cone and cylinder.doc

8/17/2019 cone and cylinder.doc

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CONE

A cone is a type of geometric shape. There are different kinds of cones. They

all have a flat surface on one side that tapers to a point on the other side.

We will be discussing a right circular cone on this page. This is a cone with a

circle for a flat surface that tapers to a point that is 90 degrees from the center

of the circle.

Terms of a Cone

In order to calculate the surface area and volume of a cone we first need to

understand a few terms

!adius " The radius is the distance from the center to the edge of the circle at

the end.

#eight " The height is the distance from the center of the circle to the tip of the

cone.

$lant " The slant is the length from the edge of the circle to the tip of the cone.

%i " %i is a special number used with circles. We will use an abbreviated

version where %i & '.(). We also use the symbol * to refer to the number pi in

formulas.

$urface Area of a Cone

The surface area of a cone is the surface area of the outside of the cone plus

the surface area of the circle at the end. There is a special formula used to

figure this out.


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$urface area & πrs + πr2

r & radius

s & slant

* & '.()

This is the same as saying +'.() , radius , slant- +'.() , radius , radius-

/,ample

What is the surface area of a cone with radius ) cm and slant cm1

$urface area & πrs + πr2

= (3.14x4x8) + (3.14x4x4)

= 100.48 + 50.24= 150.72 cm2

2olume of a Cone

There is special formula for finding the volume of a cone. The volume is how

much space takes up the inside of a cone. The answer to a volume 3uestion is

always in cubic units.

2olume & 1/3πr2h

This is the same as '.() , radius , radius , height 4 '

/,ample

5ind the volume of a cone with radius ) cm and height 6 cm1

2olume & 1/3πr2h= 3.14 x 4 x 4 x 7 ÷ 3

= 117.23 cm 3

Things to !emember$urface area of a cone & πrs + πr2

2olume of a cone & 1/3πr2h The slant of a right circle cone can be figured out using the %ythagorean

Theorem if you have the height and the radius. Answers for volume problems should always be in cubic units. Answers for surface area problems should always be in s3uare units.


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Other mathematical meaning

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co#"x co#" or * ro"c$&" co#".

Further terminology

h" "r&m"$"r o $h" *s" o * co#" &s c*!!"d $h" -d&r"c$r&x-, *#d

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co#"-6 & $h" $ru#c*$&o# !*#" &s *r*!!"! $o $h" co#"9s *s", &$ &sc*!!"d * rus$um. A# -"!!&$&c*! co#"- &s * co#" %&$h *#"!!&$&c*!

https://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Half-linehttps://en.wikipedia.org/wiki/Straight_linehttps://en.wikipedia.org/wiki/Conical_surfacehttps://en.wikipedia.org/wiki/Conical_surfacehttps://en.wikipedia.org/wiki/Conic_sectionhttps://en.wikipedia.org/wiki/Conic_sectionhttps://en.wikipedia.org/wiki/Convex_conehttps://en.wikipedia.org/wiki/Projective_conehttps://en.wikipedia.org/wiki/Directrix_(conic_section)https://en.wikipedia.org/wiki/Dandelin_sphereshttps://en.wikipedia.org/wiki/Cone#Geometryhttps://en.wikipedia.org/wiki/Radiushttps://en.wikipedia.org/wiki/Aperturehttps://en.wikipedia.org/wiki/Frustumhttps://en.wikipedia.org/wiki/Ellipsehttps://en.wikipedia.org/wiki/File:DoubleCone.pnghttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Half-linehttps://en.wikipedia.org/wiki/Straight_linehttps://en.wikipedia.org/wiki/Conical_surfacehttps://en.wikipedia.org/wiki/Conical_surfacehttps://en.wikipedia.org/wiki/Conic_sectionhttps://en.wikipedia.org/wiki/Conic_sectionhttps://en.wikipedia.org/wiki/Convex_conehttps://en.wikipedia.org/wiki/Projective_conehttps://en.wikipedia.org/wiki/Directrix_(conic_section)https://en.wikipedia.org/wiki/Dandelin_sphereshttps://en.wikipedia.org/wiki/Cone#Geometryhttps://en.wikipedia.org/wiki/Radiushttps://en.wikipedia.org/wiki/Aperturehttps://en.wikipedia.org/wiki/Frustumhttps://en.wikipedia.org/wiki/Ellipse


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5/17

*r*!!"! $o $h" "c$or , *#d *"r$ur" , &s &"# $h" &m!&c&$

"c$or "


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based on algebraic manipulation of these standard formulas.

