# A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #3 # and #5 # and the pyramid's height is #5 #. If one of the base's corners has an angle of #pi/4#, what is the pyramid's surface area?

T S A = 53.4781

Area of parallelogram base

Area of

Area of

Lateral surface area =

Total surface area =Area of parallelogram base + Lateral surface area

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The surface area of the pyramid can be calculated using the formula:

[ \text{Surface Area} = \text{Area of Base} + \text{Area of Four Triangular Faces} ]

Given that the base is a parallelogram with sides of length 3 and 5, the area of the base can be calculated using the formula for the area of a parallelogram:

[ \text{Area of Base} = \text{Base length} \times \text{Height of parallelogram} ]

The height of the parallelogram can be found using trigonometry. Since one of the corners of the base has an angle of ( \frac{\pi}{4} ) radians, the height of the parallelogram can be calculated as:

[ \text{Height of parallelogram} = \text{Base side length} \times \sin\left(\frac{\pi}{4}\right) ]

The area of each triangular face of the pyramid can be calculated using the formula for the area of a triangle:

[ \text{Area of Triangular Face} = \frac{1}{2} \times \text{Base length} \times \text{Height} ]

Substituting the given values into these formulas, we can calculate the surface area of the pyramid.

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