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AREAS AND VOLUMES




AREAS

CASE

1.Right angled triangle Area = ½ b x h
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


2. Triangle with altitude that lies within the triangle



EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Area 0f Δ ABC = ½ bh + ½ d h

= ½ h(b + d)

= ½ hL


3. A triangle where the altitude of triangle lies outside of the triangle.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Area of ABC = area of triangle BCD – area of triangle ABD

= ½ h (b+d) – ½ h d

= ½ h e – ½ h d

= ½ h (e-d)

In all the triangle the formula is the same.

Thus if you were given a triangle with a base b and its corresponding height (altitude) h, its area is equal to ½ b h


CASE II
We can also use the knowledge of trigonometrical ratios.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Area of triangle ABC = ½ b h

= ½ b a sin c

Sin A = h/c

h= c sin A

the area of triangle ABC = ½ b c sin A
Example
1. The length of two sides of a triangle are 8cm and 10 cm. find he area of the triangle, if the included angle is 30 0.


Solution;

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Area = ½ x 10 x 8 x sin 300

= 40 x ½Cm2

=20cm2

2. The area of triangle ABC with sides a,b,c.


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Area of triangle ABC

= ½ c b sin A

= ½ a c sin B

= ½ a b sin C
Example
The base of triangle PQR is 17 cm long. If corresponding height is 20cm, find the area of the triangle.

Solution;


Area of triangle PQR = ½ b h

= ½ x 17 x 20

= 170 cm2
Qn. 9
what is the area of
the paper required to make the kite shown in the figure

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


Solution


(i) = ½ x 20 x 20 x sin 40º
= 200 x 0.6428
=128.56cm2


(ii)= ½ x 10 x 10 x sin 500
= 50 x 0.7660
= 38.3 cm2



AREA OF TRAPEZIUM

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Area of trapezium ABCD = area of triangle ABC + area required of triangle ADC
= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMESL 1 h + EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES L2 h
= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES h(L1 + L2)
Examples.
1. Calculate the height of trapezium with area 84 square units and bases 16 units and 8 units as shown

Area = ½ h ( b1+ b2)
84 = ½ h ( 16+ 8)
84 = 12 h
h = 7 units.


AREA OF PARALLELOGRAM

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Area of parallelogram ABCD = area of ΔABD + ΔBCD
= ½ A B h + ½ C D h
= ½ h EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

= ½ h EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

= h x EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


Area of parallelogram = bh

AREA OF RHOMBUS
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
A rhombus is also a parallelogram.
Area = bh
We can also find the area of the rhombus by considering the diagonals of a rhombus.


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
AC and DB are the diagonals.
Area of triangle ABC = area of triangle ADC
Area of rhombus ABCD= 2 (area of triangle ABC)
OR = 2 ( area of triangle ADC)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
EXERCISE
1. 1. Calculate the area of a rhombus whose diagonals are 12dm and 10 dm.


ecolebooks.com
2. 2. Calculate the area of the trapezium ABCD shown in the figure below
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

3. ABCD is a parallelogram with A= 10cm, BAD = 600. Calculate the area of the parallelogram.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

4. 4. Find the area of trapezium ABCD shown in the figure below;
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Sin 360 = h/ s

Solution4:

h = 0.5878x 5 cm

h= 2.939 cm

area = ½ x 2.939 (7+5)

= ½ x 2.939(12)

= 17.634cm2

AREA OF A RECTANGLE


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

AREA OF SQUARE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
A square is a rectangle with equal sides.
Area of triangle ABC = area of triangle ADC
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


Also we can find the area of a square by considering the diagonals.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
ΔABC = ΔADC
Area of triangle ABC = Area of triangle ADC
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Example
1. Find the area of square in which diagonals have length of 12.5cm2

Solution
Area = ½ (length of diagonals)2
= ½ (12.5) 2
= 78.125cm 2

TOTAL SURFACE AREA OF A RIGHT CIRCULAR CONE
Right circular cone Is the one whose vertex is vertically above the center of the base of the cone.


