Geometry: Praxis I Exam
This section will familiarize you with the properties of angles, lines, polygons, triangles, and circles, as well as the formulas for area, volume, and perimeter.
Geometry is the study of shapes and the relationships among them. Basic concepts in geometry will be detailed and applied in this section. The study of geometry always begins with a look at basic vocabulary and concepts. Therefore, a list of definitions and important formulas is provided.
Geometry Terms
area  the space inside a twodimensional figure 
circumference  the distance around a circle 
chord  a line segment that goes through a circle, with its endpoints on the circle 
congruent  equal, in reference to lengths, measures of angles, or size of figures 
diameter  a chord that goes directly through the center of a circle—the longest line segment that can be drawn in a circle 
hypotenuse  the longest side of a right triangle, always opposite the right angle 
leg  either of the two sides of a right triangle that make the right angle 
perimeter  the distance around a figure 
π (pi)  the ratio of any circle's circumference to its diameter (pi is an irrational number, but most of the time it is okay to approximate π with 3.14) 
radius  a line segment from the center of a circle to a point on the circle (half of the diameter) 
surface area  the sum of the areas of all of a threedimensional figure's faces 
volume  the space inside a threedimensional figure 
Coordinate Geometry
Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal xaxis and a vertical yaxis. These two axes intersect at one coordinate point, (0,0), the origin. A coordinate point, also called an ordered pair, is a specific point on the coordinate plane with the first number representing the horizontal placement and the second number representing the vertical placement. Coordinate points are given in the form of (x,y).
Graphing Ordered Pairs (Points)
 The xcoordinate is listed first in the ordered pair and tells you how many units to move to either the left or the right. If the xcoordinate is positive, move to the right. If the xcoordinate is negative, move to the left.
 The ycoordinate is listed second and tells you how many units to move up or down. If the ycoordinate is positive, move up. If the ycoordinate is negative, move down.
 Notice that the graph is broken up into four quadrants with one point plotted in each one. Here is a chart to indicate which quadrants contain which ordered pairs based on their signs:
Example
Graph the following points: (2,3), (3,–2), (–2,3), and (–3,–2).
Lines, Line Segments, and Rays
A line is a straight geometric object that goes on forever in both directions. It is infinite in length, and is represented by a straight line with an arrow at both ends. Lines can be labeled with one letter (usually in italics) or with two capital letters near the arrows. Line segments are portions of lines. They have two endpoints and a definitive length. Line segments are named by their endpoints. Rays have an endpoint and continue straight in one direction. Rays are named by their endpoint and one point on the ray.
Parallel and Perpendicular Lines
Parallel lines (or line segments) are a pair of lines that, if extended, would never intersect or meet. The symbol  is used to denote that two lines are parallel. Perpendicular lines (or line segments) are lines that intersect to form right angles, and are denoted with the symbol .
Lengths of Horizontal and Vertical Segments
Two points with the same ycoordinate lie on the same horizontal line and two points with the same xcoordinate lie on the same vertical line. The distance between a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points.
Example
Find the lengths of line segments and
Distance of Coordinate Points
The distance between any two points is given by the formula d = , where (x_{1},y_{1}) represents the coordinates of one point and (x_{2},y_{2}) is the other. The subscripts are used to differentiate between the two different coordinate pairs.
Example
Find the distance between points A(–3,5) and B(1,–4).
Let (x_{1},y_{1}) represent point A and let (x_{2},y_{2}) represent point B. This means that x_{1} = –3, y_{1} = 5, x_{2} = 1, and y_{2} = –4. Substituting these values into the formula gives us:
Midpoint
The midpoint of a line segment is a point located at an equal distance from each endpoint. This point is in the exact center of the line segment, and is said to be equidistant from the segment's endpoints.
In coordinate geometry, the formula for finding the coordinates of the midpoint of a line segment whose endpoints are (x_{1},y_{1}) and (x_{2},y_{2}) is given by M =
Example
Determine the midpoint of the line segment with A(–3,5) and B(1,–4).
Let (x_{1},y_{1}) represent point A and let (x_{2},y_{2}) represent point B. This means that x_{1} = –3, y_{1} = 5, x_{2} = 1, and y_{2} = –4. Substituting these values into the formula gives us:
Note: There is no such thing as the midpoint of a line, as lines are infinite in length.
Slope
The slope of a line (or line segment) is a numerical value given to show how steep a line is. A line or segment can have one of four types of slope:
 A line with a positive slope increases from the bottom left to the upper right on a graph.
