## Falling_In_Katt one year ago Find the total area for the regular pyramid. T. A. = $([][][]+[][]\sqrt{[]})$

1. Falling_In_Katt

2. SolomonZelman

you want the surface area correct?

3. Falling_In_Katt

yes

4. SolomonZelman

|dw:1436465833356:dw|

5. SolomonZelman

Do you know how to find the area of the|dw:1436465959427:dw|??

6. Falling_In_Katt

no...

7. SolomonZelman

|dw:1436466095899:dw|

8. SolomonZelman

x²+6²=10² x=?

9. Falling_In_Katt

4^2

10. SolomonZelman

I guess we should start over from when I split the triangle

11. SolomonZelman

|dw:1436466297750:dw|

12. SolomonZelman

now, you can almost find the area of this small triangle (and then multiply times 2 to get the area of the full triangle), BUT you don't know the hieght.

13. SolomonZelman

U can see that the small triangle (formed by splitting the original triangle in halves) is a right triangle, and therefore we can apply a pythegorean theorem (a²+b²=c²) to find the missing height. in this case (as regards to this smaller triangle) we know a leg of 6 units a hypotenuse of 10. Therefore we will set the equation: a² + 6² = 10² (to solve for this missing height (or this missing leg)

14. SolomonZelman

can you find this missing side>?

15. Falling_In_Katt

64?

16. SolomonZelman

a² + 6² = 10² a² + 36 = 100 a² = 64 good, but that is a² (you found that if you raise this missing side to the second power, then you get 64, BUT, you haven't found the actual side) Take the square root of both sides to find what "a" is going to be.

17. Falling_In_Katt

8

18. Falling_In_Katt

?

19. SolomonZelman

yes

20. SolomonZelman

the missing height is 8.

21. SolomonZelman

|dw:1436467003677:dw|

22. SolomonZelman

can you find the area of this triangle? (shaded) |dw:1436467035260:dw|

23. Falling_In_Katt

24?

24. SolomonZelman

yes

25. SolomonZelman

and the area of the whole triangle would be?

26. SolomonZelman

(if a triangle with an area of 24 is only a half of the entire triangle, then the entire triangle is?)

27. Falling_In_Katt

48

28. SolomonZelman

yes

29. SolomonZelman

What if I told you to find the area of this:|dw:1436467254378:dw|(our 4th face in a pyramid)

30. SolomonZelman

I will split this triangle in halves as well.

31. SolomonZelman

|dw:1436467297253:dw|

32. SolomonZelman

can you find the height using the pythegorian theorem a²+b²=c² ?

33. SolomonZelman

ok, you are missing the leg or the hypotenuse here?

34. Falling_In_Katt

Leg?

35. SolomonZelman

yes

36. SolomonZelman

So, you know one of the legs is 6, and the hypotenuse is 10, but you are missing another leg. Please plug this into a²+b²=c² (note: it doesn't mater which leg is a and which leg is b - the missing or the known leg)

37. SolomonZelman

please take a shot to plug in numbers into a²+b²=c² (if you make a mistake it is ok, i will correct)

38. Falling_In_Katt

$a ^{2}+6^{2}=12^{2} ?$ $a ^{2}=108$ $a=6\sqrt{3}$

39. SolomonZelman

yes. a=√108 a=√(36•3) a=6√3 correct

40. SolomonZelman

|dw:1436467977322:dw|

41. SolomonZelman

This again shows the rule of a 30º-60º-90º, btw .... and in any case, you now have to find the area of 1/2 of this triangle

42. SolomonZelman

|dw:1436468071869:dw|

43. SolomonZelman

this is a smaller triangle, but a bigger triangle (that is twice greater) would have the area of 6•6√3, without dividing by 2.

44. SolomonZelman

So your 4th face is 36√3.

45. SolomonZelman

Now, your faces of the pyramid: you have three faces (side faces), of 48 each. AND a base (bottom face) of 36√3. So the total surface area of the pyramid (the sum of all of its face), is?

46. Falling_In_Katt

$144+36\sqrt{3}$?

47. SolomonZelman

yes

48. SolomonZelman

wolfram gives an estimate of 206.35 (or we can estimate √3 using a Taylor polynomial of function f(x)=√x, at a=4.... no I am just kidding.. would be to much....)

49. SolomonZelman

you are done, that is your answer.... (for units though, I would add ":square units" or "units²" to denote that we found the area just now - (not volume pressure or other). )

50. SolomonZelman

yw

51. Falling_In_Katt

Thank you!!