How do you find the circumference of a circle and its area?

The circumference of a circle is π × diameter, or 2 × π × radius, and the area is π × radius × radius. Pi (π) is about 3.14. A circle with a radius of 5 cm has a circumference of 2 × 3.14 × 5 = 31.4 cm and an area of 3.14 × 5 × 5 = 78.5 cm². Circumference is a length; area is in square units.

Pi (π) is the ratio of a circle's circumference to its diameter, and it is the same for every circle. Its value is about 3.14159. Pi is an irrational number, so it cannot be written as a fraction of two whole numbers, and its decimal digits never end and never repeat. School problems usually round it to 3.14 or 22/7.

How many digits are in pi?

Pi has infinitely many digits, because its decimal expansion never ends and never settles into a repeating pattern. Computers have calculated hundreds of trillions of digits, but practical work needs very few. NASA's Jet Propulsion Laboratory uses 3.141592653589793, which is 15 decimal places, for its most accurate interplanetary navigation.

Pi Day is March 14 every year, because the date written as 3/14 matches the first three digits of pi, 3.14. Schools and math clubs often mark it with circle activities, pi-digit challenges and round foods such as pie. NASA publishes a Pi Day Challenge of space-themed circle problems for students.

What is the circumference of the Earth?

The circumference of the Earth at the equator is about 40,075 km, or about 24,901 miles. NASA's Earth fact sheet gives an equatorial radius of 6,378.137 km, and 2 × π × 6,378.137 is about 40,075 km. The distance around the poles is slightly shorter, because the Earth is a little flattened.

How do you find the circumference from the area?

The circumference can be found from the area in three steps: divide the area by π, take the square root to get the radius, then multiply by 2 × π. A circle with an area of 78.5 cm² gives 78.5 ÷ 3.14 = 25, the square root of 25 is 5 cm, and the circumference is 2 × 3.14 × 5 = 31.4 cm.

Circumference of a Circle and Area, Step by Step

Your sixth grader has a worksheet of circles, a calculator and two formulas that look almost the same. The circumference of a circle is the distance around it, the area is the space inside it, and both come from one number called pi. Once you know which measurement a problem gives you (the radius or the diameter), finding the circumference and the area of a circle takes two or three steps.

What is pi, and why does every circle need it?

Pi (π) is the number of times a circle's diameter fits around its edge, and it is the same for every circle in the world. Wikipedia's article on pi gives the value as approximately 3.14159 and notes that its decimal form "never ends, nor does it enter a permanently repeating pattern."

Pi is an irrational number, so no fraction or decimal is exactly equal to it. OpenStax Prealgebra 2e tells students to "approximate π with 3.14 or 22/7 depending on whether the radius of the circle is given as a decimal or a fraction."

A circle with its diameter d in blue and its radius r in green; below it the circumference is unrolled into a straight line that holds three diameters plus a small extra piece, with C = π × d, C = 2 × π × r and d = 2 × r.

Radius vs diameter

The radius and the diameter are the two lengths every circle formula starts from. The radius (r) runs from the center to the edge, and the diameter (d) runs from edge to edge through the center, so the diameter is always twice the radius: d = 2 × r.

A pizza 12 inches across has a diameter of 12 inches and a radius of 6 inches. Before you touch a formula, write down both numbers. Most wrong answers in circle homework start with the wrong one of the two.

How to find the circumference of a circle, step by step

The circumference of a circle is π × diameter, written C = π × d, or 2 × π × radius, written C = 2 × π × r. Both formulas give the same answer, because 2 × r is the diameter. OpenStax calls circumference "the perimeter of a circle", which links it to the area and perimeter of squares and rectangles.

Find the diameter. Measure across the circle through the center, or double the radius. A dinner plate measures 26 cm across, so d = 26 cm.
Choose a value for pi. Use 3.14 for decimals and whole numbers, 22/7 when the radius is a multiple of 7, or the π key on a calculator for the most exact answer.
Multiply pi by the diameter. 3.14 × 26 = 81.64. The guide to multiplying and dividing decimals covers this step by hand.
Write the unit. The circumference is about 81.64 cm. Circumference is a length, so it uses cm, inches or feet, with no small 2.
Check the size. The answer should be a bit more than 3 diameters: 3 × 26 = 78, and 81.64 is a little more. An answer smaller than 3 diameters or larger than 4 is wrong.

A wheel with a radius of 30 cm uses the other form: C = 2 × 3.14 × 30 = 188.4 cm.

Free printable worksheetMultiplication — 6th GradersWorksheet · PDF

The part people get wrong: radius or diameter

The most common circumference mistake puts the radius into C = π × d. A circle with a radius of 5 cm then gets 3.14 × 5 = 15.7 cm, which is half of the right answer, 31.4 cm. Ask "is this number the whole way across, or half of it?" before you multiply.

The opposite mistake doubles a diameter in C = 2 × π × r. A plate 26 cm across becomes 2 × 3.14 × 26 = 163.28 cm, twice the real distance around it. The size check from step 5 catches both errors in a few seconds.

How to find the area of a circle

The area of a circle is π × radius × radius, written A = π × r². The area counts the square units inside the circle, so it always uses square units such as cm² or square inches.

A circle with a radius of 5 cm has an area of 3.14 × 5 × 5 = 3.14 × 25 = 78.5 cm². The dinner plate from the step list has a radius of 13 cm, so its area is 3.14 × 13 × 13 = 3.14 × 169 = 530.66 cm². A 12-inch pizza has a radius of 6 inches and an area of 3.14 × 36 = 113.04 square inches.

The order matters: square the radius first, then multiply by π. The small 2 in r² is an exponent, which the guide to exponents explains, and children who know their square numbers by heart (25, 36, 49, 64) finish each area in one step.

Play free in your browserSquare NumbersGrade 4–6

Why the area formula works

The area formula comes from cutting a circle into thin wedges, like a pizza, and laying them side by side. Point the wedges up and down in turn and they form a shape close to a parallelogram. The more wedges you cut, the closer the shape gets to a rectangle.

A circle cut into 16 wedges, the top half blue and the bottom half amber, rearranged into a near-parallelogram whose long edge is π × r and whose height is r, giving A = π × r × r.

The long edge of that shape is half of the circumference, which is half of 2 × π × r, or π × r. The height is one radius, r. A rectangle π × r long and r tall has an area of π × r × r, which is the formula.

Common Core standard 7.G.B.4 asks grade 7 students to "know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle", and this wedge picture is that informal derivation.

Circumference vs area vs diameter vs radius

Circumference, area, diameter and radius describe four different things about the same circle.

Measurement	What it is	Formula	Unit	When you need it
Radius (r)	Center to edge	r = d ÷ 2	cm, in	Starting point for area; compasses are set to it
Diameter (d)	Edge to edge through the center	d = 2 × r	cm, in	Measuring plates, pipes, pizzas, wheels
Circumference (C)	Distance around the circle	C = π × d = 2 × π × r	cm, in	Fencing a round bed, ribbon around a cake, wheel turns
Area (A)	Space inside the circle	A = π × r²	cm², in²	Paint, grass seed, pizza size, tablecloths

Circumference grows in step with the radius, but area grows with the radius squared. A 16-inch pizza has a radius of 8 inches and an area of 3.14 × 64 = 200.96 square inches, almost twice the 113.04 square inches of a 12-inch pizza.

Which one a problem asks for

The words in a word problem tell you which formula to use. "Around", "edge", "trim", "fence" and "how far does the wheel roll" point to circumference. "Cover", "inside", "paint", "carpet" and "how much pizza" point to area.

The guide on how to find volume uses the circle's area as the base of a cylinder: a can's volume is π × r² × h.

Kitchen-table circle activities

Kitchen-table circle activities let a child find pi with their own hands before using it in a formula. Each one takes about ten minutes with a ruler, string and a few round objects.

String around a plate: Wrap string around a dinner plate, mark it, and lay it along a ruler. A 26 cm plate gives about 81 or 82 cm, and 82 ÷ 26 is about 3.15, close to pi.
Measure jar lids: Measure the diameter and the circumference of three lids of different sizes. Divide each circumference by its diameter; every answer lands near 3.1.
Roll a can: Mark a point on the edge of a tin can, roll it one full turn along a sheet of paper, and measure the track. The track is the circumference.
Trace and count: Trace a jar lid on centimeter grid paper and count the whole and half squares inside. A lid with a radius of 4 cm covers about 50 squares, and 3.14 × 16 = 50.24.

How to check progress

Five short questions show whether a child understands circles or is copying a formula.

The vocabulary check: Point at a circle and ask for the radius, the diameter and the circumference. A child who can show all three with a finger knows what each word means.
The basic pair: Give a radius of 10 cm. The circumference is 62.8 cm and the area is 314 cm², with the right unit on each.
The diameter trap: Give a diameter of 8 inches and ask for the area. The radius is 4, so the area is 3.14 × 16 = 50.24 square inches.
The size check: Ask whether a circumference of 9 cm is possible for a circle 4 cm across. It is not, because 3 × 4 = 12 already.
The reverse question: Give a circumference of 31.4 cm and ask for the diameter: 31.4 ÷ 3.14 = 10 cm.

A child who knows the method but stalls on 3.14 × 16 needs more multiplication practice than geometry. The math game for 6th graders covers the number work, and adults relearning this can use the math game for adults for the same facts.

Common circle mistakes and edge cases

Four mistakes cause most wrong circle answers, and each has a short fix.

Writing (π × r)² for π × r²: With r = 5, (3.14 × 5)² = 15.7 × 15.7 = 246.49, more than 3 times the real area of 78.5. Square only the radius, then multiply by π.
Squaring the diameter: A circle 10 inches across gets 3.14 × 10 × 10 = 314, which is 4 times the real area of 78.5 square inches. Halve the diameter first.
Mixing up the units: A circumference in cm² or an area in cm describes the wrong kind of measurement. Circumference is a length in cm; area is in cm².
Using a rounded radius too early: A radius of 2.45 m rounded to 2 m loses about a third of the area. Round only the final answer.

Edge case: 22/7 or 3.14?

The fraction 22/7 and the decimal 3.14 are both approximations of pi, and they give slightly different answers. 22/7 is about 3.1429 and 3.14 is a little under pi, so the true value of about 3.1416 sits between them.

Use 22/7 when the radius or diameter is a multiple of 7, because the 7s cancel. A circle with a radius of 7 cm has a circumference of 2 × 22/7 × 7 = 44 cm and an area of 22/7 × 7 × 7 = 154 cm², while 3.14 gives 43.96 cm and 153.86 cm². The π key on a calculator gives 153.94 cm².

To work backward from an area to a radius, divide by π and take a square root, which the guide to square roots covers.

Your circle checklist and next steps

Radius or diameter: write down both before using any formula, with d = 2 × r.
Circumference: multiply π × d, or 2 × π × r, and give the answer in cm or inches.
Area: square the radius first, then multiply by π, and give the answer in cm² or square inches.
Value of pi: use 3.14 for most homework, 22/7 for multiples of 7, or the π key when the problem asks for it.
Size check: the circumference is a bit more than 3 diameters, and the area is a bit more than 3 radius squares.

Practise it in the MathIt app

About 90 math games with adaptive difficulty, daily practice and 2–5 player battles — free on iOS and Android.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice that keeps the square numbers and multiplication facts behind every circle problem quick. You can also challenge 1 to 4 friends or family members in real-time math battles, in any of 33 languages.

How to find the circumference of a circle and its area (and what pi is)

Key takeaways