Circular Cone 0ormulas in terms of radius r and height h-

Volume of a cone-o V = 123 π r'h

Slant height of a cone-o

s = "r' h' Lateral surface area of a cone-

o L = π rs = π r"r' h' Base surface area of a cone a circle-

o B = π r'

$otal surface area of a cone-o A = L B = π rs π r' = π rs r = π rr "r' h'

Circular Cone Calculations-

+se the follo%ing additional formulas along %ith the formulasabove. 5iven radius and height calculate the slant height/ volume/

lateral surface area and total surface area.

5iven r/ h find s/ V/ L/ Ao use the formulas above

5iven radius and slant height calculate the height/ volume/


5iven r/ s find h/ V/ L/ Ao

h = "s

'

6 r

'

5iven radius and volume calculate the height/ slant height/


5iven r/ V find h/ s/ L/ Ao h = 3 , v 2 π r'

5iven radius and lateral surface area calculate the height/

slant height/ volume and total surface area.

5iven r/ L find h/ s/ V/ Ao s = L 2 π ro h = "s' 6 r'

5iven radius and total surface area calculate the height/slant height/ volume and lateral surface area.

5iven r/ A find h/ s/ V/ Lo s = 7A 6 π r'8 2 π ro h = "s' 6 r'

5iven height and slant height calculate the radius/ volume/


5iven h/ s find r/ V/ L/ Ao r = "s' 6 h'

5iven height and volume calculate the radius/ slant height/


http://www.calculatorsoup.com/calculators/geometry-plane/circle.phphttp://www.calculatorsoup.com/calculators/geometry-plane/circle.php


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5iven h/ V find r/ s/ L/ Ao r = "7 3 , v 2 π , h 8

5iven slant height and lateral surface area calculate the

radius/ height/ volume/ and total surface area.

5iven s/ L find r/ h/ V/ Ao r = L 2 π , so

h = "s

'

6 r

'

Surface Area of a Cone

The first step in finding the surface area of a cone is to measure the radius of the circle

part of the cone. The next step is to find the area of the circle, or base. The area of a

circle is 3.14 times the radius squared (πr 2. !o", #ou "ill need to find the area of thecone itself. $n order to do this, #ou must measure the side (slant height of the cone.

%a&e sure #ou use the same form of measurement as the radius.

'ou can no" use the measurement of the side to find the area of the cone. The formula

for the area of a cone is 3.14 times the radius times the side (πrl .

So the surface area of the cone equals the area of the circle plus the area of the cone and

the final formula is gien b#)

SA = πr 2 + πrl

*here,

r is the radiush is the height

l is the slant height

The area of the cured (lateral surface of a cone + πrl !ote)

A cone does not hae uniform (or congruent crosssections. (more about conic section

here

http://www.web-formulas.com/Math_Formulas/Algebra_Conic_Section.aspxhttp://www.web-formulas.com/Math_Formulas/Algebra_Conic_Section.aspx


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-xample 1) A cone has a radius of 3cm and height of cm, find total surface area of the

cone.

Solution)

To begin "ith "e need to find slant height of the cone, "hich is determined b# using

/#thagoras, since the cross section is a right triangle.

l 2 = h2 + r 2

l 2 = 52 + 32

l 2 = 25 + 9l = √(34)l = 5.83 cm

And 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 03.1cm2

-xample 2) The total surface area of a cone is 3 square inches. $f its slant height is

four times the radius, then "hat is the base diameter of the cone se 5 + 3.

Solution)

The total surface area of a cone = πrl + πr 2 = 375 inch2

Slant height) l = 4 × radius = 4r

Substitute l = 4r and π = 33 × r × 4 r + 3 × r 2 = 37512r 2 + 3r 2 = 375

15r 2 = 375r 2 = 25

r = 25r = 5

So the base radius of the cone is inch.

