Download Area of a Parallelogram

Math 300
Basic College Mathematics
Chapter 8
Geometry
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 1 of 18
8.1 Lines and Angles
You must be able to identify line segments, rays and angles.
In your text, Basic College Mathematics, please refer to pages 511 – 516.
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 2 of 18
8.2 Plane Figures and Solids
You must be able to identify plane figures and solids.
In your text, Basic College Mathematics, please refer to pages 521 – 525.
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 3 of 18
8.3 Perimeter
Perimeter of a Rectangle
 To find the perimeter of a rectangle, you may do either of the following:
a. add all sides
Or
b. use the formula: P = 2l + 2w
(P stands for perimeter, l stands for length, and w stands for width)
Example: Find the perimeter.
2 cm
4 cm
Adding all sides
P = 4 cm + 2 cm + 4 cm + 2 cm
P = 12 cm
Or
Use the formula
P = 2l +2w
P = 2(4) + 2(2)
P=8+4
P = 12 cm
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 4 of 18
8.3 Perimeter (cont)
Perimeter of a Square
 To find the perimeter of a square, you may do either of the following:
a. add all sides
Or
b. use the formula: P = 4 · s
(P stands for perimeter and s stands for side)
Example: Find the perimeter.
10 km
10 km
Adding all sides
P = 10 km + 10 km + 10 km + 10 km
P = 40 km
Or
Use the formula
P=4·s
P = 4 · 10
P = 40 km
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 5 of 18
8.4 Area
All Area Answers will be written using Units²
Area of a Rectangle
 To find the area of a rectangle, use the following formula:
A=l·w
(A stands for area, l stands for length, and w stands for width)
Example: Find the area.
5 ft
10 ft
A=l·w
A = 10 ft · 5 ft
A = 50 ft 2
Area of a Square
 To find the area of a square, use the following formula:
A = s2
(A stands for area and s stands for side)
Example: Find the area.
12 km
A=s·s
A = 12 km · 12km
A = 144 km 2
12 km
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 6 of 18
8.4 Area (cont)
Area of a Triangle
Area of a Triangle
 To find the area of a triangle, use the following formula:
A=
1
b h
2
(A stands for area, b stands for base, and h stands for height)
Base is always perpendicular to Height.
Example: Find the area.
12 m
16 m
A=
1
b h
2
A=
1
16 12
2
A = 812
{First, multiply
1
times 16; which equals to 8}
2
{Then, multiply 8 times 12 to get the answer}
A = 96 m 2
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 7 of 18
8.4 Area (cont)
Parallelograms and Trapezoids
A parallelogram is a four-sided figure with two
pairs of parallel sides.
Area of a Parallelogram
A = b · h (Area = base times height)
Base is always perpendicular to Height.
Example: Find the area of this parallelogram.
5 km
7 km
A=b·h
A = 7 km · 5 km
A = 35 km
2
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 8 of 18
8.4 Area (cont)
A trapezoid is a polygon with four sides with one
pair of sides parallel to each other.
The parallel sides are called the bases.
Area of a trapezoid
1
A = 2 · h · (a + b)
(h stands for height, a & b stands for bases.)
Example: Find the area of this trapezoid.
12 cm
7 cm
18 cm
A = ½ h · (a + b)
A = ½ ·7· (12 + 18)
2
A = (3.5) · (30) = 105 cm
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 9 of 18
8.4 Area (cont)
Circles
Diameter of a Circle
Formula for finding the diameter of a circle is:
d=2·r
Diameter is Twice the Radius
Example: Find the length of a diameter of this
circle.
d=2·r
2
1
d = 2 · 4 ft = 4 ft
1
ft
4
1
d = 2 ft (or 0.5 ft)
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 10 of 18
8.4 Area (cont)
Radius of a Circle
A radius is a segment with one endpoint on
the center and the other endpoint on the
circle.
Formula for finding the radius of a circle is: r
d
= 2
Radius is Half of Diameter
Example: Find the length of a radius of this
circle.
d
r= 2
12
r= 2 m
12 m
r=6m
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 11 of 18
8.4 Area (cont)
Circumference of a Circle
The perimeter of a circle is called its
circumference.
Formula for finding circumference of a circle
with given diameter is
C=Π·d
22
The number Π (pi) is about 3.14 or 7
Example: Find the circumference of this circle.
Use 3.14 for pi.
C=Π·d
C  3.14 · 6 cm
6 cm
C  18.84 cm
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 12 of 18
8.4 Area (cont)
Optional Formula for finding circumference of
a circle with given radius is
C=2·Π·r
22
The number Π (pi) is about 3.14 or 7
Example: Find the circumference of this circle.
22
Use 7 for pi.
C=2·Π·r
22
C  2 · 7 · 70 in
C
3080
 7
70 in
in
C  440 in
Note: You may use C=∏d and just
double the radius to find the
140-in diameter.
