Math Is Fun Forum

I have a rectangle, with a length of 7 and a width of 4:

15. What is the perimeter? - Answer: E
A 31
B 26
C 58
D 35
E 22
F 59

16. What is the area? - Answer: A
A 28
B 65
C 79
D 13
E 10
F 19

This rectangle just became the base of a regular prism, with a height of 6:

17. What is the lateral area? - Answer: F
A 119
B 873
C 256
D 349
E 332
F 132

18. What is the total surface area? - Answer: D
A 119
B 873
C 256
D 188
E 332
F 132

19. What is the volume? - Answer: E
A 651
B 327
C 395
D 221
E 168
F 342

20. What is the area of the largest rectangular side? - Answer: D
A 65
B 32
C 36
D 42
E 16
F 34

The shape is a pyramid with:

6. a rectangular base with a width of 4
A Lateral area: 120; Volume: 98
B Lateral area: 192; Volume: 76
C Lateral area: 240; Volume: 51
D Lateral area: 176; Volume: 33
E Lateral area: 288; Volume: 75
F Lateral area: 100; Volume: 64

Volume:
a = 6 x 4
a = 24

v = 1/3 x 24 x 8
v = 64

Lateral Area:
6 + 4 + 6 + 4 = 20
20 x .50 = 10
10 x 10 = 100

Answer: F

7. a square base
A Lateral area: 120; Volume: 96
B Lateral area: 90; Volume: 64
C Lateral area: 176; Volume: 144
D Lateral area: 192; Volume: 35
E Lateral area: 288; Volume: 288
F Lateral area: 240; Volume: 75

Volume:
a = 6 x 6
a = 36

v = 1/3 x 36 x 8
v = 96

Lateral Area:
6 + 6 + 6 + 6 = 24
24 x .50 = 12
12 x 10 = 120

Answer: A

8. a rectangular base with a width of 3
A Lateral area: 192; Volume: 144 
B Lateral area: 90; Volume: 48
C Lateral area: 276; Volume: 64
D Lateral area: 176; Volume: 144
E Lateral area: 92; Volume: 96
F Lateral area: 62; Volume: 24

Volume:
a = 6 x 3
a = 18

v = 1/3 x 18 x 8
v = 48

Lateral Area:
3 + 6 + 3 + 6 = 18
18 x .50 = 9
9 x 10 = 90

Answer: B

9. a rectangular base with a width of 5

A Lateral area: 100; Volume: 48
B Lateral area: 240; Volume: 112
C Lateral area: 176; Volume: 96
D Lateral area: 110; Volume: 80
E Lateral area: 288; Volume: 144 
F Lateral area: 90; Volume: 64

Volume:
a = 5 x 6
a = 30

v = 1/3 x 30 x 8
v = 80

Lateral Area:
6 + 5 + 6 + 5 = 22
22 x .50 = 11
11 x 10 = 110

Answer: D

10. a rectangular base with a width of 7
A Lateral area: 240; Volume: 64
B Lateral area: 188; Volume: 96
C Lateral area: 176; Volume: 144
D Lateral area: 130; Volume: 112
E Lateral area: 144; Volume: 215
F Lateral area: 100; Volume: 128

Volume:
a = 6 x 7
a = 42

v = 1/3 x 42 x 8
v = 112

Lateral Area:
7 + 6 + 7 + 6 = 26
26 x .50 = 13
13 x 10 = 130

Answer: D

Well this is what my lesson says and shows:

"Regular Pyramid Area Theorem
The area L of any regular pyramid with a base that has perimeter P and with slant height l is equal to one-half the product of the perimeter and the slant height.
Formula: L = (1/2)(P)l"

"A pyramid is a polyhedron with a single base and lateral faces that are all triangular.  All lateral edges of a pyramid meet at a single point, or vertex.  A regular pyramid is a pyramid that has a base that is a regular polygon and with lateral faces that are all congruent isosceles triangles.

At any rate, the equations for area and volume are just like all the others we've done--just plug in the numbers.

If a regular pyramid has a square base with a length and width of 3, and slant height of 5, and a height of 4 then what is the area of the pyramid, and what is the pyramid's volume.