 A line with a negative slope decreases from the upper left to the bottom right on a graph.
 A horizontal line is said to have a zero slope.
 A vertical line is said to have no slope (undefined).
 Parallel lines have equal slopes.
 Perpendicular lines have slopes that are negative reciprocals of each other.
The slope of a line can be found if you know the coordinates of any two points that lie on the line. It does not matter which two points you use. It is found by writing the change in the ycoordinates of any two points on the line, over the change in the corresponding xcoordinates. (This is also known as the rise over the run.)
The formula for the slope of a line (or line segment) containing points (x_{1}, y_{1}) and (x_{2}, y_{2}):
Example
Determine the slope of the line joining points A(–3,5) and B(1,–4).
Let (x_{1},y_{1}) represent point A and let (x_{2},y_{2}) represent point B. This means that x_{1} = –3, y_{1} = 5, x_{2} = 1, and y_{2} = –4. Substituting these values into the formula gives us:
Example
Determine the slope of the line graphed below.
Two points that can be easily determined on the graph are (3,1) and (0,–1). Let (3,1) = (x_{1}, y_{1}), and let (0,–1) = (x_{2}, y_{2}). This means that x_{1} = 3, y_{1} = 1, x_{2} = 0, and y_{2} = –1. Substituting these values into the formula gives us:
Note: If you know the slope and at least one point on a line, you can find the coordinate point of other points on the line. Simply move the required units determined by the slope. For example, from (8,9), given the slope , move up seven units and to the right five units. Another point on the line, thus, is (13,16).
Determining the Equation of a Line
The equation of a line is given by y = mx + b where:
 y and x are variables such that every coordinate pair (x,y) is on the line
 m is the slope of the line
 b is the yintercept, the yvalue at which the line intersects (or intercepts) the yaxis
In order to determine the equation of a line from a graph, determine the slope and yintercept and substitute it in the appropriate place in the general form of the equation.
Example
Determine the equation of the line in the graph below.
In order to determine the slope of the line, choose two points that can be easily determined on the graph. Two easy points are (–1,4) and (1,–4). Let (–1,4) = (x_{1}, y_{1}), and let (1,–4) = (x_{2}, y_{2}). This means that x_{1} = –1, y_{1} = 4, x_{2} = 1, and y_{2} = –4. Substituting these values into the formula gives us:
Looking at the graph, we can see that the line crosses the yaxis at the point (0,0). The ycoordinate of this point is 0. This is the yintercept.
Substituting these values into the general formula gives us y = –4x + 0, or just y = –4x.
Example
Determine the equation of the line in the following graph.
Two points that can be easily determined on the graph are (–3,2) and (3,6). Let (–3,2) = (x_{1},y_{1}), and let (3,6) = (x_{2},y_{2}). Substituting these values into the formula gives us:
We can see from the graph that the line crosses the yaxis at the point (0,4). This means the yintercept is 4.
Substituting these values into the general formula gives us y = x + 4.
Angles
Naming Angles
An angle is a figure composed of two rays or line segments joined at their endpoints. The point at which the rays or line segments meet is called the vertex of the angle. Angles are usually named by three capital letters, where the first and last letter are points on the end of the rays, and the middle letter is the vertex.
This angle can be named either ABC or CBA, but because the vertex of the angle is point B, letter B must be in the middle.
We can sometimes name an angle by its vertex, as long as there is no ambiguity in the diagram. For example, in the angle above, we may call the angle B, because there is only one angle in the diagram that has B as its vertex.
But, in the following diagram, there are a number of angles that have point B as their vertex, so we must name each angle in the diagram with three letters.
Angles may also be numbered (not measured) with numbers written between the sides of the angles, in the interior of the angle, near the vertex.
Classifying Angles
The unit of measure for angles is the degree.
Angles can be classified into the following categories: acute, right, obtuse, and straight.
 An acute angle is an angle that measures between 0 and 90 degrees.
 A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex.
 An obtuse angle is an angle that measures more than 90°, but less than 180
 A straight angle is an angle that measures 180°. Thus, both of its sides form a line.
Special Angle Pairs
 Adjacent angles are two angles that share a common vertex and a common side. There is no numerical relationship between the measures of the angles.
 A linear pair is a pair of adjacent angles whose measures add to 180°.
 Supplementary angles are any two angles whose sum is 180°. A linear pair is a special case of supplementary angles. A linear pair is always supplementary, but supplementary angles do not have to form a linear pair.
