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PLANE MENSURATION- II
PERIMETER
MATH-103
Mathematics for Interior Design
Dr. Farhana Shaheen
MENSURATION
 Mensuration is to measure the quantities such
  as perimeter, area, volume, length of a closed
  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
DEFINITION: MENSURATION
 Mensuration is a branch of Mathematics which
  deals with the measurements of lengths of lines,
  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)


                                                     3
SHAPES IN 2-D
AND 3-D
PLANE MENSURATION
 Plane Mensuration deals with perimeter, length of
  sides and areas of two dimensional figures and
  shapes.
 For example,

 Circle, Semi-circle

 Rectangle

 Pentagon

 Semi-circle

 Triangles,

 Trapezium, etc.
                                                      5
PLANE MENSURATION




                    6
PERIMETER
 The measure of region enclosed in a closed figure
  (inside the figure) is called Area.
 A Perimeter is a path that surrounds an area. The
  word comes from the Greek peri (around) and
  meter (measure).
 The term may be used either for the path or its
  length - it can be thought of as the length of the
  outline of a shape.
 Calculating the perimeter has considerable practical
  applications. The perimeter can be used to
  calculate the length of fence required to surround a
                                                         7
  yard or garden.
PERIMETER OF CLOSED FIGURES
   For example, we need to find the perimeter of
    Polygons, Circles, or any other closed figures.




   http://www.helpingwithmath.com/by_subject/geomet
    ry/geo_area.htm
                                                       8
TANGRAMS
   The tangram (Chinese: "seven boards of skill") is a
    dissection puzzle consisting of seven flat shapes, called
    tans, which are put together to form shapes. The
    objective of the puzzle is to form a specific shape
    (given only an outline or silhouette) using all seven
    pieces, which may not overlap. It was originally
    invented in China.
   It is one of the most popular dissection puzzles in the
    world.
   A dissection puzzle, also called a transformation
    puzzle or Richter Puzzle, is a tiling puzzle where a set
    of pieces can be assembled in different ways to produce
    two or more distinct
     geometric shapes.                                          9
TANGRAMS




           10
PERIMETER OF A RECTANGLE:
 The perimeter is the distance around all of a
  shape.
 Perimeter of a rectangle:

  P = w + l + w + l = 2 (l + w)




                                                  11
Trapezium:
A trapezium is a shape with four sides, that
has one set of parallel sides.
What is the perimeter of a trapezium?




    http://www.calculatoredge.com/enggcalc/perim   12
    eter.htm#rhombus
Trapezium/Trapezoid Formula :

Area of Trapezium = ½×(a + b)×h
where
    a, b = sides, h = height

Perimeter of Trapezium = a + b + c + d
where
    a, b, c, d = sides of the trapezium.




                                           13
EXAMPLE: TO FIND PERIMETER
          OF A TRAPEZIUM

   To find the perimeter of the Trapezium, we should
    know the lengths of all sides.

 Ques: Find the perimeter of the Trapezium with
  sides 4 cm, 10 cm,15 cm and 7cm ?
 Solution:




                                                        14
1x8+1=9
       12 x 8 + 2 = 98
      123 x 8 + 3 = 987
     1234 x 8 + 4 = 9876
    12345 x 8 + 5 = 98765
   123456 x 8 + 6 = 987654
  1234567 x 8 + 7 = 9876543
 12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321   15
1 x 9 + 2 = 11
       12 x 9 + 3 = 111
      123 x 9 + 4 = 1111
     1234 x 9 + 5 = 11111
    12345 x 9 + 6 = 111111
   123456 x 9 + 7 = 1111111
  1234567 x 9 + 8 = 11111111
 12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111   16
PERIMETER

   The perimeter of a particular shape is the total length of its sides.
   For a triangle: (p = a + b + c)
   The perimeter is equal to the length of side a, plus the length of
    side b, plus the length of side c.
   For a square: (p = 4s)
   The perimeter is equal to 4 times the length of a side.
   For a rectangle: (p = 2 (l + w)
   The perimeter is equal to 2 times the sum of the base plus the
    height.
   For regular polygons: ( p = n l)
   The perimeter is equal to the number of sides (n) times the length
    (l) of a side.
   Note: Circles do not have sides made of line segments like
    polygons do but they do have a perimeter known as a
    circumference.
    The circumference is equal to 2 times pi times the radius (r).          17
PERIMETER




                                                          3/13/2013
 Any shape’s “perimeter” is the outside of the
  shape…like a fence around a yard.
 To calculate the perimeter of any shape, just add up
  “each” line segment of the “fence”.




                                                         18
PERIMETER

The distance around the outside of a rectangle.
PERIMETER




                                                   3/13/2013
   The distance around the outside of a shape.




