plane mensuration perimeter of polygons
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Maths for Interior Design-Plane Mensuration-Perimeter- by Dr. Farhana ShaheenTRANSCRIPT
PLANE MENSURATION- II PERIMETERMATH-103
Mathematics for Interior Design
Dr. Farhana Shaheen
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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.
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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)
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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.
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PLANE MENSURATION
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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 yard or garden.
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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/geometry/geo_area.htm
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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.
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TANGRAMS
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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)
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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/perimeter.htm#rhombus
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Trapezium/Trapezoid Formula :
Area of Trapezium = ½×(a + b)×hwhere a, b = sides, h = height
Perimeter of Trapezium = a + b + c + dwhere a, b, c, d = sides of the trapezium.
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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:
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1 x 8 + 1 = 912 x 8 + 2 = 98
123 x 8 + 3 = 9871234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 987654312345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
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1 x 9 + 2 = 1112 x 9 + 3 = 111
123 x 9 + 4 = 11111234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 1111111112345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111
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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).
PERIMETER
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”.
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PERIMETER
The distance around the outside of a rectangle.
PERIMETER
The distance around the outside of a shape.
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SQUARE/RECTANGLE FORMULA
Perimeter=2(Length+Width)P= 2(25+14)
P=2(20+20)P= 50+28 P=40+40P= 78 P=80
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2520
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OTHER SHAPES
Just add up EACH segment
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8 sides, each side 10 so 10+10+10+10+10+10+10+10=80
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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.
PERIMETERThe distance around the outside of a rectangle.
length
length
width width
Perimeter = (2 x L) + (2 x W)
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RIYAD BANK-YANBU AL SINAIYAH
How many shapes do you see in this picture?
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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
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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 = 4 s 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) Trapezium Sum of all sides
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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
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TYPES OF TRIANGLES
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TYPES OF TRIANGLES
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EQUILATERAL, ISOSCELES AND SCALENE
Equilateral Triangle
Three equal sides
Three equal angles,
always 60°
Isosceles TriangleTwo equal sides Two equal angles
Scalene TriangleNo equal sides No equal angles
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WHAT TYPE OF TRIANGLES?
Acute TriangleAll angles are less than 90°
Right TriangleHas a right angle (90°)
Obtuse TriangleHas an angle more than 90°
Triangles can also have names that tell you what type of angle is inside:
http://home.avvanta.com/~math/triangles.html
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SSSSSSSSSSSSSSSSS
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EULER DIAGRAM OF QUADRILATERAL TYPES
Kite Quadrilateral
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NAME THE SEVEN QUADRILATERALS AND FIND THEIR PERIMETERS
AREA WORD PROBLEMS
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EXAMPLE 1: FIND THE AREA AND PERIMETER OF THE GIVEN SCALENE TRIANGLE
Example: 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 be sure the "height" is measured at right angles to the "base“.
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QUESTION 1:
Find the area of the given trapezium.
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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.
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QUESTION 3:
Find area and perimeter of the shaded region.
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 IS200CM, 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?
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Can you find the perimeters of these closed figures?
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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.
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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 ______ .