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 13 of 18
8.4 Area (cont)
Area of a Circle
A=Π·r
2
Example: Find the area of this circle.
14 cm
A=Π·r
2
A  3.14 · (14cm)²
A  615.44 cm
2
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 14 of 18
8.4 Area (cont)
Example: Find the area of this circle.
10 in
First, you must find the radius of the circle. (Note: the
picture shows the diameter and the formula for area is pi
times radius squared.)
d
10
r = 2 = 2 = 5 in.
Now, you are ready to find the area.
A =   r2
A

A
 78.5 in 2
3.14  (5)2
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 15 of 18
8.7 Congruent and Similar Triangles
You must be able to determine whether two triangles are congruent, to
find the ratios of corresponding sides in similar triangles, and to find the
lengths of sides of similar triangles.
In your text, Basic College Mathematics, please refer to pages 566 – 569.
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 16 of 18
Math 300 Chapter 8
Glossary
Point – a point is a place in space. It has no length, width or height. It does have a
location.
Right triangle - a triangle that has one right angle.
Equilateral triangle - all three sides of the triangle are the same length. Also, all
three interior angles have the same measure.
Isosceles triangle – two sides have the same length. Also, angles opposite the equal
sides have an equal measure.
Scalene triangle - a triangle with no sides having the same length. None of the
interior angles have the same measure.
Hypotenuse – the side opposite the right angle in a right triangle
Legs – the other two side of a right triangle that are not the hypotenuse
Line Segment – a piece of a line. A line segment has two endpoints.
Complementary angles - two angles that have a sum of 90
Supplementary angles - two angles that have a sum of 180
Line – a set of points extending indefinitely in two directions. A line has neither
width or height but it does have length.
Polygon – a closed plane figure that lies on a plane.
Regular polygon – is a polygon that has all sides the same length and whose angles
have the same measure.
Perimeter – the distance around a polygon
Angle – an angle is formed when two rays start from the same endpoint
Vertex – the point where two rays start is called a vertex
Congruent – triangles that have the same size and shape are congruent
Area – a measure of the amount of surface in a region
Ray – part of a line with one endpoint. The ray extends indefinitely in one direction.
Square root – the square root of a number x is a number y whose square is a
Transversal –a line that intersects two or more lines at different points is called a
transversal
Straight angle - an angle that measures 180
Volume – the measure of space within a solid
Vertical angles – when two lines intersect, they form 4 angles. The angles opposite
each other are the vertical angles.
Adjacent angles – when two angles in a triangle share a common side, they are
called adjacent angles
Obtuse angle - an angle whose measure is between 90 and 180
Acute angle - an angle whose measure is between 0 and 90
Right angle - angle that measures 90
Similar triangle - when triangles have exactly the same shape. The size may be
different.
Quadrilateral - a polygon with 4 sides.
Parallelogram - a special quadrilateral with opposite sides parallel and equal in
length.
Rectangle – a special parallelogram that has 4 right angles.
Math 300 M-G Chapter 8; Rev: Aug. 2009
Page 17 of 18
Square – a special rectangle that has all 4 sides equal in length.
Rhombus – a special parallelogram that has all 4 sides equal in length.
Trapezoid - a quadrilateral with exactly one pair of opposite sides parallel.
Circle – a plane figure that consists of all points that are the same fixed distance
from a point c. The point c is called the center.
Radius – the distance from the center of a circle to any point on the circle.
Diameter – distance between two points on the circle passing through the center.
Solid – a figure that lies in space, it has length, width and height or depth.
Rectangular solid - is a solid whose 6 sides are all rectangles.
Cube – a rectangular solid whose 6 sides are all squares.
Pyramid - a solid that has a square base and heights are perpendicular to their base
(page 524).
Sphere – all points in space that are the same distance from a point c.
Cylinder – bases are circles and heights are perpendicular to the base.
Cones – base is a circle and heights are perpendicular to the base.
Properties
The sum of the measures of the interior angles of a triangle is 180.
Perimeter of a rectangle is 2(length) + 2(width)
Perimeter of a square is 4(side)
Perimeter of a triangle is side a + side b + side c
Circumference of a circle is 2()radius or (diameter)
Area of a rectangle = (length)(width)
Area of a square = (side)(side) side squared
Area of a triangle = ½ (base)(height)
Area of a parallelogram = (base)(height)
Area of a trapezoid = ½(base-1 + base-2)(height)
Area of a circle = (radius)(radius)  = 22/7 or =3.14
Congruent triangles – Angle-Side-Angle (ASA)
Congruent triangles – Side-Side-Side (SSS)
Congruent triangles – Side-Angle-Side (SAS)
Hints
When there is no possibility of confusion as to which angle is being named, you may
use the vertex alone.
Since a right triangle has one angle that equals 90, the sum of the measure of the
other two interior angles is also 90.
Math 300 M-G Chapter 8; Rev: Aug. 2009
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