                                                  20
SQUARE/RECTANGLE FORMULA




                                        3/13/2013
 Perimeter=2(Length+Width)
 P= 2(25+14)              P=2(20+20)
 P= 50+28                 P=40+40
 P= 78                    P=80


14


                                20
        25                                21
OTHER SHAPES




                                                               3/13/2013
   Just add up EACH segment




                         10


    8 sides, each side 10 so 10+10+10+10+10+10+10+10=80




                                                          22
PERIMETER OF A TRIANGLE
Measure each side of the triangle then add
            them together.
PERIMETER
To find the perimeter you need to measure the
      length of each side of the rectangle.




           then add them together.
PERIMETER
  The distance around the outside of a rectangle.
                      length




width                                          width




                       length
            Perimeter = (2 x L) + (2 x W)
RIYAD BANK-YANBU AL SINAIYAH
   How many shapes do you see in this picture?




                                                  26
PYTHAGOREAN THEOREM:
   Pythagorean theorem, a2 + b2 = c2, so A + B = C.




   Generalization for similar triangles   Generalization for
    green area A + B = blue area C          regular pentagons   27
UNIT OF MEASUREMENT OF PERIMETER

 To measure anything we first fix a unit to be used to
  measure it such as:
 To measure a length, you use a meter, feet, inch,
  centimeter as a unit (m, ft, inch, cm).

                    TABLE FOR PERIMETER
        NAME                PERIMETER
   Rectangle           2(Length+Width) = 2(l+w)
   Square/Rhombus       4 *Side         =4s
   Triangle           (a+b+c) Sum of all sides where a, b, and c are
                        the lengths of the sides of the triangle.
   Parallelogram      p = 2(b + h)                                     28
   Trapezium         Sum of all sides
TRIANGLES
   A triangle has three sides and three angles.

   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
                                                       29
TYPES OF TRIANGLES




                     30
TYPES OF TRIANGLES




                     31
EQUILATERAL, ISOSCELES AND SCALENE

   Equilateral
    Triangle
   Three equal
      sides
   Three equal
  angles, always
       60°


                   Isosceles Triangle
                    Two equal sides
                    Two equal angles

                   Scalene Triangle
                    No equal sides      32
                    No equal angles
WHAT TYPE OF TRIANGLES?
Triangles can also have names that
tell you what type of angle is inside:


   Acute Triangle
  All angles are less
       than 90°



                                     Right Triangle
                                  Has a right angle (90°)



                                   Obtuse Triangle
                               Has an angle more than 90°
                                                            33
http://home.avvanta.com/~math/triangles.html
SSSSSSSSSSSSSSSSS




                    34
EULER DIAGRAM OF QUADRILATERAL TYPES

                          Kite Quadrilateral




                                               35
NAME THE SEVEN QUADRILATERALS AND FIND
THEIR PERIMETERS




                                         36
AREA WORD PROBLEMS
EXAMPLE 1: FIND THE AREA AND PERIMETER OF
THE GIVEN SCALENE TRIANGLE


Example: What is the area of this triangle?




Height = h = 12
Base = b = 20
Area = ½ b h = ½                              20   12 = 120
To find Perimeter=?


Note: The base can be any side, Just be
sure the "height" is measured at right
angles to the "base“.                                         38
QUESTION 1:
   Find the area of the given trapezium.




                                            39
QUESTION 2:
 Find the area of the shaded regions, if
 Length of one side of square is s = 4 cm.

 Note:

 Point A is half of s.

 Point D is mid-point

  of the square.




                                              40
QUESTION 3:
   Find area and perimeter of the shaded region.




                                                    41
PERIMETER WORD
PROBLEMS
1.   A REGULAR PENTAGON HAS ONE SIDE 25CM
     LONG.
     HOW LONG IS THE PERIMETER OF THIS SHAPE?
2. A REGULAR HEXAGON HAS A PERIMETER OF
     300CM.
     HOW LONG IS THE LENGTH OF ONE SIDE?
3.   A SQUARE FIELD HAS A PERIMETER OF
                                 1200M.
HOW LONG IS THE LENGTH OF EACH SIDE OF
                             THE FIELD?
 ONE SIDE OF THE FIELD IS HEDGED. WHAT
LENGTH OF FENCING IS NEEDED TO FENCE IN
                THE OTHER THREE SIDES?
4. WHAT IS THE PERIMETER OF A SQUARE
OF SIDE 20CM?
5. IF THE PERIMETER OF SQUARE IS
200CM, THEN WHAT IS THE LENGTH OF ONE
OF THE SIDES?
6. WHAT IS THE PERIMETER OF AN
EQUILATERAL TRIANGLE OF SIDE
8.9CM?
7. A REGULAR HEPTAGON HAS A SIDE
LENGTH OF 26CM. WHAT IS ITS
PERIMETER?
8. A RECTANGLE HAS A PERIMETER OF
68CM.
ITS SHORT SIDE IS 10CM. WHAT IS THE
LENGTH OF THE LONG SIDE?
   Can you find the perimeters of these closed
    figures?




                                                  51
QUICK CHECK:
   i. The sum of all angles in a polygon is
    ___________ .


   ii. The area of a triangle is
    __________________________


   iii. The perimeter of a trapezium is
    ______________________


   iv. _____________ triangle has two sides equal.
                                                      52
QUICK CHECK:
o   v. _______________triangle has one 90 degree
    angle

o   vi. _______________ triangle has all angles and
    sides same.

   vii. ________________ triangle: Has all three
    angles and all three sides different.

   viii. The perimeter of a regular polygon with n
    sides is ______ .                                 53

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Plane Mensuration Perimeter of Polygons

  • 1. PLANE MENSURATION- II PERIMETER MATH-103 Mathematics for Interior Design Dr. Farhana Shaheen
  • 2. MENSURATION  Mensuration is to measure the quantities such as perimeter, area, volume, length of a closed 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. DEFINITION: MENSURATION  Mensuration is a branch of Mathematics which deals with the measurements of lengths of lines, 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) 3
  • 5. PLANE MENSURATION  Plane Mensuration deals with perimeter, length of sides and areas of two dimensional figures and shapes.  For example,  Circle, Semi-circle  Rectangle  Pentagon  Semi-circle  Triangles,  Trapezium, etc. 5
  • 7. PERIMETER  The measure of region enclosed in a closed figure (inside the figure) is called Area.  A Perimeter is a path that surrounds an area. The word comes from the Greek peri (around) and meter (measure).  The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape.  Calculating the perimeter has considerable practical applications. The perimeter can be used to calculate the length of fence required to surround a 7 yard or garden.
  • 8. PERIMETER OF CLOSED FIGURES  For example, we need to find the perimeter of Polygons, Circles, or any other closed figures.  http://www.helpingwithmath.com/by_subject/geomet ry/geo_area.htm 8
  • 9. TANGRAMS  The tangram (Chinese: "seven boards of skill") is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a specific shape (given only an outline or silhouette) using all seven pieces, which may not overlap. It was originally invented in China.  It is one of the most popular dissection puzzles in the world.  A dissection puzzle, also called a transformation puzzle or Richter Puzzle, is a tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes. 9
  • 10. TANGRAMS 10
  • 11. PERIMETER OF A RECTANGLE:  The perimeter is the distance around all of a shape.  Perimeter of a rectangle: P = w + l + w + l = 2 (l + w) 11
  • 12. Trapezium: A trapezium is a shape with four sides, that has one set of parallel sides. What is the perimeter of a trapezium? http://www.calculatoredge.com/enggcalc/perim 12 eter.htm#rhombus
  • 13. Trapezium/Trapezoid Formula : Area of Trapezium = ½×(a + b)×h where a, b = sides, h = height Perimeter of Trapezium = a + b + c + d where a, b, c, d = sides of the trapezium. 13
  • 14. EXAMPLE: TO FIND PERIMETER OF A TRAPEZIUM  To find the perimeter of the Trapezium, we should know the lengths of all sides.  Ques: Find the perimeter of the Trapezium with sides 4 cm, 10 cm,15 cm and 7cm ?  Solution: 14
  • 15. 1x8+1=9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321 15
  • 16. 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 +10= 1111111111 16
  • 17. PERIMETER  The perimeter of a particular shape is the total length of its sides.  For a triangle: (p = a + b + c)  The perimeter is equal to the length of side a, plus the length of side b, plus the length of side c.  For a square: (p = 4s)  The perimeter is equal to 4 times the length of a side.  For a rectangle: (p = 2 (l + w)  The perimeter is equal to 2 times the sum of the base plus the height.  For regular polygons: ( p = n l)  The perimeter is equal to the number of sides (n) times the length (l) of a side.  Note: Circles do not have sides made of line segments like polygons do but they do have a perimeter known as a circumference. The circumference is equal to 2 times pi times the radius (r). 17
