Math Is Fun Forum

WyoCowboy

Member

Registered: 2018-03-04

Posts: 7

Surface Area & Volume 12 sided

I am stuck on solving the below problem. I have been working on for a month now and can not seem to get it.

Find the surface area and volume of the pool shown below when the sides, the twelve "bumpers" making up the perimeter of the pool, are 5 ft each and the depth of the pool is 6 ft.

This is what I had:

360/5=72

2*side angle+central angle=180

2*side angle+72=180

2*side angle=108

Side angle=54

Tan(54)=H/(5/2)

H*Tan(54)=(5/2)

H=Tan(54)/(5/2)

H=5.52

A=1/2*2.5*5.52

A=6.9

6.9*2=13.8

13.8*5=69ft

This is what they came back with:

Area of _______    (Be sure to state what the shape is.)

Show every step of your work for just this shape.

Area of _______     (Be sure to state what the shape is.)

Show every step of your work for just this shape.

Other area needed for surface area

What other area of the pool would you need to calculate other than the bottom of the pool to get all of the surfaces?

Surface area calculation

Show how you use the answer above.  (You will not find separate surface areas for each shape.  Use the three answers above in your work.)

Volume calculation

Show how you use the answers above. What is the general formula for the area of any prism?

I am totally lost and have been on this particular problem for a month and getting no where and frustrated.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Surface Area & Volume 12 sided

hi WyoCowboy

Welcome to the forum.

You're the second new member who has posted this problem.  Trouble is, without a diagram I don't really know what the shape is so don't know how to advise you.

There are ways to get a diagram into a post but here's a simpler suggestion.  Look at the diagram in front of you.  Now describe it in words in a new post ... such as "The pool is made up of  a rectangle which is ... by ... and at the ends there are .... .....  It's depth is ......

Then I'll make a diagram which you can check.  Then we can make progress with the solution.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

WyoCowboy

Member

Registered: 2018-03-04

Posts: 7

Re: Surface Area & Volume 12 sided

The middle is a square, then there are 4 additional side on each end. For a total of 12 sides/bumpers, they are 5 feet each with 6' depth

https://1drv.ms/u/s!AodFpEAO3TPygyDFygEImSjzH60k

Last edited by WyoCowboy (2018-03-05 07:38:47)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Surface Area & Volume 12 sided

hi WyoCowboy

That's great.  Thanks for the link.  I've re-made part of the diagram so I can see what's going on. I've left my construction lines as it will help us work out the sizes.

First I constructed AB and made a circle, centred on A with radius AC = 5

And I made AD perpendicular to AB

Then I bisected angle DAB to fix E with EAB = 45, and bisected DAE to fix F with FAE = 22.5

Then I made a second circle, centred on F with radius 5 again to fix G.

Finally I fixed H and J by reflection in the line from G, perpendicular to AB.  I should have labelled the other end of this line.  Let's say GK where K is halfway between A and J.

So that gives us half an octagon AFGHJ.  The other end will have a similar shape.

The question doesn't say that the ends are half of a regular octagon but without this assumption there is not enough information to do the problem.

I'm assuming you have already done simpler problems involving regular polygons.  If so you'll know how to calculate the area of triangle KFA.  Multiply by 8 and you'll have the area of the octagonal part.  Add on the area of the square and you've got the total surface area.

As this pool is a prism the volume is then just surface area x depth of pool.

Hoe that helps,

Bob

I've also assumed the surface area means the area of the top (water) surface.  If the question means the total surface area of the prism you'll have to double the answer you have for the 'top' and add on 10 rectangular areas for the side walls.  You might point out the ambiguities in the question wording so it can be improved for future students. It should state that the ends are each half a regular shape and that surface area means all surfaces including the bottom and sides or just the top surface of the pool.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

WyoCowboy

Member

Registered: 2018-03-04

Posts: 7

Re: Surface Area & Volume 12 sided

Ok thank you I submitted it hopefully its correct.

WyoCowboy

Member

Registered: 2018-03-04

Posts: 7

Re: Surface Area & Volume 12 sided

Area of _______     (Be sure to state what the shape is.)

Show every step of your work for just this shape.

Area of _______     (Be sure to state what the shape is.)

Show every step of your work for just this shape.

Other area needed for surface area

Show how you calculate the part of the surface area not included already.

Surface area calculation

Show how you use the answer above.  (You will not find separate surface areas for each shape.  Use the three answers above in your work.)

Volume calculation

Show how you use the answers above. What is the general formula for the area of any prism?
My teacher wants me to answer by filling this in how would I do that?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Surface Area & Volume 12 sided

hi WyoCowboy

The middle section isn't a square even though the question says 'two regular polygons'.  It'll have to be a rectangle as it's length and width are not the same.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

WyoCowboy

Member

Registered: 2018-03-04

Posts: 7

Re: Surface Area & Volume 12 sided

Below is what I submitted, where did I mess up in solving?

