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

1. MENSURATION
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.
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4. 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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5. SHAPES IN 2-D
AND 3-D

6. 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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7. SOLID MENSURATION
 Solid mensuration deals with areas and volumes of
solid objects in three dimensions.
 For example: Polyhedrons, Cylinders, Cones,
Spheres. They are called Space Figures.
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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. EXAMPLES OF POLYHEDRONS
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10. Here is an interesting and
lovely way to look at the
beauty of mathematics, and of
God, the sum of all wonders.
The Beauty of
Mathematics
Shapes of Numbers
Wonderful World
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Trapezium Definition:
A trapezium is a shape with
four sides, that has one set
of parallel sides.
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12. 1 x 8 + 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 = 98765432112
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13. 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 13
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14. 9 x 9 + 7 = 88
98 x 9 + 6 = 888
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
Brilliant, isn’t it? 14
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15.  0 x 9 + 8 = 8
9 x 9 + 7 = 88
98 x 9 + 6 = 888
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
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16. 1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 =
12345678987654321
And look at this symmetry:
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17. TAKE A LOOK AT THIS SYMMETRY:
RIGHT ANGLED TRIANGLE
 1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
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18. PLANE MENSURATION
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19. AREA OF CLOSED FIGURES
 The measure of region enclosed in a closed figure
is called Area.
 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
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20. TANGRAMS
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21. CONCEPT OF AREA AND ITS USES IN DAILY
LIFE 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
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
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22. UNIT OF MEASUREMENT OF AREA
 To measure anything we first fix a unit to be used to
measure it such as:
 To measure a length, you use a meter as a unit.
 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
 Triangle 1/2 Base*Altitude = ½ b h
 Parallelogram Base*Altitude = b h
 Trapezium 1/2*h (Sum of two parallel side) = ½ h (a + b)
2 2
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23. 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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24. 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
No equal angles
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25. WHAT TYPE OF TRIANGLE?
Acute Triangle
All angles are less
than 90°
Right Triangle
Has a right angle (90°)
Obtuse Triangle
Has an angle more than 90°
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Triangles can also have names that
tell you what type of angle is inside:
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http://home.avvanta.com/~math/triangles.html

26. EXAMPLE: TO FIND AREA OF A TRIANGLE
 To find the area of the Triangle we should know the
base and altitude(height)
 Area of the triangle=1/2*base*altitude
Exp: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
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AREA OF A TRIANGLE
 ½ Base x Height = Area
 (It’s ½ because ½ of the “square” is missing)
Base
Height
Height
Base
8
5
½ Base x Height = Area
½ (8) x 5 = Area
4 x 5 = 20

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AREA OF PARALLELOGRAM
 Area=Base x Height = bh
 (Area=length x width)
BASE (length)
Height (width)
8
5
Base 8 x Height 5 = Area 40
The diagonal line is NOT
the height!!!
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29. TO FIND AREA OF A TRAPEZIUM
 Trapezium:
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Trapezium Definition:
A trapezium is a shape with four sides,
that has one set of parallel sides.
Trapezium/Trapezoid Formula :
Area of Trapezium = ½ (a + b)h
where
a, b = sides, h = height
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32. EXAMPLE: TO FIND AREA OF A
TRAPEZIUM
 To find the area of the Trapezium, we should know
the altitude(height)and sides
 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
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DIFFERENT NAMES/SAME IDEA
 Length x Width = Area
 Side x Side = Area
 Base x Height = Area

34. QUESTION: FIND THE AREA OF THE GIVEN
SCALENE TRIANGLE
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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“.

35.  Can you find the Areas of these closed figures?
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37. TYPES OF TRIANGLES
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39. EULER DIAGRAM OF QUADRILATERAL TYPES
Kite Quadrilateral
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40. NAME THE SEVEN QUADRILATERALS AND FIND
THEIR AREAS.
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41. QUICK CHECK:
 i. The sum of all angles in a triangle is
___________ degrees.

 ii. The area of a triangle is
__________________________

 iii. The area of a trapezium is
______________________

 iv. _____________ triangle has two sides equal.
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42. QUICK CHECK:
o v. _______________triangle has one 90 degree
angle
o 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
______ degrees.
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