Pyramids

[Radit2125]

Could You help me with the solutions to these 2 problems?
1. The base of the pyramid ABCM is the isosceles triangle ABC (AB= 40 and AC= BC=25). The surrounding edge CD=8 is perpendicular to the base. What is the area of the pyramid?
2. The base of the pyramid ABCM is the triangle ABC (AB= 13, BC= 14, AC= 15). The surrounding edge AD=16 against the medium-sized side of the base is perpendicular to the base. What is the full area (S1) of the pyramid?

The first graph is to the first problem and the second graph is to the second problem

 

[JeffM]

You have submitted 46 messages. By now you should certainly know that you should show your work.

Sometimes the first step in a problem is to identify what you know that is relevant.

Do you have a formula for the surface area of a pyramid?

If so, what is that formula?

If not, how many faces does a pyramid have?

Each face is what kind of plane figure?

Do you have a formula for the area of that kind of plane figure?

The point is that staring at a piece of paper is unlikely to do much. Sometimes just trying to specify things that may be relevant to solving the problem gives you clues.

 

[Dr.Peterson]

Could You help me with the solutions to these 2 problems?
1. The base of the pyramid ABCM is the isosceles triangle ABC (AB= 40 and AC= BC=25). The surrounding edge CD=8 is perpendicular to the base. What is the area of the pyramid?
2. The base of the pyramid ABCM is the triangle ABC (AB= 13, BC= 14, AC= 15). The surrounding edge AD=16 against the medium-sized side of the base is perpendicular to the base. What is the full area (S1) of the pyramid?

The first graph is to the first problem and the second graph is to the second problem

The first problem is straightforward, except that I don't know what "surrounding edge" means; perhaps you have translated this from another language and chose the wrong word. But just find the lengths of the edges, then find the area of each triangle, and add. A couple of those parts are harder than others, but since they can be done different ways, we'll need to see what methods you have learned before we can help.

The second problem is odd, apparently requiring a knowledge of what a "surrounding edge" is, and of the grammar of the whole second sentence. There is no information about M at all; was that a typo for D? I think we need a better translation; can you also show us the original so we can try to work it out ourselves?

 

[Radit2125]

I am sorry, the lateral edge is perpendicular to the base in the 2 problems.
In the second problem the pyramid is ABCD and AD is perpendicular to the base

 

[Dr.Peterson]

I am sorry, the lateral edge is perpendicular to the base in the 2 problems.
In the second problem the pyramid is ABCD and AD is perpendicular to the base

I can see why you might translate "lateral" as "surrounding". But we don't normally talk about "the lateral edge" either. I suppose it really means "a" lateral edge, one of three, and is just emphasizing that it is not in the base.

And "against the medium-sized side of the base" must mean that point A is "opposite" the middle-length side BC.

And presumably "area" in the first problem and "full area (S1)" in the second both mean "total surface area".

Now we need to see some work for you. Have you found any of the side lengths? How about some of the triangle areas? Once you've shown your thinking about those, we can help you continue.

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[Radit2125]

To the first problem I suppose this is a part of the decision:
Because CM is perpendicular to the base we can find out what the length of AM is using the Pythagorean theorem:
Let's look at the triangle ACM:
AC^2+MC^2= AM^2
25^2+ 8^2= AM^2=> AM^2= 689=> AM= sqrt 689

About the second problem, I think that because AD is perpendicular to the base, we can finf out the length of DC in the triangle ADC:
AC^2+AD^2= DC^2
15^2+16^2= DC^2=> DC^2= 481=> DC= sqrt 481

But from here I don't know how to continue;

 

[Dr.Peterson]

To the first problem I suppose this is a part of the decision:
Because CM is perpendicular to the base we can find out what the length of AM is using the Pythagorean theorem:
Let's look at the triangle ACM:
AC^2+MC^2= AM^2
25^2+ 8^2= AM^2=> AM^2= 689=> AM= sqrt 689

I suppose, again, that M is the same as D??

You are correct here, but it happens that AM (or AD) is the one length you don't need in order to find the areas (unless you really love Heron's formula even with irrational numbers involved).

You can easily find all the other lengths and altitudes, and use the formula K = 1/2 bh for each.

