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Understanding the radius is one of the most important elements of the geometry of the circle. The radius is a line segment with one endpoint at the center of the circle or sphere and the other at its boundary. The word may also refer to the length of such a line segment. Here is an example:

As you've previously learned, circles are defined as the set of all points in a plane at a given distance from the center. This means that all of the radii (the accepted plural form of radius) of a particular circle are the same length. Every circle has an infinite number of radii, and they can be named if specific points in the circle are labeled. Consider the following diagram:

In this example, PC, PB, and PA are all radii because they connect the origin point to a point on the outer boundary. Always put the origin point first when naming radii in this manner!

The formula
$r=\frac{C}{2\times \pi}$
may be used to calculate the radius of any circle or sdivhere, where
C equals the
circumference
of the circle and 𝜋 represents
pi
: an
irrational number
approximately equal to 3.14. Most problems will ask you to use 3.14
as the value of 𝜋 unless you are permitted to use a calculator or
asked to leave your answer in terms of 𝜋.

The diameter is also closely related to the radius in that the diameter is a line segment containing the center and both endpoints. If you know the diameter, the radius is simply half of that value. Likewise, you can calculate the diameter of any circle by doubling its radius.

If you know the
area
of a circle, you can use it to determine its radius as well. The
procedure is dividing the area by 𝜋 and then taking the
square root
of the result. This can be expressed mathematically as
$r=\sqrt{\frac{A}{\pi}}$
.

The radius is important because it can be used to determine more
information about a circle. For example, you can use it to calculate
the area of a circle. The formula is
$A=\pi {r}^{2}$
where A is the area, 𝜋 is pi, and r is the radius.

Similarly, you can use the radius to calculate the circumference of
a circle. The formula is
$C=2\pi r$
where C is the circumference and r is the radius.

Again, you can also use the radius to find the diameter of a circle. Put simply, if you have the area, circumference, diameter, or radius of a circle, you can calculate all four values with a few formulas and a little bit of math.

a. If a circle's diameter is expressed as 40 units, what is its radius?

$\mathrm{radius}=\frac{\mathrm{Diameter}}{2}=\frac{\mathrm{40}}{2}=\mathrm{20}\mathrm{units}$

b. If a circle's radius is expressed as 100 units, what is its diameter?

$\mathrm{Diameter}=2\times \mathrm{Radius}=2\times 100=200\mathrm{units}$

c. What is the radius of a circle with a circumference of 15 inches?

$\mathrm{Radius}=\frac{\mathrm{Circumference}}{2\times \pi}=\frac{15}{2\times 3.14}\approx 2.39\mathrm{inches}$

d. If AC and AB are both radii on the same circle and AB measures seven inches, what is the length of AC?

Since both AC and AB are radii of the same circle, their lengths are
equal. So,
$\mathrm{AC}=7\mathrm{inches}$

e. What is the area of a circle with a radius of 4 cm?

$\mathrm{Area}=\pi \times {\left(\mathrm{radius}\right)}^{2}=3.14\times {\left(4\right)}^{2}\approx 50.24{\mathrm{cm}}^{2}$

f. What is the circumference of a circle with a radius of 9 cm using 3.14 as the value of 𝜋?

$\mathrm{Circumference}=2\times \pi \times \mathrm{Radius}=2\times 3.14\times 9\approx 56.52\mathrm{cm}$

g. What is the radius of a circle with an area of 379.94
cm^{2}?

$\mathrm{Radius}=\sqrt{\frac{\mathrm{Area}}{\pi}}=\sqrt{\frac{379.94}{3.14}}\approx 11\mathrm{cm}$

Perpendicular Transversal Theorem

Common Core: High School - Geometry Flashcards

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

The radius is a foundational concept in geometry and students who struggle with it are likely to feel overwhelmed by subsequent coursework as well. If the student in your life is falling behind their peers or you simply wish to equip them with an extra academic edge, contact the Educational Directors at Varsity Tutors right now to learn more about how 1-on-1 tutoring is one of the best ways for students of all ages and ability levels deepen their understanding of radii and related geometry concepts.

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