 Complementary angles are two angles whose sum measures 90 degrees. Complementary angles may or may not be adjacent.
Example
Two complementary angles have measures 2x° and 3x + 20°. What are the measures of the angles?
Because the angles are complementary, their sum is 90°. We can set up an equation to let us solve for x:
2x + 3x + 20 = 90
5x + 20 = 90
5x = 70
x = 14
Substituting x = 14 into the measures of the two angles, we get 2(14) = 28° and 3(14) + 20 = 62°. We can check our answers by observing that 28 + 62 = 90, so the angles are indeed complementary.
Example
One angle is 40 more than 6 times its supplement. What are the measures of the angles?
Let x = one angle.
Let 6x + 40 = its supplement.
Because the angles are supplementary, their sum is 180°. We can set up an equation to let us solve for x:
x + 6x + 40 = 180
7x + 40 = 180
7x = 140
x = 20
Substituting x = 20 into the measures of the two angles, we see that one of the angles is 20° and its supplement is 6(20) + 40 = 160°. We can check our answers by observing that 20 + 160 = 180, proving that the angles are supplementary.
Note: A good way to remember the difference between supplementary and complementary angles is that the letter c comes before s in the alphabet; likewise 90 comes before 180 numerically.
Angles of Intersecting Lines
Important mathematical relationships between angles are formed when lines intersect. When two lines intersect, four smaller angles are formed.
Any two adjacent angles formed when two lines intersect form a linear pair, therefore they are supplementary. In this diagram, 1 and 2, 2 and 3, 3 and 4, and 4 and 1 are all examples of linear pairs.
Also, the angles that are opposite each other are called vertical angles. Vertical angles are angles that share a vertex and whose sides are two pairs of opposite rays. Vertical angles are congruent. In this diagram, 1 and 3 are vertical angles, so 1 3; 2 and 4 are congruent vertical angles as well.
Note: Vertical angles is a name given to a special angle pair. Try not to confuse this with right angles or perpendicular angles, which often have vertical components.
Example
Determine the value of y in the following diagram:
The angles marked 3y + 5 and 5y are vertical angles, so they are congruent and their measures are equal. We can set up and solve the following equation for y:
3y + 5 = 5y
5 = 2y
2.5 = y
Replacing y with the value 2.5 gives us the 3(2.5) + 5 = 12.5 and 5(2.5) = 12.5. This proves that the two vertical angles are congruent, with each measuring 12.5°.
Parallel Lines and Transversals
Important mathematical relationships are formed when two parallel lines are intersected by a third, non parallel line called a transversal.
In the preceding diagram, parallel lines l and m are intersected by transversal n. Supplementary angle pairs and vertical angle pairs are formed in this diagram, too.
Other congruent angle pairs are formed:
 Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the transversal: 3 and 6; 4 and 5.
 Corresponding angles are angles on corresponding sides of the parallel lines, on corresponding sides of the transversal: 1 and 5; 2 and 6; 3 and 7; 4 and 8.
Example
In the following diagram, line l is parallel to line m. Determine the value of x.
The two angles labeled are corresponding angle pairs, because they are located on top of the parallel lines and on the same side of the transversal (same relative location). This means that they are congruent, and we can determine the value of x by solving the equation:
4x + 10 = 8x– 25
10 = 4x – 25
35 = 4x
8.75 = x
We can check our answer by replacing the value 8.75 in for x in the expressions 4x + 10 and 8x – 25:
4(8.75) + 10 = 8(8.75) – 25
45 = 45
Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the problem would be solved in the same way.
Area, Circumference, and Volume Formulas
Here are the basic formulas for finding area, circumference, and volume. They are discussed in detail in the sections that follow.
Triangles
The sum of the measures of the three angles in a triangle always equals 180°.
Exterior Angles
An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles:
 An exterior angle and an interior angle that share the same vertex are supplementary. In other words, exterior angles and interior angles form straight lines with each other.
 An exterior angle is equal to the sum of the nonadjacent interior angles.
 The sum of the exterior angles of a triangle equals 360°.
Example
Classifying Triangles
It is possible to classify triangles into three categories based on the number of congruent (indicated by the symbol ) sides. Sides are congruent when they have equal lengths.
AngleSide Relationships
Knowing the angleside relationships in isosceles, equilateral, and right triangles is helpful.
 In isosceles triangles, congruent angles are opposite congruent sides.
 In equilateral triangles, all sides are congruent and all angles are congruent. The measure of each angle in an equilateral triangle is always 60°.
