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7/27/2019 Project in Math III
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Project In Math III
Submitted To: T. Liezl Gaspar
Submitted By: Lester Cabaron
7/27/2019 Project in Math III
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&
ITS FORMULAS
SURFACE AREA OF A SOLID
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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
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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.
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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?
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Sphere’s Surface Area Formula:
Where r is the radius.
SA = 4 π r 2
Where d is the diameter.
SA = πd 2
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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?
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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.
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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.
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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
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Sample Problems:
A cone has a radius of 3cm and height of 5cm, find total surface area of the cone.
3cm
5cmnot
given
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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
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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
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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
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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
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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
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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.
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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.
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Cylinder’s Surface Area Formula:
A = 2πr(r + h)
Where,
r is the radius
h height
A = 2πr 2 + 2πrh
or simply,
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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
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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
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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?
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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: