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 SExample: For a square with one side = 5 cm P = 4 * 5 cm = 20 cm

• Triangles: P = side a + side b + side cExample: 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 SExample: 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) / 2Example: For a triangle with base = 12 cm and height = 7 cm A = (12 cm * 7 cm) / 2 = 42 cm2

• Circles: A = p R2 = p D2/4Example: 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 cmV = 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)/4Example: 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 TrianglesRadian 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 / pradian = degree p / 180

• Examples: 23 = 23 p / 180 = 0.4014 rad3.5 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 TrianglesCompute 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 = 240A = 60 in the 4th quadrant means = 360 – A = 300

Reference and Special Angles

• Negative angles: Express a negative angle in the following form: A = 360 - Example: -35 = 360 – 35 = 325

• Angles larger than 360: Express the angle in the following form: A = - 360Example: 450 = 450 - 360 = 90