Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Section A.1: Fundamentals of Geometry
Definition of geometry
The word geometry means geo = earth and metria = measure: earth measure.
The Greek mathematician Euclid (∼ 300 B.C.) wrote a 13-volume textbook called The Elements that describes
the geometry called Euclidean geometry. This is the geometry of points, lines, and planes.
Section A.2: Points, Lines, and Planes
Definition of point, line and plane
Geometric objects, such as points, lines, and planes, represent idealizations that do not exist in the real
world.
A geometric point is imagined to have zero size. No real object has zero size, but many real phenomena
approximate geometric points. Stars, for example, appear as points of light in the sky.
A geometric line is formed by connecting two points along the shortest possible path. It has infinite length
and no thickness. Because no physical object is infinite in length, we usually work with the line segments,
or pieces of a line. One ”spaghetto“ is a good approximation to a line segment.
A geometric plane is a flat surface that has infinite length and width, but no thickness. A smooth tabletop is
a good approximation to a segment of a plane.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Definition of dimension
The dimension of an object is the number of independent directions in which you could move if you were
on the object. If you were a prisoner confined to a point, you would have no place to go, so a point has zero
dimensions. If you walk on a line, you can move only in one direction (forward and backward count as
the same direction, one positive and the other negative), so a line is one-dimensional. In a plane, you can
move in two independent directions, such as north-south and east-west (or any combination of those two
directions). Therefore, a plane is two-dimensional. In a three-dimensional space, such as the world around
us, you can move in three independent directions: north-south, east-west, and up-down.
Another way to think about dimension is by the number of coordinates required to locate a point. In a line
you need only one coordinate x to find a point, this is why the line is one-dimensional.
A plane is two-dimensional since it requires two coordinates, x and y, to locate a point.
Three coordinates x, y, and z are needed to locate a point in three-dimensional space.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Angles
The intersection of two lines or line segments forms an angle. The point of intersection is called vertex.
The most common way to measure angles is in degrees. By definition, a full circle has an angle of 360◦ , so
k
an angle of k ◦ degrees represents 360
of a circle.
•
•
•
•
A right angle measures 90◦ .
A straight angle is formed by a straight line and measures 180◦ .
An acute angle is any angle whose measure is less than 90◦ .
An obtuse angle is any angle whose measure is between 90◦ and 180◦ .
Angles can also be measured in radians. The angle that subtends a full circle is defined to be 2π radians.
Thus,
2π
360◦
1◦ =
radians and 1 radian =
∼ 57.3◦
360
2π
A right angle is 90◦ or π2 radians. A straight angle is 180◦ or π radians.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Parallel and perpendicular lines
Two lines or line segments in a plane that meet in a right angle are called perpendicular. Two lines or line
segments in a plane that are everywhere the same distance apart are called parallel. Parallel lines in a plane
never meet.
Ex.1
Find the degree and radian measure of the angles that sutend
(1) a semicircle (half a circle).
(2) a quarter circle.
(3) an eighth of a circle.
(4) a hundredth of a circle.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Section A.3: Plane Geometry
Plane geometry
Plane geometry is the geometry of two-dimensional object.
Definition of a circle
A circle is the locus of all points equidistant from a point called center. The distance between the center and
a point on the circle is called radius. The diameter of a circle is twice its radius, which means it is the distance
across the circle passing through its center.
Definition of a polygon
A polygon is any closed shape in the plane made from straight line segments. A regular polygon is a regular
polygon in which all the sides have the same length and all interior angles are equal.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Triangles
Triangles are polygons with 3 sides and 3 angles. A triangle is equilateral if it has all the sides of equal length.
A triangle is isosceles if it has 2 sides of equal length. A right triangle is a triangle that has a right angle.
The sum of the three angle measures is always 180◦ .
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Perimeter and Area
Definition of perimeter and area
The perimeter of a plane object is the length of its boundary. In the case of a circle, the perimeter is called
circumference. The area is the measure of a region bounded by a curve.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Ex.2
A window consists of a 4-foot by 6-foot rectangle capped by a semicircle. How much trim is needed to go
around the window?
Ex.3
You have built a stairway in the new house and want to cover the space beneath the stairs with polywood.
What is the area of this region?
Ex.4
A one-block city park is bounded by two sets of parallel streets. The streets along the block are each 55
yards long and the perpendicular distance between the streets is 39 yards. How much sod should you be
purchased to cover the entire park in grass?
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Section A.4: Three-Dimensional Geometry
Definition of surface area and volume
Surface area is the measure of how much exposed area a solid object has (square units) . The volume of a
solid object is how much space it occupies.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Ex.5 Water reservoir.
A water reservoir has a rectangular base that measures 30 meters by 40 meters, and vertical walls 15
meters high. At the beginning of the summer, the reservoir was filled to capacity. At the end of the
summer, the water depth was 4 meters. How much water was used?
Ex.6 Comparing volumes.
Which holds more soup: a can with a diameter of 3 inches and a height of 4 inches or a can with a diameter
of 4 inches and a height of 3 inches?
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Section A.5: Scaling Laws
Scaling
Scaling is a process by which a real object is modeled by a similar object whose dimensions are proportionally larger or smaller.
Ex.7
An architect makes a scale model of an auditorium in which all lengths are smaller by a factor of 100. The
scale factor in this case is 100.
Scaling laws
We have the following scaling laws:
• Lengths always scale with the scale factor.
• Areas always scale with the square of the scale factor.
• Volumes always scale with the cube of the scale factor.
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Ex.8
Suppose that we make an engineering model of a car with a scale factor of 10. The actual car will be
10 times as long, 10 times as wide, and 10 times as tall as the model car. How will the surface area and
volume of the actual car scale with the surface area and volume of the model?
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Chapter 10: Modeling with Geometry
Lecture notes
Math 1030 Section A
Ex.9
Suppose your size suddenly doubled, that is, your height, width, and depth doubled. For example, if you
were 5 feet tall before, you now are 10 feet tall.
(1) By what factor has your waist size increased?
(2) How much more material will be required for your clothes?
(3) By what factor has your weight changed?
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