Circles: Circumference and Area

A circle is a curved line in a flat plane. Every point on it is the same distance from one fixed point, the center. The filled region inside is called a disk. When we find the area of a circle, we measure this region.

From the center to the edge

A radius joins the center to a point on the circle. The word also means the length of that segment. All radii in one circle have the same length. “Radii” is the plural of “radius.”

A diameter goes from one point on the circle to another through the center. It contains two radii placed end to end:

For a circle with radius 4.5 cm, the diameter is 9 cm. If a circle has diameter 15 in, its radius is 7.5 in. Check where a marked segment starts and ends before using its number.

The circumference is the distance all the way around the circle. It is a length, measured in centimeters, inches, or another length unit. A string wrapped once around a circular lid measures its circumference.

Lines that meet a circle

A chord joins two points on the circle. It need not pass through the center. A diameter is a special chord: it does pass through the center, and it is the longest possible chord.

Suppose points A and B lie on a circle. The segment AB is a chord. To call it a diameter, you also need to know that it passes through the center. A line that looks nearly central in a sketch may still be just a chord.

A tangent is a straight line that touches a circle at exactly one point. That point is the point of tangency. A line that crosses the circle meets it at two points, so it is not a tangent. A tangent continues beyond the touching point; a radius ends on the circle.

Pieces of the circle

These names describe different kinds of pieces:

• An arc is part of the curved line between two points.
• A sector is a region between two radii and an arc. It looks like a pizza slice.
• A segment is a region between a chord and its arc.
• A semicircle is half a circle, cut across a diameter.
• A quarter circle is one fourth of a circle. Its two straight sides are radii meeting at a right angle.

An arc is a line, so it has length. A sector and a segment are regions, so they have area. In a semicircle, the straight side is a diameter. In a quarter circle, each straight side is a radius.

Set a compass to the radius

To draw a circle, set the gap from the compass point to the pencil tip equal to the radius. Keep the point at the center and turn the pencil all the way around.

For a circle with diameter 10 cm, first calculate the radius: 10 ÷ 2 = 5 cm. Open the compass to 5 cm. Opening it to 10 cm would make a circle twice as wide as required.

Measure the circumference and diameter of several circular objects. A large object has a larger circumference, but it also has a larger diameter. Dividing the two measurements reveals a fixed relationship.

Find the ratio

Wrap string once around a can. Mark the length, straighten the string, and measure it. Then measure straight across the can through its center. Use the same unit for both measurements.

If the circumference is 20.7 cm and the diameter is 6.6 cm:

A plate measuring 78.5 cm around and 25 cm across gives 78.5 ÷ 25 = 3.14 too. For every circle, the exact ratio is pi, written $\pi$:

A string can stretch or sit loosely, and a ruler reading has limited precision. Your measured ratios may fall a little above or below 3.14. The exact circle relationship stays the same. Because the two lengths use the same unit, their ratio has no unit.

Choose a value for π

The digits begin 3.14159265… and continue forever without a repeating pattern. Pi is irrational: it cannot be written exactly as a fraction of two integers.

Both 3.14 and 22/7 are useful approximations. The first is a little below π; the second is a little above it. A calculator's π key uses many more digits. Use the approximation requested, and keep full calculator values until the final rounding.

For a diameter of 10 cm, multiplying by 3.14 gives 31.4 cm. Using 22/7 gives about 31.43 cm, while the π key gives about 31.42 cm, both to two decimal places. The small difference comes from the chosen approximation.

An answer written with π still in it keeps the exact number. In a formula, $\pi d$ means “π times the diameter.” Pi is a known number, not an unknown length.

Check that a result makes sense

A circle's circumference is more than three diameters and less than four:

Picture a regular hexagon inside a circle. Its six equal sides each match the radius. Its perimeter is six radii, or three diameters. The curved circle around it is longer. A square fitted around the circle has four sides, each as long as the diameter. Its perimeter is longer than the circle's circumference.

For diameter 20 cm, the circumference is between 60 cm and 80 cm. It is about 62.8 cm. An answer of 6.28 cm fails this quick check.

