Math 103-Mensuration-Areas-sem-dfs

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Mathematics for Interior Design- Geometry-Plane Mensuration-Area- by Dr. Farhana Shaheen-KSA

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Math 103-Mensuration-Areas-sem-dfs

  1. 1. MENSURATIONMATH-103Mathematics for Interior DesignDr. Farhana Shaheen
  2. 2. MENSURATION Mensuration is to measure the quantities such as perimeter, area, volume, length of a closed www.brainybetty.com Free powerpoint template: geometrical figure. If we want to build a house or planning a garden, we need to know exactly the space we have and the amount of space we need. For the purpose for these we have to calculate the length of the boundary and space occupied. In other words we have to determine the perimeter and area of the plot. 2
  3. 3. Free powerpoint template:www.brainybetty.com 3
  4. 4. DEFINITION: MENSURATION Mensuration is a branch of Mathematics which deals with the measurements of lengths of lines, www.brainybetty.com Free powerpoint template: areas of surfaces and volumes of solids. Mensuration may be divided into two parts: 1. Plane Mensuration (for 2 dimension) 2. Solid Mensuration (for 3 dimension) 4
  5. 5. SHAPES IN 2-DAND 3-D
  6. 6. PLANE MENSURATION Plane Mensuration deals with perimeter, length of sides and areas of two dimensional figures and www.brainybetty.com Free powerpoint template: shapes. For example, Circle, Semi-circle Rectangle Pentagon Semi-circle Triangles, Trapezium, etc. 6
  7. 7. SOLID MENSURATION Solid mensuration deals with areas and volumes of solid objects in three dimensions. www.brainybetty.com Free powerpoint template: For example: Polyhedrons, Cylinders, Cones, Spheres. They are called Space Figures. 7
  8. 8. SPACE FIGURES Polyhedrons are space figures with flat surfaces, called faces, which are made of polygons. Prisms and pyramids are examples of polyhedrons. A cylinder has two parallel, congruent bases that are circles. A cone has one circular base and a vertex that is not on the base. A sphere is a space figure having all its points an equal distance from the center point. Note that cylinders, cones, and spheres are not polyhedrons, because they have curved, not flat, surfaces.
  9. 9. Free powerpoint template:www.brainybetty.com 9 EXAMPLES OF POLYHEDRONS
  10. 10. The Beauty ofMathematics www.brainybetty.com Free powerpoint template:Shapes of NumbersHere is an interesting andlovely way to look at thebeauty of mathematics, and ofGod, the sum of all wonders. 10 Wonderful World
  11. 11. Trapezium Definition: A trapezium is a shape withfour sides, that has one set www.brainybetty.com Free powerpoint template:of parallel sides. 11
  12. 12. 1x8+1=9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 www.brainybetty.com Free powerpoint template: 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432123456789 x 8 + 9 = 987654321 12
  13. 13. 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 www.brainybetty.com Free powerpoint template: 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111123456789 x 9 +10= 1111111111 13
  14. 14. 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 www.brainybetty.com Free powerpoint template: 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 8888888898765432 x 9 + 0 = 888888888 Brilliant, isn’t it? 14
  15. 15.  0x9+8 =8 9 x 9 + 7 = 88 98 x 9 + 6 = 888 www.brainybetty.com Free powerpoint template: 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888 987654321 x 9 - 1 = 8888888888 9876543210 x 9 - 2 = 88888888888 15
  16. 16. And look at this symmetry: 1x1=1 11 x 11 = 121 111 x 111 = 12321 www.brainybetty.com Free powerpoint template: 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 123456765432111111111 x 11111111 = 123456787654321 111111111 x 111111111 = 12345678987654321 16
  17. 17. TAKE A LOOK AT THIS SYMMETRY:RIGHT ANGLED TRIANGLE 1x1=1 11 x 11 = 121 111 x 111 = 12321 www.brainybetty.com Free powerpoint template: 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111 = 12345678987654321 17
  18. 18. Free powerpoint template: www.brainybetty.com 18PLANE MENSURATION
  19. 19. AREA OF CLOSED FIGURES The measure of region enclosed in a closed figure is called Area. www.brainybetty.com Free powerpoint template: For example, we need to find the areas of Polygons, Circles, or any other closed figures. http://www.helpingwithmath.com/by_subject/geomet ry/geo_area.htm 19
  20. 20. Free powerpoint template: www.brainybetty.com 20TANGRAMS
  21. 21. CONCEPT OF AREA AND ITS USES IN DAILYLIFE SITUATIONS There are two vacant rooms in a school numbered as 3 and 4. Students of class-VI are to take their mid-day meals. The monitor was asked to make arrangements www.brainybetty.com Free powerpoint template: for the students of his class in room number 3. He said “Students of our class cannot be accommodated in Room No.3, but they can be accommodated in Room No. 4”. What could be the difference between the two rooms so that one of these can accommodated the students of Class VI whereas the other one cannot accommodate it? The general answer will be Room No. 4 is bigger than Room No. 3. It is because the region enclosed within Room No. 4 is greater than Room No. 3. Thus, the measure of region enclosed in a closed figure is called Area. http://www.helpingwithmath.com/by_subject/geometry/g eo_area.htm 21
  22. 22. UNIT OF MEASUREMENT OF AREA To measure anything we first fix a unit to be used to measure it such as: www.brainybetty.com Free powerpoint template: To measure a length, you use a meter as a unit. 2 2 Area is measured in square unit like 3 m or 5 cm . TABLE FOR AREA NAME AREA Rectangle Length*breadth = L W Square Side*Side =s 2 Triangle 1/2 Base*Altitude = ½ b h Parallelogram Base*Altitude =bh Trapezium 1/2*h (Sum of two parallel side) = ½ h (a + b) 22
  23. 23. TRIANGLES A triangle has three sides and three angles. www.brainybetty.com Free powerpoint template: The three angles always add to 180° There are three special names given to triangles that tell how many sides (or angles) are equal. There can be 3, 2 or no equal sides/angles: Equilateral Triangle has all 3 sides equal Isosceles Triangle has 2 sides equal Scalene Triangle has No sides equal 23
  24. 24. EQUILATERAL, ISOSCELES AND SCALENE Equilateral Triangle www.brainybetty.com Free powerpoint template: Three equal sides Three equal angles, always 60° Isosceles Triangle Two equal sides Two equal angles Scalene Triangle No equal sides 24 No equal angles
  25. 25. WHAT TYPE OF TRIANGLE?Triangles can also have names thattell you what type of angle is inside: www.brainybetty.com Free powerpoint template: Acute Triangle All angles are less than 90° Right Triangle Has a right angle (90°) Obtuse Triangle Has an angle more than 90° 25http://home.avvanta.com/~math/triangles.html
  26. 26. EXAMPLE: TO FIND AREA OF A TRIANGLE To find the area of the Triangle we should know the base and altitude(height) www.brainybetty.com Free powerpoint template: Area of the triangle=1/2*base*altitudeExp:1 Find the area of the triangle which base is 6 m and altitude is 8 m? Solution: Here, Base of the triangle= 6 m Altitude of the triangle = 8 m Area of the Triangle= 1/2*base*altitude = 1/2*6*8 = 24 m2 26
  27. 27. AREA OF A TRIANGLE ½ Base x Height = Area (It’s ½ because ½ of the “square” is missing) Height 5 Base 8 Height ½ Base x Height = Area ½ (8) x 5 = Area 4 x 5 = 20 Base Free powerpoint template: www.brainybetty.com 27
  28. 28. AREA OF PARALLELOGRAM Area=Base x Height = bh www.brainybetty.com Free powerpoint template: (Area=length x width) The diagonal line is NOT 5 the height!!! Height (width) 8 BASE (length) 28 Base 8 x Height 5 = Area 40
  29. 29. Free powerpoint template: www.brainybetty.com 29TO FIND AREA OF A TRAPEZIUM Trapezium: 
  30. 30. Trapezium Definition: A trapezium is a shape with four sides,that has one set of parallel sides.Trapezium/Trapezoid Formula : www.brainybetty.com Free powerpoint template:Area of Trapezium = ½ (a + b)hwhere a, b = sides, h = height 30
  31. 31. Free powerpoint template:www.brainybetty.com 31
  32. 32. EXAMPLE: TO FIND AREA OF A TRAPEZIUM To find the area of the Trapezium, we should know the altitude(height)and sides www.brainybetty.com Free powerpoint template: Ques: Find the area of the Trapezium which height is 4 m and sides are 10 m and 12 m? Solution: Here, height of the Trapezium=h= 4 m Side of one Trapezium= a = 10 m Side of another Trapezium= b = 12 m Area of the Triangle = 1/2*h(a + b) =1/2*4(10+12) = 44 m2 32
  33. 33. DIFFERENT NAMES/SAME IDEA Length x Width = Area Side x Side = Area Base x Height = Area Free powerpoint template: www.brainybetty.com 33
  34. 34. QUESTION: FIND THE AREA OF THE GIVENSCALENE TRIANGLE www.brainybetty.com Free powerpoint template: Example: What is the area of this triangle? Height = h = 12 Base = b = 20 Area = ½ b h = ½ 20 12 = 120 Note: The base can be any side, Just be sure the "height" is measured at right angles to the "base“. 34
  35. 35.  Can you find the Areas of these closed figures? www.brainybetty.com Free powerpoint template: 35
  36. 36. Free powerpoint template:www.brainybetty.com 36
  37. 37. Free powerpoint template: www.brainybetty.com 37TYPES OF TRIANGLES
  38. 38. Free powerpoint template:www.brainybetty.com 38
  39. 39. EULER DIAGRAM OF QUADRILATERAL TYPES Kite Quadrilateral 39
  40. 40. NAME THE SEVEN QUADRILATERALS AND FINDTHEIR AREAS. www.brainybetty.com Free powerpoint template: 40
  41. 41. QUICK CHECK: i. The sum of all angles in a triangle is ___________ degrees. www.brainybetty.com Free powerpoint template: ii. The area of a triangle is __________________________ iii. The area of a trapezium is ______________________ iv. _____________ triangle has two sides equal. 41
  42. 42. QUICK CHECK:o v. _______________triangle has one 90 degree angleo vi. _______________ triangle has all angles and sides are the same. vii. ________________ triangle: Has all three angles and all three sides different. viii. The interior angles of a simple quadrilateral ABCD add up to 42 ______ degrees.

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