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  R =  D Example: For a circle with radius = 2 inches P = 2 *  * 2 in = in  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 … You can find the “  “ 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 cm 2 Squares: A = S 2 Example: For a square with one side = 5 cm A = (5 cm) 2 = 25 cm 2 Triangles: A = (b  h) / 2 Example: For a triangle with base = 12 cm and height = 7 cm A = (12 cm * 7 cm) / 2 = 42 cm 2 Circles: A =  R 2 =  D 2 /4 Example: For a circle with radius = 5 inches A =  * (5 in) 2 = in 2

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 cm 3 Cubes: V = S 3 Example: For a cube with one side = 4 cm V = (4 cm) 3 = 64 cm 3 Cylinder: V =  R 2 h = (  D 2 h)/4 Example: For a cylinder with r = 5“ and h = 10“ V =   5 2  10 = in 3

Basic Properties of Triangles Trigonometry: Trigonometry is a branch of mathematics that studies triangles. Angle (4 th, p. 384, 3 rd, 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

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’ =  (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  radians. degree = radian  180  /  radian = degree  / 180  Examples: 23  = 23   / 180  = rad 3.5 rad = 3.5  180  /  = 

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

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 cm 2

Basic Properties of Triangles

Reference and Special Angles

Quadrant angles: 0 , 90 , 180 , and 270 , w.r.t. the 1 st, 2 nd, 3 rd, and 4 th quadrant. Reference angles: Examples:  A = 60  in the 3 rd quadrant means  = 180  +  A = 240   A = 60  in the 4 th 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 =   360  Example: 450  = 450   = 90 