# Surface area, spanning angle,

• Rola
In summary: Surface Area is the total area of all the parts of a object that are visible on a flat surface. You could try and find the total area of the cone by adding up all the individual areas of the parts, or you could find the total area by multiplying the total length of the parts by the total width of the parts.
Rola

Hello,

Can you help me in this question ,

In part a I must find the volume of cup(cylinder)
In part b I must find radius but how can I find Area

In part c ,d I don't know how can I solve it??

What is spanning angle and surface area??

Note : this question from previous exam but I think this question is strange because In my course I study about (Sequences,functions,infinite series,matrices,vectors,equation of circle.

An ice cream shop offers 2 types of serving :cones and cylindrical cups . The ice cream cup is H=8cm in height , and D=6 cm in diameter...

(a) How much water the ice cream cup hold??

(b)if the ice cream cone height ,h=18 cm ,then what must be the area,A,and radius ,r,of the cone base ,for it to have the same volume as the cup??

(c)Find the general formula for the spanning angle Ф,of the sector made by the cone when it is split open on a flat surface .

(d)Find Surface Area S,of the ice cream cone with dimensions as in part (b)

if thickness,t=sqrt3/(5 pi) cm of the cone is made of edible matter such as sugar ...etc ,What is the volume of the matter that makes cone

Rola, in order for us to help you, you have to show us some work, what you have tried yourself to get these questions done!

Do you know the formulae for the volume/surface area of cones and cylinders? That would be a good place to start.

The spanning angle they are referring to is - if you imagine the curved surface of the cone as a sheet of paper, roll out the paper onto a flat surface and it will look like a sector of a circle. The spanning angle is the angle inside that sector.

Surface area is the area of the object if we were to unroll every curved part onto a flat surface, and also place the flat parts of the object onto the surface as well, and add up all the areas.

(a)The amount of water that the ice cream cup hold=the volume of ice cream cup(cylinder)

Volume=pi*r^2*h=pi*(3)^2*8=72pi cm^3

==========================================

volume of cone=1/3*pi*r^2*h

after solving

r=2*sqrt3 cm

Area of base cone=pi*r^2=2*pi*sqrt3 cm^2

========================================

(c)-

after splitting the cone

the resulting shape will be sector

but how can I find the angle??

Firstly , I think that the angle =arc length *radius but then I recognize that is not true

because this is cone not circle ..

and i still encounter difficulty in part d and e

can you guide me??

Your answer to a) is incorrect, but b) isn't. You have r correctly, but subbed it into the formula for Area incorrectly.

For c), There is a simple formula for the curved surface area of a cone, its $$SA = \pi r l$$ where r is the radius of the base of the cone, and l is the slant height. To work out the slant height, you should draw a diagram and think about Pythagoras.

## 1. What is surface area and why is it important in science?

Surface area is the measure of the total area that the surface of an object occupies. It is an important concept in science because it helps determine the amount of material needed for a given surface, the rate of heat transfer, and the amount of contact between substances in chemical reactions.

## 2. How is surface area calculated?

The calculation of surface area depends on the shape of the object. For a cube, it can be calculated by multiplying the length of one side by itself and then multiplying by six since a cube has six equal sides. For other shapes, such as a sphere or cylinder, specific formulas must be used.

## 3. What is spanning angle and how is it related to surface area?

Spanning angle is the angle at which two surfaces meet. It is related to surface area because the larger the spanning angle, the larger the surface area of the object. For example, a sphere has a spanning angle of 180 degrees, which results in a larger surface area compared to a cube with a spanning angle of 90 degrees.

## 4. Can surface area be changed without altering the shape of an object?

Yes, surface area can be changed by altering the dimensions of an object without changing its shape. For example, a piece of paper can be folded numerous times, changing its dimensions and thus its surface area, without changing its shape as a rectangular plane.

## 5. How is surface area used in real-world applications?

Surface area is used in a variety of real-world applications such as determining the amount of paint needed to cover a wall, calculating the surface area of a building for construction purposes, and measuring the amount of medication needed for a specific surface area of skin for a patient. It is also used in environmental science to calculate the surface area of land or bodies of water for conservation and management purposes.

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