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Total surface area of a cone = area of curved surface + base area.


BUT;
area of curved surface ( lateral surface ) = area of small triangles.
If we consider our cone , AB, BC , CD and DC are approximated line segments, hence we have small triangles VAB,VBC , VCD and VDE.
Hence area of
curved surface
= ½ AB x VA + ½ BC x VC + ½ CD x VC + ½ DEx VD
But VA= VB = VC = VD = VE
Area of curved surface = ½ AxBxL+ ½ BxCxL+ ½ xCx DxL+ ½ DxExL
= ½ L (AB+BC+CD+DE)
= ½ L (2πR)
= πRL
Total surface area = πR2+ πRL
= πR(R+L)

TOTAL SURFACE AREA OF A RIGHT CYLINDER

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Total surface area of a right cylinder
= area of curved surface+ bases area.
= 2πRh + πR2


TOTAL SURFACE AREA OF A RIGHT PYRAMID
A right pyramid is the one which the slant edges joining the vertex to the corner of the base are equal.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


Total surface area of a right pyramid
=area of triangle VAB+ VBC+VDC+VDA + area of the base.
= lateral surface + area of the base.
BUT
As VAB, VDC, VBC and VDA are isosceles triangles. Then VA,VB,VC and VD are slant height.
Example
Consider the data below of a right pyramid. Find the total surface area of the pyramid.
Total surface area of a pyramid
= area of laterals + base area
=area of ΔVAB+ ΔVBC+ ΔVDC+ ΔVDA+ base area


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


EXERCISE
1. The radius of a base of right circular cylinder is 7dm and height is 10 dm. find;
(a). The total surface area.


Solution:

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
The total surface area = 2πR (h+r)
= 2 x 3.14 x 7 dm (10+7)
= 74.732 dm2
2. Calculate the lateral surface area of the right cone shown below.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
= πr(r+L)
= 3.14×3.5x(3.5+1.6)
= 3.14×3.5×19.5
= 214.305cm2



THE TOTAL SURFACE AREA OF A RIGHT PRISM

A right prism is a prism in which each of the vertical edges is perpendicular to the plane of the base. an example of right prism is shown in the figure below where EABF, FBSG, HDCG and EADH are faces made up the lateral surface. and ABCD and EFGH are bases.


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
A right prism is a prism in which each of the vertical edges is perpendicular to the plane of the base. an example of right prism is shown in the figure below where EABF, FBSG, HDCG and EADH are faces made up the lateral surface. and EFGH are bases.
The total surface area of a prism ABCDEFG
= Area of lateral surface + base area
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Example

Find the total surface area of a rectangular prism 12cm long,8 cm wide, and 5 cm high.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

soln
Surface Area = BF(ABxBC)2
=5( 12 x8) 2
= 5cm x 192cm
=960 cm2

Base area = 12×8 x2
= 192 cm2 . : Total surface area = 240 cm2 +192 cm2 = 432 cm2


Exercise
1. The altitude of a rectangular prism is 4cm and the width and length of its base are 12cm and 3 cm respectively. Calculate the total surface area of the prism.

2. One side of a cube is 4dm. calculate
a. The lateral surface area.
b. Total surface area.

3. Figure below shows a right triangular prism whose base is a right angles triangle. Calculate its total surface area.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
4. The altitude of a square pyramid is 5units long and a side of the base is 5 units long. Find the area of a horizontal cross-section at distance 2 units above the base.

Solution

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


Answers

Solution1(a)

= 4(2+3) 2 +2(2×3)
= 4×10 + 12

= 40 + 12

= 53cm2


Solution2.
(a) Lateral area = 2 (4+4+4+4)

=2x 16

=32dm2

(b)Total surface area = 32 + 2(4+4)

= 48 dm2



Solution3.