How polygons helped estimate π

In the third century BCE, Archimedes used polygons inside and outside a circle. With polygons of up to 96 sides, he found:

These limits are about 3.1408 and 3.1429. The polygons gave a way to narrow down π without measuring a curved line directly.

The circumference is the whole distance around a circle. Since π tells us how many diameters fit into this distance, multiply the diameter by π:

The diameter is twice the radius. Replacing the diameter with two radii gives the same rule in another form:

Choose the matching formula

If the diameter is given, use $C=\pi d$. If the radius is given, use $C=2\pi r$. Both formulas measure the same boundary. The factor of 2 is already included when you use the diameter.

A dinner plate has diameter 24 cm. Write the formula, put in the diameter, then calculate:

The calculator gives 75.398… cm. To the nearest tenth, the circumference is 75.4 cm. The next digit after the tenths place is 9, so 75.3 rounds up to 75.4.

Now take a flower bed with radius 3.5 m. The distance across it is 7 m, so its circumference must be a little more than 21 m. Calculate directly from the radius:

Putting 3.5 into the diameter formula would give half this distance. Doubling an already given diameter would give twice the correct circumference.

Give an exact answer

“In terms of π” asks you to leave π in the answer. For radius 6 m:

This is exact. It does not need a decimal. Write the number before π. A rounded form, such as 37.7 m, is useful for measurement but loses a little precision.

Calculate without the π key

Some questions ask you to use 3.14 or 22/7. For the flower bed, 22/7 makes the arithmetic simple:

For a circle with diameter 5 cm, using 3.14 gives:

These are answers using the chosen approximation. Using 3 as a value of π is useful for an estimate, but too rough for an answer requested to the nearest tenth.

Keep the unit and round once

A bicycle wheel with diameter 26 in has circumference $26\pi$ in, or about 81.7 in. A diameter in inches produces a circumference in inches. Circumference uses length units, never square units.

Enter the complete product before rounding. Keep π or the full calculator value until the final step.

You can wrap a tape around a tree trunk even when you cannot measure straight through it. Treating its cross-section as a circle lets you use that distance around to find its diameter.

Undo multiplication by π

The circumference formula multiplies the diameter by π. To recover the diameter, divide by π:

The radius is half the diameter, so you can find it in two steps or divide by the whole product $2\pi$:

On a calculator, enter the divisor in parentheses: $C÷(2\times \pi )$. This keeps both 2 and π underneath the circumference.

A circle measuring 50 cm around

First find the diameter:

Its diameter is about 15.9 cm to the nearest tenth. To find the radius, halve the full calculator value:

The radius is about 7.96 cm to the nearest hundredth. If you halve the already rounded 15.9, you get 7.95 cm. That small difference comes from rounding too soon.

Check by putting the unrounded diameter back into the formula. Multiplying 15.915… by π returns 50 cm. A rough check also helps: the diameter should be a little less than one third of 50 cm, so around 16 cm is sensible.

Work with exact π values

If the circumference is $18\pi$ m, divide it by π:

Then $r=18÷2=9$ m. Pi cancels, so neither result needs a decimal approximation.

A common mix-up is to divide by $2\pi$ and label the answer “diameter.” That division finds the radius. Write the quantity's name beside each result so the final answer matches the question.

Use the radius to draw the circle

Suppose a circle must have circumference 22 cm. Its compass setting comes from:

Open the compass to about 3.5 cm, place its point at the chosen center, and draw the circle. Setting the compass to the diameter would make the drawn circumference twice as large.

A circle's area measures the flat region inside its edge. Square units describe how much surface it covers. An area of 20 cm² means the same amount of surface as twenty squares, each 1 cm by 1 cm.

Put the circle between two squares

Imagine a square fitted around a circle of radius $r$. Its side is the circle's diameter, $2r$, so its area is $4r^2$. The circle leaves the four corners empty and has less area than this square.

Now put a square inside the circle, with all four corners on the circle. Its diagonals are both diameters, each $2r$ long. Using half the product of the diagonals gives its area:

The circle covers this inner square and more. Its area lies between the two:

For radius 5 cm, the limits are 50 cm² and 100 cm². A grid estimate, combining partly covered squares, gives about 78 cm². The estimate lies between the limits.