  • 18. PERIMETER 3/13/2013  Any shape’s “perimeter” is the outside of the shape…like a fence around a yard.  To calculate the perimeter of any shape, just add up “each” line segment of the “fence”. 18
  • 19. PERIMETER The distance around the outside of a rectangle.
  • 20. PERIMETER 3/13/2013  The distance around the outside of a shape. 20
  • 21. SQUARE/RECTANGLE FORMULA 3/13/2013  Perimeter=2(Length+Width)  P= 2(25+14) P=2(20+20)  P= 50+28 P=40+40  P= 78 P=80 14 20 25 21
  • 22. OTHER SHAPES 3/13/2013  Just add up EACH segment 10 8 sides, each side 10 so 10+10+10+10+10+10+10+10=80 22
  • 23. PERIMETER OF A TRIANGLE Measure each side of the triangle then add them together.
  • 24. PERIMETER To find the perimeter you need to measure the length of each side of the rectangle. then add them together.
  • 25. PERIMETER The distance around the outside of a rectangle. length width width length Perimeter = (2 x L) + (2 x W)
  • 26. RIYAD BANK-YANBU AL SINAIYAH  How many shapes do you see in this picture? 26
  • 27. PYTHAGOREAN THEOREM:  Pythagorean theorem, a2 + b2 = c2, so A + B = C.  Generalization for similar triangles Generalization for green area A + B = blue area C regular pentagons 27
  • 28. UNIT OF MEASUREMENT OF PERIMETER  To measure anything we first fix a unit to be used to measure it such as:  To measure a length, you use a meter, feet, inch, centimeter as a unit (m, ft, inch, cm). TABLE FOR PERIMETER NAME PERIMETER  Rectangle 2(Length+Width) = 2(l+w)  Square/Rhombus 4 *Side =4s  Triangle (a+b+c) Sum of all sides where a, b, and c are the lengths of the sides of the triangle.  Parallelogram p = 2(b + h) 28  Trapezium Sum of all sides
  • 29. TRIANGLES  A triangle has three sides and three angles.  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 29
  • 32. EQUILATERAL, ISOSCELES AND SCALENE Equilateral Triangle Three equal sides Three equal angles, always 60° Isosceles Triangle Two equal sides Two equal angles Scalene Triangle No equal sides 32 No equal angles
  • 33. WHAT TYPE OF TRIANGLES? Triangles can also have names that tell you what type of angle is inside: Acute Triangle All angles are less than 90° Right Triangle Has a right angle (90°) Obtuse Triangle Has an angle more than 90° 33 http://home.avvanta.com/~math/triangles.html
  • 35. EULER DIAGRAM OF QUADRILATERAL TYPES Kite Quadrilateral 35
  • 36. NAME THE SEVEN QUADRILATERALS AND FIND THEIR PERIMETERS 36
  • 38. EXAMPLE 1: FIND THE AREA AND PERIMETER OF THE GIVEN SCALENE TRIANGLE Example: What is the area of this triangle? Height = h = 12 Base = b = 20 Area = ½ b h = ½ 20 12 = 120 To find Perimeter=? Note: The base can be any side, Just be sure the "height" is measured at right angles to the "base“. 38
  • 39. QUESTION 1:  Find the area of the given trapezium. 39
  • 40. QUESTION 2:  Find the area of the shaded regions, if  Length of one side of square is s = 4 cm.  Note:  Point A is half of s.  Point D is mid-point of the square. 40
  • 41. QUESTION 3:  Find area and perimeter of the shaded region. 41
  • 43. 1. A REGULAR PENTAGON HAS ONE SIDE 25CM LONG. HOW LONG IS THE PERIMETER OF THIS SHAPE?
  • 44. 2. A REGULAR HEXAGON HAS A PERIMETER OF 300CM. HOW LONG IS THE LENGTH OF ONE SIDE?
  • 45. 3. A SQUARE FIELD HAS A PERIMETER OF 1200M. HOW LONG IS THE LENGTH OF EACH SIDE OF THE FIELD? ONE SIDE OF THE FIELD IS HEDGED. WHAT LENGTH OF FENCING IS NEEDED TO FENCE IN THE OTHER THREE SIDES?
  • 46. 4. WHAT IS THE PERIMETER OF A SQUARE OF SIDE 20CM?
  • 47. 5. IF THE PERIMETER OF SQUARE IS 200CM, THEN WHAT IS THE LENGTH OF ONE OF THE SIDES?
  • 48. 6. WHAT IS THE PERIMETER OF AN EQUILATERAL TRIANGLE OF SIDE 8.9CM?
  • 49. 7. A REGULAR HEPTAGON HAS A SIDE LENGTH OF 26CM. WHAT IS ITS PERIMETER?
  • 50. 8. A RECTANGLE HAS A PERIMETER OF 68CM. ITS SHORT SIDE IS 10CM. WHAT IS THE LENGTH OF THE LONG SIDE?
  • 51. Can you find the perimeters of these closed figures? 51
  • 52. QUICK CHECK:  i. The sum of all angles in a polygon is ___________ .   ii. The area of a triangle is __________________________   iii. The perimeter of a trapezium is ______________________   iv. _____________ triangle has two sides equal. 52
  • 53. QUICK CHECK: o v. _______________triangle has one 90 degree angle o vi. _______________ triangle has all angles and sides same.  vii. ________________ triangle: Has all three angles and all three sides different.  viii. The perimeter of a regular polygon with n sides is ______ . 53