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.
Surface Area of a Pyramid
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
$\pi {r}^{2}+\pi r\sqrt{{r}^{2}+{h}^{2}}$ .
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
$\frac{1}{2}\times $ base
$\times $ height(of a small triangle) which is
$\frac{1}{2}\times 2\pi r\times a=\pi ra$
$a$ can be calculated by using the Theorem of Pythagoras. Therefore:
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.
Find the total volume of the object.
Find the area of each face of the pyramid.
Find the total surface area of the object.
We use the formula for the volume of a triangular prism:
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.
Calculate the volumes and surface areas of the following solids: *Hint for (e): find the perpendicular height using Pythagoras.
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
Is there any normative that regulates the use of silver nanoparticles?
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
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?
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