Rearrange the surface

Cut a circle into many equal slices. Arrange the slices in a row, alternating points up and down. They fill a shape that looks nearly like a parallelogram. Using more, thinner slices makes its edges closer to those of a rectangle.

Half of the curved edge runs along the top; the other half runs along the bottom. The long side therefore approaches half the circumference:

The height approaches the radius, $r$. Moving the pieces keeps their total area. Multiplying the rectangle's length by its height gives:

For radius 5 cm, this gives $25\pi \approx 78.5$ cm², close to the grid estimate. The exponent 2 means the radius is multiplied by itself.

The same picture also gives another useful link:

Using a full circumference for that long side would count the circle's curved edge twice.

Build the circle from triangles

You can also draw a regular polygon inside the circle. Join its corners to the center. This divides the polygon into equal triangles. Each triangle has a side of the polygon as its base and the center-to-side distance as its height.

Adding their areas gives half the polygon's perimeter times that height. As the polygon gets more sides, its perimeter approaches $2\pi r$ and its height approaches $r$. Its area approaches $\frac{1}{2}\times 2\pi r\times r=\pi r^2$.

The outer square remains useful as a check. Since $\pi ÷4\approx 0.785$, the circle fills about 79% of that square. The missing corners explain why using $4r^2$ overestimates the circle's area.

To find the area, start with the radius. Square that length, then multiply by π:

Here $r^2$ means $r\times r$. It does not mean double the radius. Area uses square units because the calculation multiplies two lengths.

Calculate from a radius

A circular rug has radius 7 ft. Square the radius first:

Then multiply by π:

The decimal is rounded to the nearest tenth. If the question requests 22/7, use $\frac{22}{7}\times 49=154$ ft². The result differs slightly because 22/7 is an approximation.

On a calculator, enter π × 7², or calculate 7² first. Squaring the entire product π × 7 would square π as well and give the wrong area.

Calculate from a diameter

A circular pond has diameter 10 m. That length goes across the whole pond, so first halve it:

Now square 5 and multiply by π:

Using 10 as the radius would give $100\pi$ m². The mistaken length is twice the radius, so its square is four times too large.

Exact areas and small radii

An exact answer leaves π in place. For radius 6 cm, the area is $36\pi$ cm². A radius can also be a fraction. For radius $\frac{1}{2}$ m:

Squaring a positive number less than 1 makes it smaller. Here $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$. Doubling the fraction would give 1, which does not follow the area formula.

Convert lengths before squaring

Suppose the radius is 50 cm and the answer is wanted in square meters. Convert the radius first: 50 cm = 0.5 m. Then:

You could instead find the area in cm², but the conversion changes. A square meter contains 100 rows of 100 square centimeters, so 1 m² = 10,000 cm². Dividing an area in cm² by only 100 would leave it 100 times too large.

Keep length and area separate

For radius 2 cm, circumference is $4\pi$ cm and area is $4\pi$ cm². Their numerical values happen to match in these units. They still measure different things: one is a length around the edge, and the other is a surface. Read the requested measurement and its unit before choosing a formula.

An area tells you how much surface a circle covers. To find its radius, undo the two operations in the area formula in reverse order.

Divide, then take a square root

The formula $A=\pi r^2$ first squares the radius and then multiplies by π. Divide the area by π to recover the squared radius:

Then take the positive square root, since a radius is a positive length:

If $A=49\pi$ m², dividing by π gives $r^2=49$. The radius is 7 m, and the diameter is 14 m. Stopping at 49 confuses a squared length with the length itself.

For area 200 cm², enter the whole quotient under the square root:

The radius is about 7.98 cm; doubling the unrounded value gives a diameter of about 15.96 cm. Check by squaring the unrounded radius and multiplying by π. It returns 200 cm².

Taking the square root of 200 first and then dividing by π gives a different result. The square-root sign must cover the area divided by π. Halving the squared radius is also wrong: a square root undoes squaring, while halving undoes doubling.

Move between area and circumference

The radius links the two formulas. Using 3.14, a circumference of 31.4 cm gives:

Then $A=3.14\times 5^2=78.5$ cm². Going the other way, start from area, divide by π, take the square root, and put the radius into $C=2\pi r$.

