Quadrilateral

QUADRILATERAL

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

TYPE of quadrilateral Square Rectangle Rhombus

Parallelogram

Kite Trapezoid Cyclic Quadrilateral Irregular Quadrilateral

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

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

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

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 h255m

² h = 255m² / 17m= 15m

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

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

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:

e

eᴺ

EXAMPLE

= 4 x 1/2 x BC/2 x AD/2= 1/2 ( AD x BC )= 1/2 ( 30 x 30 )= 450 m²

30 m

30 m

When AD=BC=30m, what is the area of rhombus ABCD ?

Solution: Area

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

AREA OF PARALLELOGRAM

FORMULAE

Area = base x height = b x h

HOW TO FIND h ?from Theorem hypotenuse, Thus, h= x AD

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 hh =

= 7

h

KITE

two pairs of equal length sides that are adjacent to each other Diagonals are perpendicular to each other Same opposite angle Area = x KM x JL Perimeter = 2a+2b Actually the area of kite is half of the area of rectangle.

a

ba

b

EXAMPLE 1


What is its area of the following kite?

Solution: Area = x sum of two diagonal of the kite= x ( 6m + 13m )= 7.5m²

EXAMPLE 2


If the area of rectangle is 56cm².What is its area of the kite in the rectangle?

Solution: Area = x area of rectangle= x 56cm²= 28cm²

TRAPEZOID / TRAPEZIUM

Two opposite parallel sides of different length The other two sides unparalleled and either same or different

length Area = x height Perimeter = AB + BC + CD + DA

EXAMPLEWhat 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= 11m h

Area = (15 + 24) / 2 x 11

= 214.5 m²

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’

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

IRREGULAR QUADRILATERAL

Any quadrilateral which does not fit into any of the above. Area can be found by dividing it into several of triangular.

EXAMPLE 1In a quadrilateral the diagonal is 42cm and the two perpendiculars on it from the other vertices are 8cm and 9cm respectively. Find the area ofthe quadrilateral.

Solution: Area of ABCD = Area of ∆ABC + Area of ∆ACD= x 9cm x 42cm + x 8cm x 42cm= 189cm² + 168cm²= 357cm²

EXAMPLE 2Find 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 = x base x height

= x 9 x 40= 180 m²

PR = √PQ² + QR²= √9² + 40²= √1681= 41 m

s = = 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

PRISM

TYPE OF PRISM

TrianglePrism

Square Prism

PentagonalPrism

Hexagonal Prism

Rectangular

Prism

ObliquePrism

All prism is a polyhedron, which means all faces are flat.

PRISMS NETSDifferent types of nets of prism

REGULAR PRISMS

Cross- Section

It is a prism that has a regular Cross Section, with equal edge lengths and equal angles.

Cross- Section Square Prism Triangular Prism

Cross- Section Cross- Section Pentagonal Prism Cube

IRREGULAR PRISMSIt is a prism that has an irregular Cross Section, with different edges length and angles.

Irregular Pentagon Prism

Cross- Section

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

Late

ral e

dge

H

SURFACE AREA OF RIGHT PRISM

SURFACE AREA = 2b + ph

b = area of basep = perimeter of baseh = 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:

Example 2: 7 CM

12 CM

10 CM

14 CM6 CM

AREA OF HEXAGON = 3√3 ¯¯¯ 2

× x²

b = 3√3 ¯¯¯ = 127.31 CM²p = 2 (6 x 7) + 6 (7) = 126 CMh =12 CM

2 × 7² TOTAL SURFACE AREA= 2b + ph= 2(127.31) + (126)(12)= 1766.62 CM²

b = 1 ¯ = 15 CM²p = 4(7) + 2(6) + 3(14) = 82 CMh = 14 CM

7 CM

× 3 × 102TOTAL SURFACE AREA= 2b + ph= 2(15) + 82(14)= 1928 CM²

x = length of sides

Example 3;

SURFACE AREA OF OBLIQUE PRISM

SURFACE AREA = pl+2b

p = Perimeterl = Lateral Edge b = Area of Base

8 CM5 CM

15 C

M

p = 4(8 cm) + 4(5 cm) + 4(15 cm) = 112 CMl = 15 CMb = 8 CM x 5 CM = 40 CM²

Total Surface Area = 112(15) + 2(40) = 1760 CM²

VOLUME

Volume = b x h b = area of baseh = heightExample 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

12cm

Base area of prism = (b x h)/2`

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 baseh = height of prism

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

VOLUMEIV) VOLUME OF PENTAGONAL PRISM

Example 1:

Where , a = apothem lengthb = sideh = height

V = [ 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

VOLUMEV) VOLUME OF THE CUBE PRISM

Volume = s 3

Example 1:

Volume = s 3

= (3cm) 3

= 27 cm3 Volume units are always

cubed.

VOLUMEVI) VOLUME OF THE SQUARE PRISM

Volume = s 2 h

V= Area of base × Height of prism

s 2

h

Example 1:

5cm

3cm

Volume = s 2 x h = (3cm) 2 x 5 cm

= 45 cm3

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 :

FRUSTUM OF

PRISM

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.

Difference Between Cross-Section and FrustumPrism 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

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

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

2H1+h2 X AB X BC 2

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 ?????

Lateral Surface Area Lateral Surface Area of the Frustum is the combination of rectangle

and trapeziums whose area can be calculated separately

}LateralSurface

Area

Shaded parts

are the

Lateral Surface Area If the cutting plane is inclined at an angle θ to the horizontal then

from figure we have = COS θ

OR x = COS θ ( as BC = FG )

OR = COS θ

OR Area of section EFGH =

Total Surface AreaTotal Surface Area = Area of the base + Area of the section + Lateral Surface Area

Total Surface Area

= AB.BC + + Area of Trapezium and Rectangle

Let’s Do Exercise !

EXAMPLESA 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.

SOLUTION

THE END THANK YOU

Presented byGoh Xingxin 0325587Na Yong Yi 0324458Tan Kai Xuan 0325066Tan Chin Werng 0324408Lee Jia Min 0324126Tee Wan Nee 0325074Yap Foong Mei 0324867

Education

Quadrilateral