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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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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