Compare a circle with an enlarged copy

A circle of radius 3 cm has circumference $6\pi$ cm and area $9\pi$ cm². Doubling the radius to 6 cm gives circumference $12\pi$ cm and area $36\pi$ cm².

The circumference doubles. The area becomes four times as large because both radius factors double:

More generally, a radius multiplied by $k$ gives a circumference multiplied by $k$ and an area multiplied by $k^2$. Tripling the radius gives nine times the area. Halving it leaves one fourth of the area.

Pizza sizes usually name the diameter. A 16-inch pizza has twice the diameter of an 8-inch pizza, so it also has twice the radius. Its area is four times as large, assuming both are circular. Adding or comparing diameters alone misses the squared change in surface.

A semicircle has half a circle's surface. A quarter circle has one fourth. Use the radius of the original circle to find the full area, then take the needed fraction.

Take a fraction of the area

For a semicircle:

For a quarter circle:

A semicircle with diameter 10 cm has radius 5 cm. Its area is:

Halving the radius would make a smaller full circle with only one fourth of the original area. To take half the surface, halve the full area.

Follow a semicircle's boundary

Start at one end of its straight side. Follow the curve to the other end, then return along the diameter. That is the whole perimeter.

The curved part is half the circumference. The straight part is one diameter:

For the same semicircle of diameter 10 cm, the curve is $5\pi$ cm, about 15.71 cm. Add the straight 10 cm:

An answer of 15.7 cm measures only the curve. It leaves out the straight edge that closes the semicircle.

Follow a quarter circle's boundary

A quarter circle has a quarter of the circumference plus two straight radii. Its perimeter is:

With radius 8 cm, the curved edge is $4\pi$ cm. The two straight sides total 16 cm, so:

Its area is $\frac{1}{4}\times \pi \times 8^2=16\pi$ cm², about 50.3 cm². Both straight edges are 8 cm; adding only one would leave the boundary unfinished.

Read what is being measured

A semicircle's straight side is the diameter. A quarter circle's straight sides are radii. Use those facts to choose the length for the formula.

If a question asks for the curved edge, calculate just the matching fraction of the circumference. If it asks for perimeter, include the straight edges too. For area, take a fraction of the full surface and use square units. A ribbon along a curved window top and a frame around the whole window need different lengths.

A composite shape combines simpler shapes. For area, split the surface into pieces that do not overlap. For perimeter, follow every edge of the finished shape. A line used to split an area may be inside the shape and add nothing to its perimeter.

Join a rectangle and a semicircle

A garden bed is a rectangle 10 m long and 6 m wide. A semicircle is attached along one whole 6 m end, extending beyond the rectangle. The shared side is the semicircle's diameter, so its radius is 3 m.

Write an area plan: “rectangle + semicircle.” Then calculate each part:

The total area is $(60+4.5\pi )$ m², about 74.1 m². Keep the exact expression until the final rounding.

For perimeter, trace two long sides, one exposed short side, and the semicircular curve:

The shared 6 m side lies inside the bed. Adding the complete perimeters of the rectangle and semicircle would count that hidden join twice.

Subtract a missing region

A square tile has side 12 cm. A circle with diameter 12 cm is painted inside it, touching all four sides. The unpainted area is the square minus the circle:

The circle's diameter comes from the square's side. Its radius is only half that length.

Now cut a semicircle of diameter 8 cm inward from one side of an 8 cm square. Its area is subtracted:

The removed straight side is replaced by a curved edge. The remaining perimeter is three square sides plus that curve:

Removing surface does not mean removing all its edges from the perimeter. The exposed curve is now part of the boundary.

Include both edges of a ring

A ring, or annulus, lies between two circles with the same center. With outer radius 5 cm and inner radius 2 cm, subtract the two circle areas:

The boundary contains both circles, so add their circumferences:

The ring's width is 3 cm, but $\pi \times 3^2$ is the area of a small disk, not this ring. Square the two radii separately before subtracting.

