Project in Math III

7/27/2019 Project in Math III

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Project In Math III

Submitted To: T. Liezl Gaspar

Submitted By: Lester Cabaron



&

ITS FORMULAS

SURFACE AREA OF A SOLID



What is surface area and its volumes?

1.) Surface Area Of Sphere.

2.) Surface Area Of Cone.

3.) Surface Area Of Cylinder.

4.) Surface Area of An Ellipsoid.

5.) Surface Area Of a Cuboids

6.) Surface Area Of A Cube



What is Surface Area?

Surface area is the total area of the faces and curved

surface of a solid figure. Mathematical description of the

surface area is considerably more involved than the

definition of arc length or polyhedral (objects with flatpolygonal faces) the surface area is the sum of the areas of

its faces. Smooth surfaces, such as a sphere, are assigned

surface area using their representation as parametric

surfaces. This definition of the surface area is based on

methods of infinitesimal calculus and involves partial

derivatives and double integration.



1.) Sphere

A sphere is a three-dimensional space, such as

the shape of a football. A sphere is a body bounded by asurface whose every point is equidistant (i.e. the same

distance) from a fixed point, called the centre or the

origin of the sphere.

Like a circle in three dimensions, all points from the

center are constant. The distance from the center to any

points on boundary is known as the radius of the sphere. The

maximum straight distance through the sphere is known as

the diameter of the sphere. One-half of a sphere is called a

hemisphere.

What is a Sphere?



Sphere’s Surface Area Formula:

Where r is the radius.

SA = 4 π r 2

Where d is the diameter.

SA = πd 2



Sample Problems:

Surface area of the sphere:

SA = 4 × π × r 2

SA = 4 × π × (5.5)2

SA = 4 × 3.14 × 30.25

SA = 379.94

Thus the surface area of the

sphere is 379.94m2.

What is the total surface area of a

sphere whose radius is 5.5 meters?



Find the surface area of a sphere whose

diameter is 12cm?

Surface area of the sphere:

SA = π × d 2

SA = π × 122

SA = 3.14 × 144

SA = 452.16cm2

Thus the surface area of the

sphere is 379.94 m2.



2.) Cone

What is a Cone?

Formally, it is the solid figure formed by the locus of

all straight line segments that join the apex to the base. The

term "cone" is sometimes used to refer to the surface or the

lateral surface of this solid figure (the lateral surface of a cone is

equal to the surface minus the base).

The axis of a cone is the straight line (if any), passing

through the apex, about which the base has a rotational

symmetry.

A cone is an n-dimensional geometric shape that

tapers smoothly from a base (usually flat and circular) to a point

called the apex or vertex.



Cone’s Surface Area Formula:

Where,

r is the radius

h is the height

l is the slant height

SA = πr 2 + πrl

The area of the curved /lateral

surface of a cone = πrl



Sample Problems:

A cone has a radius of 3cm and height of 5cm, find total surface area of the cone.

3cm

5cmnot

given



Solution:

To begin with we need to find slant height of

the cone, which is determined by using

Pythagoras, since the cross section is a right

triangle.

l 2 = h2 + r 2

l 2 = 52 + 32

l 2 = 25 + 9

l = √(34)

l = 5.83 cm



So, the total surface area of the cone is:

SA = πr 2 + πrl

SA = π · r · (r + l)

SA = π · 3 · (3 + 5.83)

SA = 83.17 cm2

Therefore, the total surface area of the cone is

83.17cm2



The slant height of a cone is 20cm. the diameter of the base is 15cm.

Find the curved surface area of cone.

20cm

15cm

not

given



Solution:

Given that,

Slant height: l = 20cm

Diameter: d = 15cm

Step 1:

Find the radius:r = d/2 = 15/2 = 7.5cm

Step 2:

Curved surface area = πrl

CSA = πrl

CSA =π · 7.5 · 20 CSA =471.24cm2

So, the curved surface area of the cone = 471.24cm2



Height and radius of the cone is 5 yard and 7 yard.

Find the lateral surface area of the given cone.

not given

7 yard

5 yard



Solution:

Step 1:

Slant height of the cone:l 2 = h2 + r 2

l 2 = 7 2 + 52

l 2 = 49 + 25

l = 8.6

Step 2:

Lateral surface area:

LSA = πrl

LSA = 3.14 × 7 × 8.6

LSA =189.03 yd 2

So, the lateral surface area of the cone = 189.03 squared yard.



3.) Cylinder

What is a Cylinder?

A cylinder (from Greek κύλινδρος – kulindros, "roller,

tumbler") is one of the most basic curvilinear geometric shapes,

the surface formed by the points at a fixed distance from a

given line segment, the axis of the cylinder. The solid enclosed by

this surface and by two planes perpendicular to the axis is also

called a cylinder. The surface area and the volume of a cylinderhave been known since deep antiquity.



Cylinder’s Surface Area Formula:

A = 2πr(r + h)

Where,

r is the radius

h height

A = 2πr 2 + 2πrh

or simply,



Sample Problems:

Find the surface area of a cylinder with a radius of 4 cm, and a height of 3 cm.

4 cm

3 cm



Solution:

SA = 2 × π × r 2 + 2 × π × r × h

SA = 2 × 3.14 × 42 + 2 × 3.14 × 4 × 3

SA = 6.28 × 16 + 6.28 × 12

SA = 100.48 + 75.36

SA = 175.84

Surface area = 175.84 cm2



4.) Ellipsoid

The ellipsoid got its name because its crosssections parallel to the xy, xz and yz planes are all

ellipses. It has the interesting property that it is

regular everywhere except at the north and the

south poles.

What is an Ellipsoid?



A prolate spheroid has surface area

defined as:

where,

is the angular eccentricity of the prolatespheroid and e = sin(α) is its (ordinary)

eccentricity.

Ellipsoid’s Surface Area Formula:



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