2. WHAT is quadrilateral ?
A quadrilateral is a four-sided polygon with four angles.
The sum of interior angle is 360°
Area of quadrilateral can be found by dividing it into triangles
3. TYPE of quadrilateral
Square Rectangle Rhombus Parallelogram
Kite Trapezoid Cyclic Quadrilateral Irregular Quadrilateral
4. SQUARE
All side are equal in length.
Four right angles that is 90°
Same diagonal length and perpendicular to each other.
4 symmetrical lines.
Area = a²
Perimeter = 4a
a
a
a
a
5. RECTANGLE
Opposite sides are equal in length
Four right angles that is 90°
Same diagonal length and perpendicular to each other.
2 symmetrical lines
Area = a x b
Perimeter = 2a + 2b
a
bb
a
6. RHOMBUS
All side are equal in length.
Opposite sides are parallel
same opposite angle
diagonals are unequal, bisect and perpendicular
to each other
Area = base x height
= side x height
= a x h
Perimeter = 4a
a
a
a
a h
7. EXAMPLE
17m
If the area of rhombus ABCD is 255m², what is the value of h?
h m
Solution: Area = base x height
= BC x h
= 17m x h255m²
h = 255m² / 17m
= 15m
8. AREA OF RHOMBUS
When one side, ‘a’ and an interior angle, ‘θ’ are given:
Diagonal AC divides the rhombus into two equal triangles,
therefore area of the rhombus is given as:
Area of rhombus = 2 x area of ∆ ADC
= 2 x 1/2 ( a x a x sin θ )
= a² sin θ
9. EXAMPLE
When a=17m and θ=60°, what is the area of rhombus ABCD ?
a=17m
60°
Solution: Area = 2 x area of ∆ ADC
= 2 x 1/2 (17 x 17 x sin 60°)
= 250 m²
10. AREA OF RHOMBUS
Diagonals AC and BD divide the rhombus into four equal triangles,
therefore area of rhombus is given as:
Area of rhombus = 4 x 1/2 x BC/2 x AD/2
= 1/2 ( AD x BC )
= 1/2 ( e x eᴺ )
When length of two diagonals are given:
11. EXAMPLE
= 4 x 1/2 x BC/2 x AD/2
= 1/2 ( AD x BC )
= 1/2 ( 30 x 30 )
= 450 m²
When AD=BC=30m, what is the area of rhombus ABCD ?
Solution: Area
12. PARALLELOGRAM
Opposite sides are equal in length and parallel
Diagonals are unequal and bisect each other
Same opposite angle
Area = base x height
= b x h
Perimeter = 2a+2b
a
b
h
13. AREA OF PARALLELOGRAM
FORMULAE
Area = base x height
= b x h
HOW TO FIND h ?
from Theorem hypotenuse,
𝑠𝑖𝑛𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒
=
ℎ
𝐴𝐷
Thus,
h= 𝑠𝑖𝑛𝜃 x AD
14. EXAMPLE
FORMULAE
Area = base x height
= b x h
If the area of parallelogram EFGH is 112m² , what must the value of h be?
Solution: Area = base x height
= HG x h
112m² = 16m x h
h =
112𝑚²
16𝑚
= 7
h
15. KITE
two pairs of equal length sides that are adjacent to each other
Diagonals are perpendicular to each other
Same opposite angle
Area =
1
2
x KM x JL
Perimeter = 2a+2b
Actually the area of kite is half of the area of rectangle.
a
ba
b
16. EXAMPLE 1
FORMULAE
Area = base x height
= b x h
What is its area of the following kite?
Solution: Area =
1
2
x sum of two diagonal of the kite
=
1
2
x ( 6m + 13m )
= 7.5m²
17. EXAMPLE 2
FORMULAE
Area = base x height
= b x h
If the area of rectangle is 56cm².What is its area of the kite in the
rectangle?
Solution: Area =
1
2
x area of rectangle
=
1
2
x 56cm²
= 28cm²
18. TRAPEZOID / TRAPEZIUM
Two opposite parallel sides of different length
The other two sides unparalleled and either same or different length
Area =
𝐴𝐵+𝐶𝐷
2
x height
Perimeter = AB + BC + CD + DA
19. EXAMPLE
What is the area of trapezium ABCD ?
Solution:
A B
CD
B’
15m
24m
50°
h m
= 24m - 15m
= 9m
B’C
tan 50° = h/9
= 11mh
Area = (15 + 24) / 2 x 11
= 214.5 m²
20. CYCLIC QUADRILATERAL
Quadrilateral which inscribed in a circle.
a + b = 180° / c + d = 180°
Area = (S – A) (S – B) (S – C) (S – D)
where S = half perimeter of quadrilateral.A
B
C
D
The converse of this theorem too can be used as a
theorem. Hence if the exterior angle formed by
producing a side of a quadrilateral is equal to the
interior opposite angle, then the quadrilateral
is a cyclic quadrilateral.
