How to Determine the Geometry of a Circle

Radius and Diameter

The radius is a line from the center point of a circle to any part of the circle. This is probably the simplest concept related to measuring circles but possibly the most important.

The diameter of a circle, by contrast, is the longest distance from one edge of the circle to the opposite edge. The diameter is a special type of chord, a line that joins any two points of a circle. The diameter is twice as long as the radius, so if the radius is 2 inches, for example, the diameter would be 4 inches. If the radius is 22.5 centimeters, the diameter would be 45 centimeters. Think of the diameter as if you are cutting a perfectly circular pie right down the center so that you have two equal pie halves. The line where you cut the pie in two would be the diameter.

Circumference

The circumference of a circle is its perimeter or distance around it. It is denoted by C in math formulas and has units of distance, such as millimeters, centimeters, meters, or inches. The circumference of a circle is the measured total length around a circle, which when measured in degrees is equal to 360°. The "°" is the mathematical symbol for degrees.

To measure the circumference of a circle, you need to use "Pi," a mathematical constant discovered by the Greek mathematician  Archimedes. Pi, which is usually denoted with the Greek letter π, is the ratio of the circle's circumference to its diameter, or approximately 3.14. Pi  is the fixed ratio used  to calculate the circumference of the circle

You can calculate the circumference of any circle if you know either the radius or diameter. The formulas are:

C = πd
C = 2πr

where d is the diameter of the circle, r is its radius, and π is pi. So if you measure the diameter of a circle to be 8.5 cm, you would have:

C = πd
C = 3.14 * (8.5 cm)
C = 26.69 cm, which you should round up to 26.7 cm

Or, if you want to know the circumference of a pot that has a radius of 4.5 inches, you would have:

C = 2πr
C = 2 * 3.14 * (4.5 in)
C = 28.26 inches, which rounds to 28 inches

Area

The area of a circle is the total area that is bounded by the circumference. Think of the area of the circle as if you draw the circumference and fill in the area within the circle with paint or crayons. The formulas for the area of a circle are:

A = π * r^2

In this formula, "A" stands for the area, "r" represents the radius, π is pi, or 3.14. The "*" is the symbol used for times or multiplication.

A = π(1/2 * d)^2

In this formula, "A" stands for the area, "d" represents the diameter, π is pi, or 3.14. So, if your diameter is 8.5 centimeters, as in the example in the previous slide, you would have:

A = π(1/2 d)^2 (Area equals pi times one-half the diameter squared.)

A = π * (1/2 * 8.5)^2

A = 3.14 * (4.25)^2

A = 3.14 * 18.0625

A = 56.71625, which rounds to 56.72

A = 56.72 square centimeters

You can also calculate the area if a circle if you know the radius. So, if you have a radius of 4.5 inches:

A = π * 4.5^2

A = 3.14 * (4.5 * 4.5)

A = 3.14 * 20.25

A = 63.585 (which rounds to 63.56)

A = 63.56 square centimeters

Arc Length

The arc of a circle is simply the distance along the circumference of the arc. So, if you have a perfectly round piece of apple pie, and you cut a slice of the pie, the arc length would be the distance around the outer edge of your slice.

You can quickly measure the arc length using a string. If you wrap a length of string around the outer edge of the slice, the arc length would be the length of that string. For the purposes of calculations in the following next slide, suppose the arc length of your slice of pie is 3 inches.

Sector Angle

The sector angle is the angle subtended by two points on a circle. In other words, the sector angle is the angle formed when two radii of a circle come together. Using the pie example, the sector angle is the angle formed when the two edges of your apple pie slice come together to form a point. The formula for finding a sector angle is:

Sector Angle = Arc Length * 360 degrees / 2π * Radius

The 360 represents the 360 degrees in a circle. Using the arc length of 3 inches from the previous slide, and a radius of 4.5 inches from slide No. 2, you would have:

Sector Angle = 3 inches x 360 degrees / 2(3.14) * 4.5 inches

Sector Angle = 960 / 28.26

Sector Angle = 33.97 degrees, which rounds to 34 degrees (out of a total of 360 degrees)

Sector Areas

A sector of a circle is like a wedge or a slice of pie. In technical terms, a sector is a part of a circle enclosed by two radii and the connecting arc, notes study.com. The formula for finding the area of a sector is:

A = (Sector Angle / 360) * (π * r^2)

Using the example from slide No. 5, the radius is 4.5 inches, and the sector angle is 34 degree, you would have:

A = 34 / 360 * (3.14 * 4.5^2)

A = .094 * (63.585)

Rounding to the nearest tenth yields:

A = .1 * (63.6)

A =  6.36 square inches

After rounding again to the nearest tenth, the answer is:

The area of the sector is 6.4 square inches.

Inscribed Angles

An inscribed angle is an angle formed by two chords in a circle which have a common endpoint. The formula for finding the inscribed angle is:

Inscribed Angle =  1/2 * Intercepted Arc

The intercepted arc is the distance of the curve formed between the two points where the chords hit the circle. Mathbits gives this example for finding an inscribed angle:

An angle inscribed in a semicircle is a right angle. (This is called Thales theorem, which is named after an ancient Greek philosopher, Thales of Miletus. He was a mentor of famed Greek mathematician Pythagoras, who developed many theorems in mathematics, including several noted in this article.)

Thales theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle ∠ABC is a right angle. Since AC is the diameter, the measure of the intercepted arc is 180 degrees—or half the total of 360 degrees in a circle. So:

Inscribed Angle = 1/2 * 180 degree

Thus:

Inscribed Angle = 90 degrees.