About the second problem, I think that because AD is perpendicular to the base, we can finf out the length of DC in the triangle ADC:
AC^2+AD^2= DC^2
15^2+16^2= DC^2=> DC^2= 481=> DC= sqrt 481

Here you are right again, and that may turn out to be needed, because the altitude of BCD looks like it will be harder to find, though I haven't pursued it. The other three areas are easy. (Using Heron for ABC yields nice numbers, unlike BCD.)

Ah! You can use the area of ABC to find the altitude from A to BC, and use that to find the altitude of BCD. Give that a try.

This time, please don't show minimal work as you did here; do all that you can do.

 

[Radit2125]

I’ m sorry but I still don’t understand how to solve these problems. So, I would ask how I should find the height in the equation You gave me. Which triangle should I use? Could You give more explanations and directions so I can understand what to do, please?

 

[Dr.Peterson]

I’ m sorry but I still don’t understand how to solve these problems. So, I would ask how I should find the height in the equation You gave me. Which triangle should I use? Could You give more explanations and directions so I can understand what to do, please?

I can do that; but I'll also want more information about what you have learned, in order to be sure what will help you.

Let's focus on the first problem (corrected as I understand it):
1. The base of the pyramid ABCD is the isosceles triangle ABC (AB= 40 and AC= BC=25). The edge CD=8 is perpendicular to the base. What is the total surface area of the pyramid?

The total surface area is the sum of the areas of triangles ABC, ABD, BCD, and ACD.

Triangle ABC is isosceles. If you take the base as AB, you can draw a perpendicular from C to the midpoint M of AB which will divide the triangle into two right triangles. Length MC can be found by the Pythagorean theorem, and is the altitude of the triangle. Find the area.

Triangles BCD and ACD are congruent right triangles. You can easily find their areas.

Triangle ABD is another isosceles triangle; can you see why? You can use the same method as for ABC.

Then add the four areas.

Does that help?

 

[Radit2125]

Yes, now I understand. Here is what I wrote with your explanation:
Let’ s look at the triangles BCD and ADC which are congruent right triangles:
1) DC is perpendicular to AC so edge DCA= 90 degrees and edge BCD= 90 degrees because the two edges are adjacent edges
2) BC= AC because ABC is isosceles triangle
3) DC is a side in the both triangles
=> both triangles are congruent=> AD= DB
Now, let’s look at triangle BCD where we can find the side BD using the Pythagorean theorem:
BC^ 2 + CD^2 = BD^ 2
625+ 64= 689=> BD= AD= sqrt(689)
Because AD=BD=> triangle ABD is also isosceles.
Now let’s connect the edge D with the middle K in the triangle ABD:
Because KD will be a median in the triangle ABD the edges AKD and BKD are equal to 90 degrees.
Now let’s find the length of KD using the triangle AKD:
AK^2 + KD^ 2= AD^2
400 + KD^ 2= 689 => KD= 17
And now in the same way we build CM in the triangle ABC and find its length.
In the end we sum up the areas of the triangles in the pyramid and find the full area surface of the pyramid.
And I have one more question here:
If I am supposed to find the area surface of the pyramid too I would use this formula: S= P* k/2
P= 2* 25+ 40= 90 using the triangle ABC
And I am not sure but MK = 17 is not only a height or median but also it is the apothem of the pyramid. Am I right?
And after we put the values into the formula we can find the surface area.
About the second problem, could You give me such a good explanation as in the first one so I can understand it?

 

[Dr.Peterson]

You haven't stated any areas, so I can't be sure you have actually solved the first problem.

Also, I'm curious why you say, "we sum up the areas of the triangles in the pyramid and find the full area surface of the pyramid", but then say "If I am supposed to find the area surface of the pyramid too". Aren't those the same thing? I think the latter is meant to be only the lateral surface, excluding the base. Am I right?

The formula S= P* k/2 for the lateral surface area applies only to a pyramid with a regular pyramid base, so that there is one number k that is the altitude of all the lateral faces. That can be called the slant height; I don't know what you mean by "apothem" of a pyramid.

As for the second problem, you will learn much more if you do as much as possible on your own following the first example, and then when the same method stops working, try something, then show me all your work as far as you get. I gave you suggestions in post #7.