# A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #4 #, its base's sides have lengths of #8 #, and its base has a corner with an angle of #(7 pi)/8 #. What is the pyramid's surface area?

T S A = 115.0021

AB = BC = CD = DA = a = 8

Height OE = h = 4

OF = a/2 = 8/2 = 4

Area of

Lateral surface area

Area of base ABCD

T S A

T S A

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To find the surface area of the pyramid:

- Calculate the area of the rhombus base.
- Determine the area of each triangular face.
- Add the areas of all faces together.

First, find the area of the rhombus base using its diagonals (d_1) and (d_2):

[ A_{\text{base}} = \frac{d_1 \times d_2}{2} ]

Since it's a rhombus, (d_1 = d_2 = 8) (because all sides are equal).

[ A_{\text{base}} = \frac{8 \times 8}{2} = 32 , \text{square units} ]

Now, calculate the area of one triangular face. Each triangular face is an isosceles triangle with base (b) (which is a side of the rhombus) and height (h).

[ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} ]

Given that the base side length is (8) and the height of the pyramid is (4), the area of one triangular face is:

[ A_{\text{triangle}} = \frac{1}{2} \times 8 \times 4 = 16 , \text{square units} ]

Since there are four identical triangular faces, the total area of all four triangular faces is (4 \times 16 = 64) square units.

The total surface area of the pyramid is the sum of the base area and the area of the four triangular faces:

[ \text{Surface Area} = A_{\text{base}} + 4 \times A_{\text{triangle}} ]

[ \text{Surface Area} = 32 + 4 \times 16 = 32 + 64 = 96 , \text{square units} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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