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)

SHAPES IN 2-D AND 3-D

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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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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 ______ .