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ET-314
Week 9
Basic Geometry - Perimeters
• Rectangles: P = 2 (L + W)
Example: For a rectangle with length = 4 cm and width = 7 cm
P = 2 (4 cm + 7 cm) = 22 cm
• Squares: P = 4 S
Example: For a square with one side = 5 cm
P = 4 * 5 cm = 20 cm
• Triangles: P = side a + side b + side c
Example: For a triangle with sides = 12 cm, 7 cm, and 8 cm
P = 12 cm + 7 cm + 8 cm = 27 cm
• Equilateral triangles: P = 3 S
Example: For a equilateral triangle with sides = 12 inches
P = 3 * 12” = 36”
Basic Geometry - Perimeters
• Circles: P = 2 p R = p D
• Example: For a circle with radius = 2 inches
P = 2 * p * 2 in = 12.566 in
• p is a constant used for calculations involving
circular objects.
• It is the ratio of the circumference of a circle
to its diameter.
• It has a numerical value of 3.14159… You can
find the “p “ key in your calculator.
Basic Geometry - Area
• Rectangles: A = L  W
Example: For a rectangle with length = 4 cm and width = 7 cm
A = 4 cm * 7 cm = 28 cm2
• Squares: A = S 2
Example: For a square with one side = 5 cm
A = (5 cm)2 = 25 cm2
• Triangles: A = (b  h) / 2
Example: For a triangle with base = 12 cm and height = 7 cm
A = (12 cm * 7 cm) / 2 = 42 cm2
• Circles: A = p R2 = p D2/4
Example: For a circle with radius = 5 inches
A = p * (5 in) 2 = 78.54 in2
Basic Geometry - Volume
• Boxes: V = L  W  H
Example: For a rectangular box with L = 4 cm, W = 7 cm, and
H = 5 cm
V = 4 cm * 7 cm * 5 cm = 140 cm3
• Cubes: V = S 3
Example: For a cube with one side = 4 cm
V = (4 cm)3 = 64 cm3
• Cylinder: V = p R2 h = (p D2 h)/4
Example: For a cylinder with r = 5“ and h = 10“
V = p  52  10 = 785.4 in3
Basic Properties of Triangles
• Trigonometry: Trigonometry is a branch of
mathematics that studies triangles.
• Angle (4th, p. 384, 3rd, p. 321):
• An angle is formed whenever two straight
lines meet at a point.
• The magnitude of an angle is a measure of the
difference in the directions of the sides only –
it has no bearing on the lengths of the sides.
Basic Properties of Triangles
•
•
•
•
•
Right angle –formed by two perpendicular lines = 90.
Acute angle –smaller than a right angle.
Obtuse angle –greater than a right angle.
Straight angle – a straight line = 180
Complementary angles – Two angles whose sum equals
to a right angle.
• Supplementary angles – Two angles whose sum equals
to a straight angle.
• Vertical angles – opposite angles formed by two
intersecting straight lines and are equal.
• Perpendicular lines: the vertical angles equal to 90
(right angle).
Basic Properties of Triangles
Basic Properties of Triangles
Angular system:
• The angular system is the most widely used angular
measurement system.
• It divides a complete revolution into 360 degrees, each
degree into 60 minutes, and each minute into 60
seconds.
• However, minutes and seconds are usually expressed in
terms of decimal degrees for convenience.
Example: 23 15’ = 23.25 (15 / 60 = 0.25)
Basic Properties of Triangles
Radian system:
• The circular, or natural, system is usually used in
mathematical calculations and derivations when
trigonometric functions are involved.
• It divides a complete revolution into 2 p radians.
degree = radian  180 / p
radian = degree  p / 180
• Examples:
23 = 23  p / 180 = 0.4014 rad
3.5 rad = 3.5  180 / p = 200.54
Basic Properties of Triangles
• The sum of the internal angles of a triangle equals to 180:
Example: If two angles are 58 and 70, the third angle is: 180 – 58
– 70 = 52
• Triangles:
– Acute triangle: contains three acute angles.
– Obtuse triangle: contains one obtuse angle.
• Right triangles:
– A right triangle: one of its angles equals to a right angle (90).
– Any triangle can be constructed using two right triangles.
Basic Properties of Triangles
Basic Properties of Triangles
Basic Properties of Triangles
Basic Properties of Triangles
Compute the area of a right triangle:
The area of a right triangle equals to the product of
the base and the altitude divided by 2:
Area = (1/2) a  b
Example:
If a = 7 cm and b = 5 cm, Area = (7  5) / 2 = 17.5 cm2
Basic Properties of Triangles
Basic Properties of Triangles
Reference and Special Angles
Reference and Special Angles
• Quadrant angles: 0, 90, 180, and 270, w.r.t. the 1st,
2nd, 3rd, and 4th quadrant.
• Reference angles:
Examples:
A = 60 in the 3rd quadrant means  = 180 + A = 240
A = 60 in the 4th quadrant means  = 360 – A = 300
Reference and Special Angles
• Angles larger than 360:
Express the angle in
the following form:
A =  - 360
Example:
450 = 450 - 360 =
90
• Negative angles:
Express a negative
angle in the following
form:
A = 360 - 
Example: -35 = 360
– 35 = 325