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When the length of one of the sides is multiplied by a constant the effect is to multiply the original volume by that constant, as for the example in [link] .

Right pyramids, right cones and spheres

A pyramid is a geometric solid that has a polygon base and the base is joined to a point, called the apex. Two examples of pyramids are shown in the left-most and centre figures in [link] . The right-most figure has an apex which is joined to a circular base and this type of geometric solid is called a cone. Cones are similar to pyramids except that their bases are circles instead of polygons.

Examples of a square pyramid, a triangular pyramid and a cone.

Surface Area of a Pyramid

Khan academy video on solid geometry volumes

The surface area of a pyramid is calculated by adding the area of each face together.

If a cone has a height of h and a base of radius r , show that the surface area is π r 2 + π r r 2 + h 2 .

  1. The cone has two faces: the base and the walls. The base is a circle of radius r and the walls can be opened out to a sector of a circle.

    This curved surface can be cut into many thin triangles with height close to a ( a is called the slant height ). The area of these triangles will add up to 1 2 × base × height(of a small triangle) which is 1 2 × 2 π r × a = π r a

  2. a can be calculated by using the Theorem of Pythagoras. Therefore:

    a = r 2 + h 2
  3. A b = π r 2
  4. A w = π r a = π r r 2 + h 2
  5. A = A b + A w = π r 2 + π r r 2 + h 2
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Volume of a Pyramid: The volume of a pyramid is found by:

V = 1 3 A · h

where A is the area of the base and h is the height.

A cone is like a pyramid, so the volume of a cone is given by:

V = 1 3 π r 2 h .

A square pyramid has volume

V = 1 3 a 2 h

where a is the side length of the square base.

What is the volume of a square pyramid, 3cm high with a side length of 2cm?

  1. The volume of a pyramid is

    V = 1 3 A · h ,

    where A is the area of the base and h is the height of the pyramid. For a square base this means

    V = 1 3 a · a · h

    where a is the length of the side of the square base.

  2. = 1 3 · 2 · 2 · 3 = 1 3 · 12 = 4 c m 3
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We accept the following formulae for volume and surface area of a sphere (ball).

Surface area = 4 π r 2 Volume = 4 3 π r 3

A triangular pyramid is placed on top of a triangular prism. The prism has an equilateral triangle of side length 20 cm as a base, and has a height of 42 cm. The pyramid has a height of 12 cm.

  1. Find the total volume of the object.
  2. Find the area of each face of the pyramid.
  3. Find the total surface area of the object.

  1. We use the formula for the volume of a triangular prism:
    V = 1 2 b h 2 = 1 2 20 42 2 = 17640
  2. We use the formula for the volume of a triangular pyramid:
    V = 1 6 b h 2 = 1 6 20 42 2 = 5880
  3. We note that we can simply add the volumes of each of the two shapes. So we obtain: 17640 + 5880 = 23520 . This is the answer to part a.
  4. We note that we have four triangles that make up the pyramid. So the area of each face is simply:
    Area = 1 2 b h = 1 2 20 42 = 420
    This is the answer to part b.
  5. The total area of the pyramid is simply 4 × 420 = 1680
  6. The surface area of the prism is:
    Surface area = b × h + 2 × H × S + H × b = 20 × 20 + 2 × 12 × 20 + 12 × 20 = 1120
  7. To find the total surface area, we must subtract the area of one face of the pyramid from the area of the prism. We must also subtract the area of one of the triangular faces of the prism. Doing this gives the total surface area as: 1120 - 420 + 1680 - 420 = 1960 This is the answer to part c.
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Surface area and volume

  1. Calculate the volumes and surface areas of the following solids: *Hint for (e): find the perpendicular height using Pythagoras.
  2. Water covers approximately 71% of the Earth's surface. Taking the radius of the Earth to be 6378 km, what is the total area of land (area not covered by water)?

Questions & Answers

what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
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How can I make nanorobot?
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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