The radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.
In Spanish
radio
Also called
r, radii (plural)
How to say it
RAY-dee-us
Grade level
6-8
Grade 6Grade 7Grade 8CCSS 7.G.B.4GEDSATACT
Radius in a picture
A radius runs from the center to the circle. Every radius is the same length, and two of them in a straight line make the diameter, d = 2r.
Formal definition and notation
A circle with center O is the set of all points in a plane at the same distance r from O; that distance is the radius, and any segment OP with P on the circle is also called a radius. The diameter is d = 2r, so r = d2. The circumference is C = 2π r and the area is A = π r2. A sphere has a radius too: the distance from its center to any point on its surface. The plural of radius is radii.
Symbol
Read it as
Meaning
r
r
the radius of a circle or sphere
OP
segment O P
a radius from the center O to the point P on the circle
d = 2r
d equals two r
the diameter is twice the radius
Key facts
A circle has infinitely many radii, and they are all the same length.
Two radii pointing in opposite directions make a diameter, so d = 2r.
Area grows with the square of the radius: doubling the radius multiplies the area by 4, and tripling it multiplies the area by 9.
The opening of a compass sets the radius of the circle it draws; the spokes of a bicycle wheel are radii.
On a coordinate plane, a circle with center (h, k) and radius r is (x − h)2 + (y − k)2 = r2.
Finding the radius
Finding the radius
r = d2 = C2π = √Aπ
d
the diameter
C
the circumference, the distance around the circle
A
the area inside the circle
Use whichever measurement you are given. From a diameter of 26, r = 13. From a circumference of 18π, r = 9. From an area of 49π, r = 7.
Try it: drag the radius
Interactive
Drag the blue point or use the slider to change the radius. Watch the diameter, circumference and area update; notice how much faster the area grows.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Radius examples, step by step
Example 1: From diameter to radius
Warm-up
A round table has a diameter of 26 inches. What is its radius?
The diameter goes all the way across the table through its center: d = 26 inches.
The radius is half the diameter: r = d2.
r = 262 = 13.
Answer: The radius is 13 inches.
Example 2: From circumference to radius
Core
A circle has a circumference of 18.84 cm. Use 3.14 for π to find its radius.
Start from C = 2π r: 18.84 = 2 × 3.14 × r.
2 × 3.14 = 6.28, so 18.84 = 6.28r.
Divide: r = 18.84 ÷ 6.28 = 3.
Answer: The radius is 3 cm.
Example 3: From area to radius
Test-style
A circular rug has an area of 154 square feet. Using π ≈ 227, what is its radius?
A chord joins any two points on the circle. A radius has one end at the center, so a radius is never a chord; the diameter is the longest chord.
Quick check: 3 questions
Question 1 of 3
A circle has a radius of 4.5 cm. What is its diameter?
Show answer and explanation
Answer: B.d = 2r = 2 × 4.5 = 9 cm. Choice A halves the radius instead of doubling it, C repeats the radius, and D is close to π × 4.5.
Question 2 of 3
A circle has an area of 36π square units. What is its radius?
Show answer and explanation
Answer: A.π r2 = 36π, so r2 = 36 and r = 6. Choice B halves 36 instead of taking the square root, C is r2, and D is the diameter.
Question 3 of 3
If the radius of a circle is tripled, the area is multiplied by
Show answer and explanation
Answer: C.π(3r)2 = 9π r2, so the area becomes 9 times as large. Choice A is what happens to the circumference and diameter, and B doubles the 3 instead of squaring it.
Radius: frequently asked questions
What is the plural of radius?
Radii, pronounced RAY-dee-eye. Radiuses is also accepted in everyday English, but math books use radii. Every circle has infinitely many radii, one to each point on the circle, and all of them have the same length. When a problem says the radius, it means that common length.
Is the radius always half the diameter?
Yes, for every circle and every sphere. The diameter is made of two radii pointing in opposite directions from the center, so d = 2r and r = d2. A circle with diameter 30 meters has radius 15 meters. If you know either one, you know the other.
How do you find the radius from the circumference or the area?
From the circumference, divide by 2π: C = 31.4 gives r = 31.4 ÷ 6.28 = 5. From the area, divide by π and then take the square root: A = 50.24 gives r2 = 16, so r = 4. Both examples use 3.14 for π. Double the radius to get the diameter.
CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7DiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7, Grade 8CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. 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A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1ChordA chord is a line segment whose two endpoints both lie on a circle. The diameter is the longest chord, because it passes through the center. A chord at distance d from the center of a circle with radius r has length 2√r2 − d2, so chords closer to the center are longer.Geometry · Grade 7, GeometryMeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5, Grade 6AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5Area of a CircleThe area of a circle is the amount of flat space inside the circle, measured in square units. It equals pi times the radius squared: A = π r2. 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It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x2 + y2 = r2.Geometry · Geometry, Algebra 2, PrecalculusPiPi (π) is the number you get when you divide any circle’s circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 227.Geometry · Grade 7, Grade 8, Geometry