the base comes to πr and the height is r — that arrangement is the proof of the formula, not an illustration of it.
Raise the slice count and check the bar beneath. It shows how far the straight-edged figure still falls short, which is the honest measure of an argument that is exact only in the limit.
Take a circle of radius 4. Square the radius first, then multiply by π: A = π × 4² = 16π. That is the exact answer. As a decimal it is about 50.265482, and no decimal is exactly right, because π has no finite decimal form. The circumference is 2πr = 8π, about 25.13. Notice the difference between the two formulas: the area has one π and two radii, the circumference has one π and one radius. That single difference is why they are so easily confused, and why their units differ — a radius in centimetres gives a circumference in centimetres and an area in square centimetres. Now start from the circumference instead. If C = 10, then r = 10 ÷ 2π = 5/π, and the area is π × (5/π)² = 25/π, about 7.9577. The π ends up in the denominator and stays there. It is worth noticing that this is not a strange answer: 25/π is exact, and 7.9577 is not. Now backwards from an area. If A = 16π then r = √(16π ÷ π) = √16 = 4 exactly. This works because the area was given as a multiple of π with a square coefficient, which is how exercises are written. Give the area as 50 instead and the radius is √(50 ÷ π) ≈ 3.989, which is irrational and not a multiple of π — a real answer, just not an exact one. Finally, why πr² at all. Cut the circle into 16 equal slices and lay them alternately, points up and down. Eight point upward, and each contributes its arc — one sixteenth of the way round — to the base. Eight sixteenths is half the circumference, which is πr. Each slice is as tall as the radius. So the shape is πr wide and r tall, and its area is πr². The catch is that the top edge is scalloped rather than straight, so the figure is not truly a rectangle. Measured honestly: the straight-edged polygon through the slice tips gives 3.00000 in units of r² at 12 slices against π = 3.14159, which is 4.5% short. At 96 slices — the figure Archimedes worked by hand — it reaches 3.13935, still 0.07% short. It takes 26 slices to get within 1% and 257 to get within 0.01%. The rearrangement is a proof about the limit, and the limit is where it becomes exact.
| Rule | What it says | Why |
|---|---|---|
| Area | A = πr² | Square the radius first, then multiply by π. The order matters. |
| From the diameter | A = πd² ÷ 4 | Because r is d/2, and halving before squaring divides by four. |
| From the circumference | A = C² ÷ 4π | The π ends up underneath. It cancels only when C already carries one. |
| Circumference | C = 2πr | One π and one r, where the area has one π and two rs. |
| Area from both | A = C × r ÷ 2 | Exactly what the sliced rearrangement above shows. |
| The square law | double r, quadruple A | A 1% error in the radius becomes a 2.01% error in the area. |
| The answer is irrational | πr² is never a fraction | π is transcendental, so 16π is the answer and 50.2655 is a rounding. |
| Circle inside a square | π ÷ 4 ≈ 78.54% | A circle fills just under four fifths of the smallest square holding it. |
| Units | area is squared, length is not | A radius in cm gives an area in cm² and a circumference in cm. |
| 22/7 | off by 0.04% | On a radius of 10 that is an area 0.13 too large. |
| 355/113 | off by 0.0000085% | Far better, and no harder to remember than the date it looks like. |
| Semicircle | πr² ÷ 2 | Half the area, but the perimeter is πr + 2r, not half the circumference. |
| Ring between two circles | π(R² − r²) | Subtract the areas, never the radii. |
| Sector of angle θ | πr² × θ ÷ 360° | A slice is the same fraction of the area as its angle is of a turn. |
A = πr², where r is the radius — the distance from the centre to the edge. Square the radius first and multiply by π afterwards; doing it the other way round gives (πr)², which is π times too large. If you have the diameter instead, halve it first, because the diameter is twice the radius.
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Last updated: September 1, 2026 · Exact values throughout, π kept as π · The area of a circle of radius 4 is 16π, and 50.2655 is a rounding of it.