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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