Combine two matching semicircles

A track has two straight sides of 80 m and two semicircular ends of diameter 60 m. The curves together make one full circumference:

The enclosed surface is a rectangle plus a full circle:

When lengths are missing, use how the pieces fit. Three circles of diameter 4 cm touching in one row span 12 cm. A rectangle fitted tightly around that row is 12 cm by 4 cm.

Read what the job needs before choosing a formula. A fence follows an edge, while a net covers a surface. The same circle can give both a circumference and an area, but only one may answer the question.

Around the edge or over the surface?

A pond has diameter 6 m. A fence around it needs:

A net covering its water needs radius 3 m and area:

The two answers use different units because one measures an edge and the other measures a surface.

Buy enough whole units

If fencing is sold only by the whole meter, buy 19 m for the pond. Buying 18 m leaves a gap. This is rounding up to the next whole unit.

For a reported measurement, round to the accuracy requested. The pond’s exact circumference is $6\pi =18.8495\dots$ m, so it rounds to 18.8 m to the nearest tenth. Use the full value: rounding an already rounded 18.85 m again would give 18.9 m.

Turn a wheel into a distance

When a wheel rolls without slipping, one full turn moves it forward one circumference. A wheel of diameter 26 in travels $26\pi$ in per turn. After 100 turns:

There are 12 inches in a foot. Divide by 12 to get about 680.7 ft. Multiplying the rounded one-turn distance 81.7 in by 100 would lose accuracy.

To find turns instead, divide the distance by the circumference. A wheel of diameter 70 cm has circumference $70\pi$ cm. Convert that to meters, $0.7\pi$ m, before comparing with 1 km = 1,000 m:

That is 454 complete turns followed by part of another. If asked for the turns needed to reach or pass the distance using whole turns, the answer would be 455.

Match units first

Use 100 cm = 1 m and 1,000 m = 1 km. For customary lengths, use 12 in = 1 ft and 3 ft = 1 yd. A distance divided by a circumference gives a turn count only when both lengths use the same unit.

For a tabletop of radius 0.6 m, trim needs $1.2\pi \approx 3.77$ m. Varnish for the top needs $0.36\pi \approx 1.13$ m². Labeling the requested job beside each calculation keeps the two measurements separate.

A multistep problem uses a circle measurement in another calculation. Decide what the final answer should measure: a price, a distance, a turn count, or value for money. That helps you choose the steps and their units.

Find a cost from area

A circular lawn has diameter 10 m. Turf costs $8.50 per square meter. Halve the diameter to get radius 5 m, then find the area:

Multiply the full area by the price per square meter:

The cost is $667.59 to the nearest cent. The square-meter units cancel when an area is multiplied by dollars per square meter.

If turf is sold only in whole square meters, the purchase changes. The lawn covers 78.539… m², so buy 79 m². That costs $79\times 8.50=671.50$ dollars. Rounding up here is a buying decision. Rounding the area early in a calculation with no whole-unit rule would change the cost incorrectly.

Compare value using area

A 12-inch pizza costs $12 and a 16-inch pizza costs $18. These sizes give the diameters. Their radii are 6 in and 8 in, so their areas are $36\pi$ in² and $64\pi$ in².

Divide each price by its area:

The larger pizza has the lower price per square inch, so it gives more pizza for each dollar. A lower total price alone does not show better value.

For a quantity comparison, one 16-inch pizza has area $64\pi$ in². Two 10-inch pizzas have total area $2\times 25\pi =50\pi$ in². The single larger pizza has more surface, even though the two smaller diameters add to 20 inches.

Use distance per turn or per hour

A clock's minute hand is 10 cm long. That length is the radius of the tip's path. In one hour, the tip completes one circle:

Over 24 hours it travels $480\pi$ cm, about 15.1 m. The radius stays fixed; the number of complete trips changes.

A sprinkler reaching 4 m covers $16\pi$ m². One reaching 8 m covers $64\pi$ m². Doubling the reach multiplies the covered area by four, which affects both the amount watered and any cost based on that area.

Check the size of the result

For the lawn, using 3 as a rough value of π gives about 75 m². Multiplying by roughly $8.50 gives a cost near $640. The calculated $667.59 is sensible. A result near $67 or $6,676 would call for a check of the decimal point, radius, and units.