∠a = ∠a’
a
a’
21. EXAMPLE
A
B
C
D
Given the sides of the quadrilateral are A=43m , B=50m , C=45m , D=38m.
Solution: Perimeter = 1/2 (43+50+45+38)m
= 88 m
Area
= (88-43) (88-50) (88-45) (88-38)
= (S-A) (S-B) (S-C) (S-D)
= 1917 m²
22. IRREGULAR QUADRILATERAL
Any quadrilateral which does not fit into any of the above.
Area can be found by dividing it into several of triangular.
23. EXAMPLE 1
In a quadrilateral the diagonal is 42cm and the two perpendiculars on it
from the other vertices are 8cm and 9cm respectively. Find the area of
the quadrilateral.
Solution: Area of ABCD = Area of ∆ABC + Area of ∆ACD
=
1
2
x 9cm x 42cm +
1
2
x 8cm x 42cm
= 189cm² + 168cm²
= 357cm²
24. EXAMPLE 2
Find the area of a quadrilateral whose sides are 9 m, 40 m, 28 m and
15 m respectively and the angle between first two sides is a right angle.
Solution:
Area of ∆PQR =
1
2
x base x height
=
1
2
x 9 x 40
= 180 m²
PR = √PQ² + QR²
= √9² + 40²
= √1681
= 41 m
s =
15+28+41
2
= 42 m
Area of ∆PSR = √s(s-a)(s-b)(s-c)
= √42(42-15)(42-28)(42-41)
= 126 m²
Area of PQRS = 180 m² + 126 m²
= 306 m²
28. REGULAR PRISMS
Cross- Section
It is a prism that has a regular Cross Section, with equal edge lengths
and equal angles.
Cross- SectionSquare Prism Triangular Prism
Cross- SectionCross- Section Pentagonal PrismCube
29. IRREGULAR PRISMS
It is a prism that has an irregular Cross Section, with different edges
length and angles.
Irregular Pentagon Prism Cross- Section
30. RIGHT AND OBLIQUE PRISM
RIGHT PRISM OBLIQUE PRISM
A prism in which the joining edges
and faces are perpendicular to
the base faces
A prism with bases that are not
aligned one directly above the
other.
H
H
31. SURFACE AREA OF
RIGHT PRISM
SURFACE AREA = 2b + ph
b = area of base
p = perimeter of base
h = height of the prism
b = ½ (4) (6+12)
= 36 cm²
P = 6+5+12+5
=28 cm
h = 10 cm
Total Surface Area
= 2(36) + 28(10)
= 352 cm²
Example 1:
32. Example 2:
12 CM
10 CM
6 CM
AREA OF HEXAGON = 3√3
¯¯¯2
× x²
b = 3√3
¯¯¯
= 127.31 CM²
p = 2 (6 x 7) + 6 (7)
= 126 CM
h =12 CM
2
× 7² TOTAL SURFACE AREA
= 2b + ph
= 2(127.31) + (126)(12)
= 1766.62 CM²
b = 1
¯
= 15 CM²
p = 4(7) + 2(6) + 3(14)
= 82 CM
h = 14 CM
× 3 × 10
2
TOTAL SURFACE AREA
= 2b + ph
= 2(15) + 82(14)
= 1928 CM²
x = length of sides
Example 3;
33. SURFACE AREA OF
OBLIQUE PRISM
SURFACE AREA
= pl+2b
p = Perimeter
l = Lateral Edge
b = Area of Base
5 CM
p = 4(8 cm) + 4(5 cm) + 4(15 cm)
= 112 CM
l = 15 CM
b = 8 CM x 5 CM
= 40 CM²
Total Surface Area = 112(15) + 2(40)
= 1760 CM²
34. VOLUME
Volume = b x h
b = area of base
h = height
Example 1:
I) VOLUME OF A TRIANGULAR PRISM
Base Area = (8)(3)/2
= 12cm 2
Volume = 12 x 12
= 144cm 3
Given: b = 8cm, height = 3cm and length = 12cm
3 cm
8 cm
Base area of prism = (b x h)/2`
35. VOLUME
Example 1:
II) VOLUME OF THE OBLIQUE PRISM
Volume of the oblique prism
= [ ½ x ( 8+4 ) x 9 ] x 15
= 54 x 15
= 810 cm 2
Volume = B x h
B = area of base
h = height of prism
36. VOLUME
Example 1:
III) VOLUME OF TRAPEZOID PRISM
Area of Trapezium Base = ½ x ( a + b) x h
= ½ x (1.5 x 6.5 ) x 4.2
= 16.8 cm 2
Volume = Area x Height between trazepium ends
= 16.8 x 8
= 134.4 cm3
= 134 cm3
Volume = Area x Height between trapezium ends
37. VOLUME
IV) VOLUME OF PENTAGONAL PRISM
Example 1:
Where ,
a = apothem length
b = side
h = height
V = [
1
2
x 5 x side x apothem] x height of the prism.