And the base diameter of the cone + 2 6 radius + 2 6 + 17 inch.

-xample 3) *hat is the total surface area of a cone if its radius + 4cm and height + 3 cm.

Solution)


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As mentioned earlier the formula for the surface area of a cone is gien b#)

SA = πr 2 + πrl SA = πr(r + l)

As in the preious example the slant can be determined using /#thagoras)

l 2 = h2 + r 2

l 2 = 32 + 42

l 2 = 9 + 16

l = 5

$nsert l = 5 "e "ill get) SA = πr(r + l)

SA = 3.14 · 4 · (4+5) SA = 113.4 cm2

-xample 4) The slant height of a cone is 27cm. the diameter of the base is 1cm. 8ind the

cured surface area of cone.

Solution)

9ien that,

Slant height) l = 2cm:iameter) d = 15cm;adius) r = d!2 = 15!2 = 7.5cm

Cured surface area + πrl "SA = πrl "SA =π · 7.5 · 2

"SA =471.24 cm2

-xample )


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l 2 = h2 + r 2

l 2 = 7 2 + 52

l 2 = 49 + 25l = 8.6

Step 2) =ateral surface area)

#SA = πrl #SA = 3.14 × 7 × 8.6

#SA =189.3 $d 2

So, the lateral surface area of the cone + 10>.73 squared #ard.

-xample ?) A circular cone is 1 inches high and the radius of the base is 27 inches *hat

is the lateral surface area of the cone

Solution)

The lateral surface area of cone is gien b#)

#SA = π × r × l

#SA =3.14 × 2 × 15 #SA = 942 i%ch2

-xample ) 8ind the total surface area of a cone, "hose base radius is 3 cm and the

perpendicular height is 4 cm.

Solution)

9ien that)

r + 3 cm

h + 4 cm

To find the total surface area of the cone, "e need slant height of the cone, instead the

perpendicular height.

The slant height l can be found b# using /#thagoras theorem.

l 2 = h2 + r 2l 2 = 32 + 42

l 2 = 9 + 16 l = 5

The total surface area of the cone is therefore)

SA = πr(r + l) SA = 3.14 · 3 · (3+5) SA = 75.36 cm2


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$urface area of a right cone

Definition: The number of square units that will exactly cover the surface of a

cone.

Try this 7rag the orange dots to ad8ust the radius and height of the cone and

note how the area changes.

!ecall that a cone can be broken down into two parts " the top part with slanted

sides and the circular disc making the base. We can find the total surface area

by adding these together.

The base is a circle of radius r . The area of as circle is given

by 5or more see Area of a circle.

The top section has an area given by where r is the radius at the

base and s is the slant height.

$ee also 7erivation of cone area.

The slant height is the distance along the cone surface from the top to the

bottom rim. If you are given the perpendicular height you can find the slant

height using the %ythagorean Theorem. 5or more see $lant height of a cone.

:y adding these together we get the final formula This can be

simplified by combining some terms but we usually keep it this way because

sometimes we want the area of each piece separately. +$ee the e,ample below-.

/,ample

5ind the area of roof material needed to cover the conical roof shown below.

http://www.mathopenref.com/circlearea.htmlhttp://www.mathopenref.com/coneareaderivation.htmlhttp://www.mathopenref.com/coneslantheight.htmlhttp://www.mathopenref.com/circlearea.htmlhttp://www.mathopenref.com/coneareaderivation.htmlhttp://www.mathopenref.com/coneslantheight.html


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:ecause we are not going to cover the

circular base we only need the area of the top sloping part of the cone.

5rom the above we see that the area of of the sloping top is given by

The radius r of the cone at its base is 'ft +half the diameter- and

the slant height s is (;ft. $ubstituting these into the formula we get

Things to try

In the top figure click


13/17

Terms of a Cylinder

In order to calculate the surface area and volume of a cylinder we first need to

understand a few terms

!adius " The radius is the distance from the center to the edge of the circles at

each end.

%i " %i is a special number used with circles. We will use an abbreviated

version where %i & '.(). We also use the symbol * to refer to the number pi in

formulas.