L = (1/2)(P)l
L = (1/2)(12)5
L = 30

V = (1/3)Bh
V = (1/3)9(4)
V = 12
"

So I guess the Slant Height is used for finding the Lateral Area. I have a question though, is the Perimeter the same thing as the area of the Base?

And I would try to do #6 my self but I am visiting my grandparents at the moment so it will have to wait for now

h^2 + 1.5^2 = 4^2 x 1.5^2
h^2 + 1.5 = 16 x 1.5

I am really confused I have no idea what I'm doing.

Ok I will use Pythagorean theorem to find the height.

h^2 + 1.5^2 = 6^2
h^2 + 2.25 = 36
h^2 = 33.75
h = 5.80

v = 1.5 x 5.80 x 8
v = 69.6

I still don't know what's going wrong. Could you give me an example of how to find the area of this triangle?

Also, I had an idea for getting the lateral area:

2 x 6 x 8 = 96
1 x 3 x 8 = 24

96 + 24 = 120
L = 120

This is what I created for #5. Am I correct with this image?

Sorry if it's so huge I'm trying to figure out how to work this image upload thing

2. a rectangular base with a width of 3
A Lateral area: 192; Volume: 144 sqrt(3)
B Lateral area: 144; Volume: 144
C Lateral area: 90; Volume: 240
D Lateral area: 120; Volume: 288
E Lateral area: 192; Volume: 64
F Lateral area: 144; Volume: 18*sqrt(15)

So the base of the rectangle has sides of 3 and 6. So:

Volume:
A = 3 x 6 This is the area of the end rectangle  (18).  Pythagoras and square roots are unnecessary as you have the area at this point.
A = 18

V = 18 x 8= 144

Lateral Area:
So would it be:

2 x 6 x 8 = 96
2 x 3 x 8= 48
96 + 48 = 144 again... which means answer should be B?

3. a square base
A Lateral area: 176; Volume: 240   
BLateral area: 90; Volume: 64
CLateral area: 120; Volume: 144
DLateral area: 288; Volume: 144 sqrt(3)
E Lateral area: 144; Volume: 18*sqrt(15)
F Lateral area: 192; Volume: 288

Since it's a square base, all sides are equal, which means all sides measure 6. So:
Volume:
A = 6 x 6 = 36
V = 36 x 8 = 288

Lateral Area: (there are FOUR lateral faces for a square )
6 x 8 = 48
4 x 48 = 192.

Answer is F.

Ok let me try it.
Volume:
h = sqrt{6^2 - 3^2}
h = sqrt{27}
h = sqrt{9 x 3}
h = 3 {sqrt 3}

area = 1/2 x (6) x 3 {sqrt 3} = 9 {sqrt 3}

volume = 9 {sqrt 3} x 8 = 72 {sqrt 3}

I think I got it this time

Lateral Area:
6 x 8 = 48
3 x 48 = 144

Now for #1, I am looking at D as my answer.

I've done some further researching and I think I am understanding it more now. First I need to get the area of the triangle shape. I believe that would be:
6 x 3 = 18.
Now:
V = A x H
V = 18 x 8 | v = 288
Is that correct?

a^2 + b^2 = c^2
6^2 + 8^2 = c^2
36 + 64 = c^2
100 = c^2
c = 10

A = (1/2)bh
A = (1/2)(6)(8)
A = (1/2)48
A = 24

10 x 24 = 240. Is that the volume? \(^o^)/

Ok, from the link I understood how to get the area. So the base they give me is 6 as the base and the height is 8. So i have done:
A = .50 x 6 x 8 = 24

So now I have the area, I times it with height to get: V = 24 x 8 = 192
Now I'm not sure if I'm doing it EXACTLY right. I'm trying to figure out the answer for #1. To me it seems as if I got the lateral area which I feel is wrong:
The shape is a right prism with:

1. an equilateral triangle as the base
A Lateral area: 90; Volume: 288
B Lateral area: 192; Volume: 64
C Lateral area: 120; Volume: 240
D Lateral area: 144; Volume: 72 sqrt(3)
E Lateral area: 176; Volume: 18*sqrt(15)
F Lateral area: 100; Volume: 288

Sent them in, 20/20. I thank you VERY much Mr. Bob for your help!