 In a right triangle, the side opposite the right angle is called the hypotenuse and the other sides are called legs. The box in the angle of the 90degree angle symbolizes that the triangle is, in fact, a right triangle
Pythagorean Theorem
The Pythagorean theorem is an important tool for working with right triangles. It states: a^{2} + b^{2} = c^{2}, where a and b represent the legs and c represents the hypotenuse.
This theorem makes it easy to find the length of any side as long as the measures of two sides are known. So, if leg a = 1 and leg b = 2 in the triangle below, it is possible to find the measure of the hypotenuse, c.
Pythagorean Triples
Sometimes, the measures of all three sides of a right triangle are integers. If three integers are the lengths of the sides of a right triangle; we call them Pythagorean triples. Some popular Pythagorean triples are:
 3, 4, 5
 5, 12, 13
 8, 15, 17
 9, 40, 41
The smaller two numbers in each triple represent the lengths of the legs, and the largest number represents the length of the hypotenuse.
Multiples of Pythagorean Triples
Wholenumber multiples of each triple are also triples. For example, if we multiply each of the lengths of the triple 3, 4, 5 by 2, we get 6, 8, 10. This is also a triple
Example
If given a right triangle with sides measuring 6, x, and a hypotenuse 10, what is the value of x?
3, 4, 5 is a Pythagorean triple, and a multiple of that is 6, 8, 10. Therefore, the missing side length is 8.
Comparing Triangles
Triangles are said to be congruent (indicated by the symbol ) when they have exactly the same size and shape. Two triangles are congruent if their corresponding parts (their angles and sides) are congruent. Sometimes, it is easy to tell if two triangles are congruent by looking at them. However, in geometry, we must be able to prove that the triangles are congruent.
There are a number of ways to prove that two triangles are congruent:
SideSideSide (SSS)  If the three sides of one triangle are congruent to the three corresponding sides of another triangle, the triangles are congruent. 
SideAngleSide (SAS)  If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, the triangles are congruent. 
AngleSideAngle (ASA)  If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, the triangles are congruent. 
Used less often but also valid:
AngleAngleSide (AAS)  If two angles and the nonincluded side of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, the triangles are congruent. 
HypotenuseLeg (HyLeg)  If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, the triangles are congruent. 
Example
Determine if these two triangles are congruent.
Although the triangles are not aligned the same, there are two congruent corresponding sides, and the angle between them (150°) is congruent. Therefore, the triangles are congruent by the SAS postulate.
Example
Determine if these two triangles are congruent.
Although the triangles have two congruent corresponding sides and a corresponding congruent angle, the 150° angle is not included between them. This would be SSA, but SSA is not a way to prove that two triangles are congruent.
Area of a Triangle
Area is the amount of space inside a twodimensional object. Area is measured in square units, often written as unit^{2}. So, if the length of a triangle is measured in feet, the area of the triangle is measured in feet^{2}.
A triangle has three sides, each of which can be considered a base of the triangle. A perpendicular line segment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height. It measures how tall the triangle stands.
It is important to note that the height of a triangle is not necessarily one of the sides of the triangle. The correct height for the following triangle is 8, not 10. The height will always be associated with a line segment (called an altitude) that comes from one vertex (angle) to the opposite side and forms a right angle (signified by the box). In other words, the height must always be perpendicular to (form a right angle with) the base. Note that in an obtuse triangle, the height can be outside the triangle, and in a right triangle the height is usually one of the sides.
The formula for the area of a triangle is given by , where b is the base of the triangle, and h is the height.
Example
Determine the area of the following triangle.
Volume Formulas
A prism is a threedimensional object that has matching polygons as its top and bottom. The matching top and bottom are called the bases of the prism. The prism is named for the shape of the prism's base, so a triangular prism has congruent triangles as its bases.
Note: This can be confusing. The base of the prism is the shape of the polygon that forms it; the base of a triangle is one of its sides.
Volume is the amount of space inside a threedimensional object. Volume is measured in cubic units, often written as unit^{3}. So, if the edge of a triangular prism is measured in feet, the volume of it is measured in feet^{3}.
The volume of ANY prism is given by the formula V = A_{b}h, where A_{b} is the area of the prism's base, and h is the height of the prism.
Example
Determine the volume of the following triangular prism:
The area of the triangular base can be found by using the formula bh, so the area of the base is (15)(20) = 150. The volume of the prism can be found by using the formula V = A_{b}h, so the volume is V= (150)(40) = 6,000 cubic feet.
A pyramid is a threedimensional object that has a polygon as one base, and instead of a matching polygon as the other, there is a point. Each of the sides of a pyramid is a triangle. Pyramids are also named for the shape of their (nonpoint) base.
The volume of a pyramid is determined by the formula A_{b}h.
Example
Determine the volume of a pyramid whose base has an area of 20 square feet and stands 50 feet tall.
Because the area of the base is given to us, we only need to replace the appropriate values into the formula.
The pyramid has a volume of 33 cubic feet.
Polygons
A polygon is a closed figure with three or more sides—for example, triangles, rectangles, and pentagons.
Terms Related to Polygons
 Vertices are corner points, also called endpoints, of a polygon. The vertices in the above polygon are A, B, C, D, E, and F, and they are always labeled with capital letters.
 A regular polygon has congruent sides and congruent angles
 An equiangular polygon has congruent angles.
Interior Angles
To find the sum of the interior angles of any polygon, use this formula:
 S = (x – 2)180°, where x = the number of sides of the polygon.
Example
Find the sum of the interior angles in this polygon:
The polygon is a pentagon that has five sides, so substitute 5 for x in the formula:
S = (5 – 2) × 180°
S = 3 × 180°
S = 540°
Exterior Angles
Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees.
Similar Polygons
If two polygons are similar, their corresponding angles are congruent and the ratios of the corresponding sides are in proportion.
Example
These two polygons are similar because their angles are congruent and the ratios of the corresponding sides are in proportion
Quadrilaterals
A quadrilateral is a foursided polygon. Because a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360°.
Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides.
In the figure, . Parallel lines are symbolized with matching numbers of triangles or arrows
A parallelogram has:
 opposite sides that are congruent
 opposite angles that are congruent (A = C and B = D)
 consecutive angles that are supplementary (A + B = 180°, B + C = 180°, C + D = 180°, D + A = 180° )
 diagonals (line segments joining opposite vertices) that bisect each other (divide each other in half)
Special Types of Parallelograms
 A rectangle is a parallelogram that has four right angles
 A rhombus is a parallelogram that has four equal sides.
 A square is a parallelogram in which all angles are equal to 90 degrees and all sides are congruent. A square is a special case of a rectangle where all the sides are congruent. A square is also a special type of rhombus where all the angles are congruent
Diagonals of Parallelograms
In this diagram, parallelogram ABCD has diagonals and that intersect at point E. The diagonals of a parallelogram bisect each other, which means that .
In addition, the following properties hold true:
 The diagonals of a rhombus are perpendicular.
 The diagonals of a rectangle are congruent.
 The diagonals of a square are both perpendicular and congruent.
Example
In parallelogram ABCD, the diagonal = 5x + 10 and the diagonal = 9x. Determine the value of x.
Because the diagonals of a parallelogram are congruent, the lengths are equal. We can then set up and solve the equation 5x + 10 = 9x to determine the value of x.
5x + 10 = 9x Subtract x from both sides of the equation. 10 = 4x Divide 4 from both sides of the equation. 2.5 = x
Area and Volume Formulas
The area of any parallelogram can be found with the formula A = bh, where b is the base of the parallelogram, and h is the height. The base and height of a parallelogram is defined the same as in a triangle.
Note: Sometimes b is called the length (l) and h is called the width (w) instead. If this is the case, the area formula is A = lw.
A rectangular prism (or rectangular solid) is a prism that has rectangles as bases. Recall that the formula for any prism is V = A_{b}h. Because the area of the rectangular base is A = lw, we can replace lw for A_{b} in the formula giving us the more common, easier to remember formula, V = lwh. If a prism has a different quadrilateralshaped base, use the general prisms formula for volume.
Note: A cube is a special rectangular prism with six congruent squares as sides. This means that you can use the V = lwh formula for it, too.
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 = πr^{2}, 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 = πr^{2}, 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 = A_{b}h. Because the area of the circular base is A = πr^{2}, we can replace πr^{2} for A_{b} in the formula, giving us V = πr^{2}h, where r is the radius of the circular base, and h is the height of the cylinder.
A sphere is a threedimensional 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 = πr^{3}.
 Example
 Determine the volume of a sphere whose radius is 1.5'.
 Replace 1.5' in for r in the formula V = πr^{3}.
 V = πr^{3}.
 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 = πr^{2}h.
 V = πr^{2}h
 V = π(2)^{2}(6)
 V = 24π ≈ 75.40 cubic feet

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