#eight " The height or length of the cylinder.

$urface Area of a Cylinder

The surface area of a cylinder is the surface area of both circles at each end

plus the surface area of the outside of the tube. There is a special formula used

to figure this out.

$urface area & 2πr2 + 2πrh

r & radius

h & height

* & '.()

This is the same as saying +; , '.() , radius , radius- +; , '.() , radius ,

height-

/,ample

What is the surface area of a cylinder with radius ' cm and height @ cm1

$urface area & 2πr2 + 2πrh

= (2x3.14x3x3) + (2x3.14x3x5)

= 5>.52 + ?4.2

= 150.72 cm2

2olume of a Cylinder

There is special formula for finding the volume of a cylinder. The volume is

how much space takes up the inside of a cylinder. The answer to a volume

3uestion is always in cubic units.


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2olume & πr2h

This is the same as '.() , radius , radius , height

/,ample

5ind the volume of a cylinder with radius ' cm and height @ cm1

2olume & πr2h

= 3.14 x 3 x 3 x 5

= 141.3 cm 3

Things to !emember$urface area of a cylinder & 2πr2 + 2πrh

2olume of a cylinder & πr2h ou need to know the radius and height to figure both the volume and

surface area of a cylinder.

Answers for volume problems should always be in cubic units.

Answers for surface area problems should always be in s3uare units.

Circular C&linder Shape

r = radius

h = height

V = volume

L = lateral surface area

$ = top surface areaB = base surface area

A = total surface area π = pi = 3.141!

" = s#uare root

About this Calculator

$his online calculator %ill calculate the various properties of a

c&linder given ' (no%n values. $his is a right circular c&linder

%here the top and bottom surfaces are parallel but it is

commonl& referred to as a )c&linder).


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, +nits- ote that units are sho%n for convenience but do not

affect the calculations. $he units are in place to give an

indication of the order of the results such as ft/ ft' or ft3. 0or

e*ample/ if &ou are starting %ith mm and &ou (no% r and h in

mm/ &our calculations %ill result %ith V in mm3/ L in mm'/ $ in

mm'/ B in mm' and A in mm'.

Belo% are the standard formulas for a c&linder. Calculations arebased on algebraic manipulation of these standard formulas.

C&linder 0ormulas in terms of r and h-

Calculate volume of a c&linder-o V = π r'h

Calculate the lateral surface area of a c&linder 9ust the

curved outside,,-o L = ' π rh

Calculate the top and bottom surface area of a c&linder '

circles-o $ = B = π r'

$otal surface area of a closed c&linder is-o A = L $ B = ' π rh ' π r' = ' π rhr

,, $he area calculated is onl& the lateral surface of the outer

c&linder %all. $o calculate the total surface area &ou %ill need to

also calculate the area of the top and bottom. :ou can do this

using the circle calculator.

C&linder Calculations-

+se the follo%ing additional formulas along %ith the formulas

above. 5iven radius and height calculate the volume/ lateral surface

area and total surface area.

Calculate V/ L/ A ; 5iven r/ ho use the formulas above

5iven radius and volume calculate the height/ lateral surface


Calculate h/ L/ A ; 5iven r/ Vo h = V 2 π r'

5iven radius and lateral surface area calculate the height/

volume and total surface area.

Calculate h/ V/ A ; 5iven r/ Lo h = L2' π r

5iven height and lateral surface area calculate the radius/

volume and total surface area.

Calculate r/ V/ A ; 5iven h/ Lo r = L2' π h

5iven height and volume calculate the radius/ lateral surface

http://www.calculatorsoup.com/calculators/geometry-plane/circle.phphttp://www.calculatorsoup.com/calculators/geometry-plane/circle.phphttp://www.calculatorsoup.com/calculators/geometry-plane/circle.phphttp://www.calculatorsoup.com/calculators/geometry-plane/circle.php


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Calculate r/ L/ A ; 5iven h/ Vo


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Ps c!dr&c*! coord*$"s, $h" o!um" c*# " c*!cu!*$"d

$"r*$&o# o"r

cone and cylinder.doc

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