=(AB2) + (BC2) = ( AC2)

=62 + 82 = AC2

AC=10

= 8x10cm2= 80 cm2



Area of triangle = ½ b h

= ½ x 6 x 8 x 2

= 48cm2



Area of rectangle b = 10cmx10cm

=100cm2

Total surface area = 100cm2 + 48 cm2+ 80cm2


=228cm2


Solution4.
a2+b2=c2

2.52+ b2= 52

b2= 25-6.25

b= 4.33=h

a2+b2=c2

a2=251

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


A1= (½ x 5×4.33) x2

=21.65cm2

A2= x 2 x4.89×2

A2= 9.798

Area = 9.798+21.65 cm2


=31.448cm2

AREA OF A CIRCLE
Consider a circle with several radii (r).

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Re-arrange those pieces from a circle to form a parallelogram.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
A= bh
= ½ c r
½ c r = 2πr
Area of a circle = πr2
Area of a sphere = 4 πr2



LENGTH AND PERIMETER OF A RECTANGULAR POLYGON INSCRIBED IN A CIRCLE.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Consider triangle EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

AO is perpendicular to AB
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES



Sin 1800 /n = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
s= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
s= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
p=ns
p= n(dsin180Ëš)



AREA OF A RECTANGULAR POLYGON INSCRIBED IN A CIRCLE


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Area of a ΔAOB = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMESsinY
AOB = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES r x r EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Area of regular polygon inscribed in a circle.= n (EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMESr2 EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES )


Exercise
1. Find the length of one side of a regular nine –sided polygon inscribed in a circle of radius 10 cm2.

2. Find the radius of a circle which inscribes an equilateral triangle with perimeter 24 cm.

3. Find the area of a 9-sided polygon inscribed in a circle with radius 5 cm.

4.Find the area between two concentric circles.

Answers

Solution 1.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES



Solution2.


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Solution3.

Area = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES n r2 sin EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

= ½ x9 x 25 x sin 40 0
=11.25 x 0.6428
=72.315cm2


Solution4.
Area = πr2
= 3.14 x6x6
=113.04 cm2
Area = πr2
=3.14x 4×4
=50.24cm2
Area between circles = 113.04cm2 – 50.24cm2
=62.80cm2



AREAS OF SIMILAR FIGURES

Similarity
Two polygons are similar when their corresponding angles are equal and corresponding sides are proportional.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Similarity of polygons
If corresponding angles are equal, also if the corresponding side are proportional.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES =EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES = 500
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES= 600
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES = 70 0
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES= 10cm = 2
5cm

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Area of triangle ABC = ½x a x c sin B
Area of triangle PQR = ½ r p sin Q

Area of triangle ABC = ½ ac sin B
Area of triangle PQR ½ r p sin Q
But Sin B = sin Q
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Exercise.
1. Two triangles are similar, A side is 6cm long. The corresponding side to the other is 20cm. if the area of the first is 90 cm2 . what is the area of the second?

2. The ratios of the areas of two circles is 50: 72. If the radius of the smaller circle is 15 cm, find the radius of the larger circle

3. Two triangles are similar. A side of one is 2 units long . the corresponding side of the other is 5 units long. What is the ratio of their areas?

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
5. Two polygons are similar. A side of one is 8 cm long . the corresponding side of the other is18 cm . the area of the first is 16cm2. Find the area of the second.

6. The ratio of the area of two circles is 50: 72. If the radius of the smaller circle is 15 cm, find the radius of the larger circle.


Answers
Solution1.

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES


Area of triangle ABC = K2
Area of triangle PQR

Where K = 6/20
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
X=100cm2

90cm = 3 2
X 10


Solution 2
Area of small circle = K2
Area of large circle

Where K = 15/X

50 = 152
72 X
X= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
X = 18cm


Solution 3.
Area of triangle 1 = K2
Area of triangle 2
= (EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES)2
= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES
Areas = 4:25


Solution4.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES = K2
K = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

The ratio of corresponding sides = 5:4.

Solution5.
Area of triangle 1 = K 2
Area of triangle 2


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES

Solution6.
Area of small circle = K 2
Area of big circle
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - AREAS AND VOLUMES





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