Area of rectangle   (Be sure to state what the shape is.)

A = wl    The distance across the pool is incorrect.  You will need to do work with the octagon to calculate the distance across the pool.  I'm not sure which value is the distance along the edge of the pool, so I'm not sure if you are correct on that measure.

Area of octagon   (Be sure to state what the shape is.)

360/8=45

2*side+45=180

2*side=135

Side=67.5

Tan(67.5)=H/(5/2)

Tan(67.5)=H/2.5

You have not correctly solved this equation for H.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Surface Area & Volume 12 sided

hi WyoCowboy

This problem is also causing difficulties for another MIF member.  Here's a copy of what I have just sent to Kayla.  I'll answer your latest post below the quote.

hi Kayla,

I'm not surprised you are having difficulty.  I think the person who set the question made a mistake and until we get that person to change the question you won't be able to do it.  I'll give my reasoning and send a copy of this post to WyoCowboy who is also struggling to complete this.

Here's the picture you sent me:

and here's the one WyoCowboy sent.  I've added some to the picture to show the pool split into two shapes.

This 12 sided pool is not a regular polygon because the inner angles are not all equal. You will need to break the base of the pool down into 2 regular polygons.

As you can see the two ends do not make a hexagon.  They make an octagon and it certainly looks regular so no problem so far.

But what's left is a rectangle, not a square!  So it's not a regular shape.  Here's my accurate construction of one end.  Each square of my grid represents one foot.  You can clearly see that the diagonal of the octagon, which would be one of the middle shape measurements is not 10 feet.  It's closer to 13. 

Please raise this with your teacher and ask them for clarification.  You may copy from this post if it helps but I would recommend that you do not tell them it's from MIF.

Bob

Area of octagon   (Be sure to state what the shape is.)

360/8=45

2*side+45=180

2*side=135

Side=67.5

Tan(67.5)=H/(5/2)

Tan(67.5)=H/2.5

You have not correctly solved this equation for H.

You need to calculate the area of triangle AFK and then multiply by 8 for the whole octagon.

base = 5

base angle = 67.5

So height, H = 0.5 x 5 x tan(67.5) (because in the right angled triangle 0.5 x 5 is the adjacent and H is the opposite)

You have written "Tan(67.5)=H/2.5" which is correct!

Hope this helps.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

WyoCowboy

Member

Registered: 2018-03-04

Posts: 7

Re: Surface Area & Volume 12 sided

She is still telling me I am doing all of this wrong, any suggestions to go about this differently? I have attached the whole problem this is the last response I got from her: H does not equal 2.5  divided by tan(67.5)?

Find the surface area and volume of the pool shown below when the sides, the twelve "bumpers" making up the perimeter of the pool, are 5 ft each and the depth of the pool is 6 ft.

Use your logic and the formulas you have learned so far for the area of polygons, surface area, and volume to calculate the surface area of the inside of the pool in the picture (the pool liner) and the volume of the pool if it was filled all the way to the top.  Use the shape of the pool and include formulas that you have learned in class. (You cannot add or take out water to find the volume.)

- This 12 sided pool is not a regular polygon because the inner angles are not all equal. You will need to break the base of the pool down into 2 regular polygons.

- Show your work step-by-step just as you have done for #1-4 above.  You may need to include some written explanations for what you are doing in each step.  Show all of the work to find all of the areas necessary for the surface area and all of the work to find the volume.

Libera

Member

Registered: 2018-03-07

Posts: 16

Re: Surface Area & Volume 12 sided

She is still telling me I am doing all of this wrong

What "all of this"? Your first version, I suppose.
Find out H now
H = ...
and go on.
The picture will inspire you.

Last edited by Libera (2018-03-15 11:33:45)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Surface Area & Volume 12 sided

hi WyoCowboy

H does not equal 2.5  divided by tan(67.5)?

I agree with your teacher.  The triangle has an angle of 67.5 and it also has a right angle.  The height is opposite to that angle and 2.5 is adjacent.  opp = adj x TAN(angle) so you shouldn't be dividing.  If you compare the answer you got by doing that with my diagram you'll easily see it's not the correct value.

Bob

Whenever you are doing a trig question work through these steps carefully:

step 1.  Is the triangle right angled?  The formulas only work if it is.  There are more advanced formulas for cases of any triangle but these are not covered by your course.

step 2.  Identify which angle you are using and which sides are Opp, Adj ad Hyp.

step 3.  Choose the correct part of S=O/H  C=A/H / T=O/A

step 4.  Rearrange the formula to make the subject the side you don't know.

step 5.  Do the calculation.  Think about whether your answer 'looks right' for that triangle.  If you can compare with an approximate drawing do so.  It'll help you to see of you've done it correctly.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob