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# Geometry: Praxis I Exam (page 6)

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### Circles

#### Terminology

A circle is formally defined as the set of points a fixed distance from a point. The more sides a polygon has, the more it looks like a circle. If you consider a polygon with 5,000 small sides, it will look like a circle, but a circle is not a polygon. A circle contains 360 degrees around a center point.

• The midpoint of a circle is called the center.
• The distance around a circle (called perimeter in polygons) is called the circumference.
• A line segment that goes through a circle, with its endpoints on the circle, is called a chord.
• A chord that goes directly through the center of a circle (the longest line segment that can be drawn) in a circle is called the diameter.
• The line from the center of a circle to a point on the circle (half of the diameter) is called the radius.
• A sector of a circle is a fraction of the circle's area.
• An arc of a circle is a fraction of the circle's circumference.

#### Circumference, Area, and Volume Formulas

The area of a circle is A = πr2, where r is the radius of the circle. The circumference (perimeter of a circle) is 2πr, or πd, where r is the radius of the circle and d is the diameter.

Example
Determine the area and circumference of this circle:

We are given the diameter of the circle, so we can use the formula C = πd to find the circumference.
C = πd
C = π(6)
C = 6π = 18.85 feet
The area formula uses the radius, so we need to divide the length of the diameter by 2 to get the length of the radius: 6 ÷ 2 = 3. Then we can just use the formula.
A = π(3)2
A = 9π = 28.27 square feet.

Note: Circumference is a measure of length, so the answer is measured in units, whereas the area is measured in square units.

#### Area of Sectors and Lengths of Arcs

The area of a sector can be determined by figuring out what the percentage of the total area the sector is, and then multiplying by the area of the circle.

The length of an arc can be determined by figuring out what the percentage of the total circumference of the arc is, and then multiplying by the circumference of the circle.

Example
Determine the area of the shaded sector and the length of the arc AB.

Because the angle in the sector is 30°, and we know that a circle contains a total of 360°, we can determine what fraction of the circle's area it is: of the circle.
The area of the entire circle is A = πr2, so A = π(4)2 = 16π.
So, the area of the sector is square inches.
We can also determine the length of the arc AB, because it is of the circle's circumference.
The circumference of the entire circle is C = 2πr, so C = 2π(4) = 8π.
This means that the length of the arc is inches.

A prism that has circles as bases is called a cylinder. Recall that the formula for the volume of any prism is V = Abh. Because the area of the circular base is A = πr2, we can replace πr2 for Ab in the formula, giving us V = πr2h, where r is the radius of the circular base, and h is the height of the cylinder.

A sphere is a three-dimensional object that has no sides. A basketball is a good example of a sphere. The volume of a sphere is given by the formula V = πr3.

Example
Determine the volume of a sphere whose radius is 1.5'.
Replace 1.5' in for r in the formula V = πr3.
V = πr3.
V = π(1.5)3
V = (3.375)π
V = 4.5π ≈ 14.14
The answer is approximately 14.14 cubic feet.
Example
An aluminum can is 6" tall and has a base with a radius of 2". Determine the volume the can holds.
Aluminum cans are cylindrical in shape, so replace 2" for r and 6" for h in the formula V = πr2h.
V = πr2h
V = π(2)2(6)
V = 24π ≈ 75.40 cubic feet

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