<< Chapter < Page Chapter >> Page >

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
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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 does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 10 maths [caps]' conversation and receive update notifications?

Ask