V= Ah
Area of Base(A) h
6cm
7cm
10cm
A = ½ ( 5 x 6 x 7 )
= 105 cm 2
V = A x h
= 105 cm 2 x 10 cm
= 1050 cm 3
38. VOLUME
V) VOLUME OF THE CUBE PRISM
Volume = s 3
Example 1:
Volume = s 3
= (3cm) 3
= 27 cm3
Volume units are always cubed.
39. VOLUME
VI) VOLUME OF THE SQUARE PRISM
Volume = s 2 h
V= Area of base × Height of prism
s 2
h
Example 1:
3cm
Volume = s 2 x h
= (3cm) 2 x 5 cm
= 45 cm3
40. VOLUME – EXAMPLE 1
The diagram shows a cross-section of a cuboid after a cube is cut
out from it.
Area of cross-section
= (7x12) – (3x4)
= 84 – 12
= 72m 2
Volume of prism
= 72 x 5
= 360 m3
What is the volume of this prism ?
Solution :
42. What is FRUSTUM of Prism ?
Plane section is taken of a right prism parallel to its end.
Section is known as a CROSS-SECTION of the prism and the two positions of
the prism are still prisms.
43. Difference Between
Cross-Section and Frustum
Prism Differences Frustum Of Prism
Parallel Cutting Plane Not parallel to the ends
2 X base area +
( Base X Height )
* This formula works for all
prism regardless of base
shape
Surface Area Area of the base + Area of
the section + Lateral
surface area
Area of cross section X
Length
Volume Average height X Area of
the base
44. Perpendicular Cut
Cross Section
Non-parallel cut
Frustum
In figure, ABCEFGHI represents a
frustum of a prism whose cutting
plane EFGH is inclined at angle θ
to the horizontal. In this case,
the frustum can be taken as a
prism with base ABEF and
height BC.
FRUSTUM of Prism
45. Volume of Frustum of
Prism
In figure ABCEFGHI represents a frustum of a prism whose cutting plane EFGH
is inclined at an angle θ to the horizontal. In this case, the frustum can be
taken as a prism with base ABEF and height BC.
Volume of the frustum =area ABEF × BC
ABEF is a trapezium whose area is AF+BE X AB
Volume of Frustum is AF+BE X AB X BC
=
In conclusion, Volume of frustum = Average height X Area of Base
2
2
H1+h2 X AB X BC
2
46. What is Total Surface
Area ?
Total Surface Area = Area of the base + Area of the section + Lateral Surface Area
Do you know what is
Lateral Surface Area
?????
47. Lateral Surface Area
Lateral Surface Area of the Frustum is the combination of rectangle and
trapeziums whose area can be calculated separately
} Lateral
Surface
Area
Shaded parts
are the
48. Lateral Surface Area
If the cutting plane is inclined at an angle θ to the horizontal then from figure
we have
𝐴𝐵
𝐸𝐹
= COS θ
OR
𝐴𝐵
𝐸𝐹
x
𝐵𝐶
𝐹𝐺
= COS θ
( as BC = FG )
OR
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝐸𝐹𝐺𝐻
= COS θ
OR Area of section EFGH =
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒
𝐶𝑂𝑆 𝜃
49. Total Surface Area
Total Surface Area = Area of the base + Area of the section + Lateral Surface Area
Total Surface Area
= AB.BC +
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵𝑎𝑠𝑒
cos θ
+ Area of Trapezium
and Rectangle
51. EXAMPLES
A hexagonal right prism, whose base is inscribed in a circle of radius 2m, is cut by
a plane inclined at an angle Find the volume of the frustum and the area of the
section when the heighof 45∘ to the horizontal. ts of the frustum are 8m and 6m
respectively.
53. THE END
THANK YOU
Presented by
Goh Xingxin 0325587
Na Yong Yi 0324458
Tan Kai Xuan 0325066
Tan Chin Werng 0324408
Lee Jia Min 0324126
Tee Wan Nee 0325074
Yap Foong Mei 0324867
