what`s the surface area formula for ..?

Question

a rectangular pyramid ?

Answer 1

Find the area of each triangular side (is it 1/2BxH?), and the area of the rectangular bottom, and add.

There may be a more advanced formula to find it quicker, but that should work for you

Answer 2

DutchProf has the right idea here. Since this question asks for the surface area of a rectangular pyramid, we cannot assume the base is square, and we must take that fact into consideration when calculating the surface area. Because in a rectangular based pyramid one side is longer than another, one pair of faces will lean more to span the distance between them. In this case, it will be the faces with bases equal to the width of the rectangle. The faces with bases along the length of the rectangle will lean less because they have less distance to span between them. For this reason, the slant height of each pair of faces will be different, unless the base is also a square. So we must calculate the slant height of each pair of faces separately before finding the area of each pair of triangles.

Let W = width, L = length, H = height of pyramid, S = slant height of faces along the width, and S ' = slant height of faces along the length.

Looking at the rectangle from a face whose base is the length of the rectangle, then this is the case:

S = √[(L/2)² + H²] = √(L²/4) + (4H²/4) = √(L² + 4H²) /√4).
√(LW² + 4H²) /√4) = ½ √(L² + 4H²).

So S = ½ √(L² + 4H²).

Looking at the rectangle from a face whose base is the width of the rectangle, then this is true:

S' = √[(W/2)² + H²] = √(W²/4) + (4H²/4) = √(W² + 4H²) /√4).
√(W² + 4H²) /√4) = ½ √(W² + 4H²).

So S' = ½ √(W² + 4H²).

The area of each face along the width is:

A (W) = ½ WS, and since there are two of them,
2 A (W) = 2 (½ WS) = WS.

Using this same logic for the faces along the length:

A (L) = ½ LS', and since there are two of them,
2 A (L) = 2 (½ LS') = LS'.

The area of the base is A (B) = LW.

Adding all these up and substituting in the appropriate values, we get this for the final formula for our area:

A (T) = A (B) + A (W) + A (L)
*** A (T) = LW + WS + LS'
A (T) = LW + W [½ √(L² + 4H²)] + L [½ √(W² + 4H²)]
A (T) = LW + ½ [W √(L² + 4H²) + L √(W² + 4H²)].

The last equation above is the area formula for a rectangular pyramid.

If the rectangular base is also a square, then S' = S and L = W, which we can now redefine as E, the edge of the square base of the pyramid, and using the triple starred equation directly above, our formula collapses to this:

A (T) = E² + ES + ES
A (T) = E² + 2 ES

To verify this last formula, let’s evaluate the area directly from a pyramid with a square base.

The area of the base:

** A (B) = E x E = E², where E is the edge of the square base.

The area of each triangular face is given by:

A (F) = ½ x b x h = ½ x E x S = ½ ES, where E is the base of each triangle and also an edge of the square, and S is the slant height of each triangle.

Since there are 4 triangles, the total area of the faces is:

** 4 A (F) = 4 (½ ES) = 2 ES.

Adding the two double starred equations above, we get:

A (B) + 4 A (F) = A (T)
E² + 2 ES = A (T), which confirms our earlier derived equation.

Answer 3

Let L be the length, W be the width, and H the height. Assume that the top of the pyramid is located straight above the center.

The surface consists of one rectangle, two larger isosceles triangles and two smaller isosceles triangles.

The height of the triangles is
... (larger) sqrt( H^2 + (W/2)^2 ) = 1/2 sqrt( 4H^2 + W^2)
... (smaller) sqrt( H^2 + (L/2)^2 ) = 1/2 sqrt( 4H^2 + L^2)

Therefore, their areas are
... (larger) L/4 sqrt( 4H^2 + W^2 )
... (smaller) W/4 sqrt( 4H^2 + L^2)

For the total surface area, you find
... A = L W + L/2 sqrt( 4H^2 + W^2) + W/2 sqrt( 4H^2 + L^2).

... (smaller) H/4 sqrt

Answer 4

SA=1/3bh
SA is Surface area.
b is the base of the pyramid.
h is the height of a pyramid.

Answer 5

A = area of the base + 1/2 permiter of the base * slant height of one of the triangle

Answer 6

1/3 perimeter of base x height
= 1/3 x( L + W ) x 2 h

L= length w = width h = height

Answer 7

Just consider each surface individually and then add them all together in the end.

Answer 8

2(1/2hl)+2(1/2hw)+ lw

h - height
l - length
w - width

Answer 9

SA= 2(pi) (L*w) + 2(h*w) + 2(L*h)