MATHEMATICS PRESENTATIONMATHEMATICS PRESENTATION

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4455 66 77 88 99

Mr. Kibet Novestus

TOPIC: SURFACE AREA OF A SOLIDSUB TOPIC: SURFACE AREA OF A CONE / FRUSTRUM

PRIO KNOWLEDGE REQUIRED :-

Length of an arc (circles) form one.

Similarity (triangles) form two (previous topic)

• Pythagoras theorem form two.

Mr. Kibet. Novestus

CONE•Cone is a special solid with circular base

base

base

Mr. Kibet Novestus

7cm

CONSTRUCTION OF A CONE.

• Draw a circle of radius r ( 7cm)

Mr. Kibet Novestus

A

B

Cut out a sector whose arc subtends an angle of 150o

150o7cm

Mr. Kibet Novestus

7cm

•Find the length of the remaining arc AB

1500

2100

A

B

SOLnS

FORMULA:

l = Ø 2∏ r 360

Where:-

l = length of the arc AB

Ø = 210O

r = 7cm∏ = 22/7l

Therefore:

210 x 22 x 7 x 2

360 7

AB ( l ) = 25.67cm

= 25.67cm

Mr. Kibet Novestus

base

•Fold the sector to form an open cone

A

B7CM

A B

7CM

Join point A to B as shown to form a circular base

The circumference of the circular base is equal to the length of the arc AB

l

to

C = 25.67 cm

Length of the arc AB

Mr. Kibet Novestus

•NOW WE CAN DEDUCE THE RADIUS OF THE CIRCULAR BASE R AND THE HEIGHT h

rA

B

h

• C = ∏ 2 R

• r = C • 2 ∏

O

p

• OPA makes a right angled triangle as

shown

• OA is the hypotenuse which is the

radius of the initial circle (sector).

• Applying Pythagoras theorem

• h = 72 – 4.082

7cm

r = 25.67

2 ∏

r = 4.08cm

= 5.6 cm

h = 5.6 cmMr. Kibet.

• Area of the curved surface

A

B

7cm210o

• Sector that forms the cone

BA

r

AB

Area of sector above

210o x 22 x 72

360 7= 89.83cm2

The radius of the sector becomes the slant height of the cone

7cm

7cm

7cm

p1

p2

p3

The radius of the circular base is

r = 4.08 cm

Area of sector above

∏ r l

22 x 4.08 x 7

7= 89.76cm2

Mr. Kibet.

•THEREFORE THE AREA OF A CURVED SURFACE OF A CONE

Is given by: A = ∏rlWhere : r is the radius of the base

l is the slant side

l

r

h

Mr. Kibet.

•THE TOTAL SURFACE AREA OF A CLOSED CONE

∏ r l + ∏ r2

∏ r2 is the area of the circular base

Mr. Kibet.

SURFACE AREA OF A FRUSTRUMIf a cone is cut through a plane parallel to the base and

the top part forms a smaller cone, the bottom part forms a frustum

frustrumSmall cone

formedMr. Kibet.

The surface area steps:

• Complete the cone as follows

Example:

12 cm

10 cm

15 cm

x cm

• let the height be x cm

Mr. Kibet.

This will form a smaller cone( 1 )and a larger cone (2)

12 cm

15 cm

x cm

From the knowledge of similar triangles

There are two triangles A & B

15 cm

(12 – x )cm

A

X cm

10 cm

B

1

2

Mr. Kibet.

FROM THE ABOVE DIANGRAMS

x = x + 12

10 15

x = 24 cm

Surface area of frustrum

= Area of curved surface of bigger

cone

-Area of curved

surface of smaller cone

Mr. Kibet.

Surface area

= ∏ R L - ∏ r l

22/7 x 15 x 362 + 152 – 22/7 x 10 x 242 + 102

= 1021cm2

Mr. Kibet.

THE END

Mr. Kibet.