PLANE MENSURATION- IIPERIMETERMATH-103Mathematics for Interior DesignDr. 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
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
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
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, thathas 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)×hwhere a, b = sides, h = heightPerimeter of Trapezium = a + b + c + dwhere 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 = 98765432123456789 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 = 111111111123456789 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
PERIMETERThe distance around the outside of a rectangle.
PERIMETER 3/13/2013 The distance around the outside of a shape. 20
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 TRIANGLEMeasure each side of the triangle then add them together.
PERIMETERTo 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. lengthwidth 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
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 thattell 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° 33http://home.avvanta.com/~math/triangles.html
EXAMPLE 1: FIND THE AREA AND PERIMETER OFTHE GIVEN SCALENE TRIANGLEExample: What is the area of this triangle?Height = h = 12Base = b = 20Area = ½ b h = ½ 20 12 = 120To find Perimeter=?Note: The base can be any side, Just besure the "height" is measured at rightangles 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
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. WHATLENGTH OF FENCING IS NEEDED TO FENCE IN THE OTHER THREE SIDES?
4. WHAT IS THE PERIMETER OF A SQUAREOF SIDE 20CM?
5. IF THE PERIMETER OF SQUARE IS200CM, THEN WHAT IS THE LENGTH OF ONEOF THE SIDES?
6. WHAT IS THE PERIMETER OF ANEQUILATERAL TRIANGLE OF SIDE8.9CM?
7. A REGULAR HEPTAGON HAS A SIDELENGTH OF 26CM. WHAT IS ITSPERIMETER?
8. A RECTANGLE HAS A PERIMETER OF68CM.ITS SHORT SIDE IS 10CM. WHAT IS THELENGTH 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 angleo 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