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 #2 #, its base's sides have lengths of #5 #, and its base has a corner with an angle of #( pi)/6 #. What is the pyramid's surface area?
T S A = 20.5387
AB = BC = CD = DA = a = 5
Height OE = h = 2
OF = a/2 = 1/2 = 2.5
Area of
Lateral surface area
diagonal
#OB = d_2/2 = BCsin (C/2)=5sin(pi/12)= 1.294
#OC = d_1/2 = BC cos (C/2) = 5* cos (pi/12) = 4.8295
Area of base ABCD
T S A
T S A
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To find the surface area of the pyramid, we first need to calculate the area of each face and then sum them up.

Calculate the area of the rhombus base: The area of a rhombus can be found using the formula: Area = base * height. Since the base's sides have lengths of 5, and one angle of the rhombus is π/6, we can use trigonometry to find the height of the rhombus. The height (h) of the rhombus can be calculated as: h = side * sin(angle). Substituting the given values, we get: h = 5 * sin(π/6) ≈ 5 * 0.5 = 2.5. So, the area of the rhombus base is: Area_base = 5 * 2.5 = 12.5 square units.

Calculate the area of each triangular face: Each triangular face has a base length equal to the side length of the rhombus (5 units) and a height equal to the height of the pyramid (2 units). The area of each triangular face is: Area_triangle = (1/2) * base * height. Substituting the given values, we get: Area_triangle = (1/2) * 5 * 2 = 5 square units for each face.

Since there are four triangular faces, the total area of all triangular faces is: Total_area_triangles = 4 * Area_triangle = 4 * 5 = 20 square units.

The total surface area of the pyramid is the sum of the area of the base and the total area of the triangular faces: Total_surface_area = Area_base + Total_area_triangles = 12.5 + 20 = 32.5 square units.
Therefore, the surface area of the pyramid is 32.5 square units.
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When evaluating a onesided 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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