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## AREAS AND VOLUMES

AREAS

CASE

1.Right angled triangle Area = ½ b x h

= ½ h(b + d)

= ½ hL

= 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

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

Area = ½ x 10 x 8 x sin 300

= 40 x ½Cm2

=20cm2

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

Area of triangle ABC

= ½ c b sin A

= ½ a c sin B

= ½ a b sin C

Solution;

= ½ x 17 x 20

= 170 cm2

the paper required to make the kite shown in the figure

AREA OF TRAPEZIUM

Area = ½ h ( b1+ b2)

84 = ½ h ( 16+ 8)

84 = 12 h

h = 7 units.

AREA OF PARALLELOGRAM

Area of parallelogram ABCD = area of ΔABD + ΔBCD

= ½ A B h + ½ C D h

= ½ h

= ½ h

= h x

Area of parallelogram = bh

AREA OF RHOMBUS

A rhombus is also a parallelogram.

Area = bh

We can also find the area of the rhombus by considering the diagonals of a rhombus.

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)

4. 4. Find the area of trapezium ABCD shown in the figure below;

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

AREA OF SQUARE

A square is a rectangle with equal sides.

Area of triangle ABC = area of triangle ADC

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

Area of triangle ABC = Area of triangle ADC

Solution

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.

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

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.

= 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.

The total surface area of a prism ABCDEFG

Find the total surface area of a rectangular prism 12cm long,8 cm wide, and 5 cm high.

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.

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

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