Surface Area and Volume of a Sphere, Hemisphere and Composite Shape

SUBJECT: MATHEMATICS

CLASS: SS 3

TERM: FIRST TERM

WEEK 6  Date:……………..

Topic

 Surface Area and Volume of a Sphere, Hemisphere and Composite Shape

Content

A solid in geometry is a shape with length, width, and height. It is not a flat shape. Some of the common solids are cuboid, cube, cylinder, cone, square based pyramids, sphere etc.

Below are sketches of common solids listed above.

Cuboid                                                   Cube

Cylinder                                                Cone                                        sphere

Sphere

Typical examples of sphere are football, oranges etc

Area of a Sphere

Fig 4.2

Fig. 4.2 represents a solid sphere of radius r.

Current surface area = 4πr².

Volume of Sphere = 4πr³

3.

Example 1.

FIFA 2010 world cup ball(Jabulani) is made with a radius of 7cm. Find

1. the curved surface area of the ball
2. the volume (Take π= 22/7)

Solution

1. Surface area = 4 πr²

r = 7cm

Surface area = 4 x ²²/₇ x 7²

= 4 x 22 x 7  = 616 cm²

1. Volume = 4/3 πr³

= ⁴/₃  x ²²/₇  x 7 ³/₁

=  4 x 22 x 7²= 1437.33cm³

3.

Evaluation

1. How many lead balls, each of radius 3cm, can be ,made from a lead sphere of radius 12cm?
2. A student was asked to paint an earth globe of radius 21m.

(a) What is the surface area of the globe painted by the student?

(b)  If the globe is filled up with the liquid paint, what is the volume of the paint used in filling the globe?

(take π = 22/7)

Reading Assignment

1. New General Mathematics for SS3, chapter 3 pg 15-17
2. Essential Mathematics for SS3 by AJS Oluwasanmi Chapter 7 pages 86-87

Hemisphere

A solid hemisphere is half of a sphere.

( a )                                                                       (b)

Sphere                                                        Hemisphere    = ½ of Sphere.

F,g 4.3

Area of hemisphere = ½ of area of sphere

= ½ x 4 πr² = 2 πr².

Volume of hemisphere

= ½ of sphere.

= ¹/⁷     x 4²/₃πr³

=  ²/₃ πr³

Example 1:

Calculate the surface area and volume of a hemisphere which has a radius of 2cm(take π = 22/7)

Solution:

Curved surface area of hemisphere = 2πr²

= 2 x 22   x 2²

7

= 25.14cm.

Volume of hemisphere = ²/₃ πr³

= ²/₃ x ²²/₇ x 2³

Example 2:

Calculate the total surface area of a solid hemisphere of radius 3.5m (take π = ²²/₇)

Total surface area = curved surface area + plane surface area

C.S.A.   = 2πr² = 2 x ²²/₇   x (⁷/₂)²

= 22  x 7   = 77cm2

2

Plane surface area = πr² = ²²/₇  x   (⁷/₂)²

= ²²/₇ x 7²   = ⁷⁷/₂cm²

2²

Total surface area = (77 + 38.5)cm²  = 115.5cm².

Example 3:

A solid shown below is made up of a cylinder with a hemisphere on top. Calculate

1. the surface area b.  the volume of the solid

Total surface area = Area of hemisphere + area of cylinder

Area of hemisphere  = ½ ( 4 πr²)  = 2πr²

= 2 x π x 7² = 98πcm².

Area of cylinder πr² + 2πrh

= π(7)² + 2 x π x 7 x 20

= 49π + 280π

= 329 πcm²

total surface area = 329πcm² + 98πcm²

= 427 π cm²

= 427 x 22   = 1342cm²

7

1. Volume of solid = πr²h  + ½ ( 4/3πr³)

= π x 72 x 20 + ²/_3π x 73

= 980π + 686 π

3

=   3800cm³

Example 4

Fig. 2.3 shows a wooden structure in the form of a cone, mounted on a hemispherical base.  The vertical height of the cone is 24cm and the base radius 7cm. Calculate, correct to 3 significant figures, the surface area of the structure (take π = ²²/₇).

Solution:

Let the slant height of the cone be l . using Pythagoras rules:

L² = 24² + 7²

L²  √625.,  L= 25cm

curved surface area of cone = πrl

= 22/7 x 7 x 25., = 550cm²

Surface area of hemisphere

= ½ x 4 πr²  = 2 πr²

=  2 x ²²/₇ x 7²= 308cm².

Surface area of structure =  (550+ 308) cm², == 858cm²

Evaluation

1.  Calculate the total surface area of a solid hemisphere of radius 7.7cm. (Take π = 22/7)
2. Calculate the volume and surface area of a hemisphere of diameter 9cm.

General   Evaluation

1.  A hemisphere bowl has an external radius 0f 18cm and is made of wood 3cm thick. Calculate the volume of wood in the bowl.
2. A  measuring  cylinder  of  radius  3cm  contains  water  to  a  height  of   49cm.If  this  water  is  poured  into  similar  cylinder  of  radius  7cm, what  will  be  the  height  of  the water  column
3. From a cylindrical object of diameter 70cm and height 84cm, a right solid cone having its base as one of the circular ends of the cylinder and height 84cm is removed. Calculate

1. the volume of the remaining solid object expressing your answer in the form of a x 10ⁿ
2. the surface area of the remaining solid object.

Reading Assignment:

New General Mathematics, chapter 3, pages 16 -19.

Essential Mathematics for SS3, by AJS Oluwasanmi, chapter 7, pages 88 -90

Weekend Assignment

1   What is the surface area of a sphere, radius 1cm to 3s.f.?

2   Find the volume of a hemisphere of diameter 4cm (a) 16.8cm²(b) 16.8cm³  (c ) 15.8cm³     (d) 15.8cm²

3. The volume of a sphere is 4190cm³. What is its radius? (a) 15cm (b)14cm  (c) 12cm        (d) 10cm.
4. What is the diameter of a sphere of surface area 804cm²? (a) 18cm (b) 17cm (c) 19cm  (d) 16cm
5. What is the volume of a hemisphere of diameter 9cm?(a) 190cm³(b) 191cm³(c ) 192cm³ (d) 193cm³

Theory

1. A sphere has a volume of 1000cm³(a) Calculate its radius correct to 3.s.f.

(b) Find the surface area of the sphere.

2. A hemisphere bowl has an external radius of 24cm and is made of wood 3cm thick calculate the volume of wood in the bowl.