Find the surface area of a rectangular pyramid with those measurements. t = 8 cm m = 4cm and h = 2cm Express your answer as a decimal rounded to the nearest hundredth (1 point) cos^ 1
To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the four triangular faces.
1. Area of the base: Since the base is a rectangle, we can use the formula for the area of a rectangle, which is length times width. The length (l) and width (w) of the rectangle can be found using the given measurements t and m.
l = t = 8 cm
w = m = 4 cm
Area of the base = l * w = 8 cm * 4 cm = 32 cm^2
2. Area of the triangular faces: Each triangular face can be considered as a right triangle. The area of a right triangle can be found using the formula 1/2 * base * height.
The base of each triangle is the same as the width of the base rectangle, which is 4 cm.
The height of each triangle can be found using the given measurement h.
a) The height (h1) of the first triangular face:
h1 = h = 2 cm
Area of the first triangular face = 1/2 * 4 cm * 2 cm = 4 cm^2
b) The height (h2) of the second triangular face:
h2 = sqrt(h^2 + (t/2)^2) = sqrt(2^2 + (8/2)^2) = sqrt(4 + 16) = sqrt(20) ≈ 4.47 cm
Area of the second triangular face = 1/2 * 4 cm * 4.47 cm ≈ 8.94 cm^2
c) The height (h3) of the third triangular face:
h3 = sqrt(h^2 + (m/2)^2) = sqrt(2^2 + (4/2)^2) = sqrt(4 + 4) = sqrt(8) ≈ 2.83 cm
Area of the third triangular face = 1/2 * 4 cm * 2.83 cm ≈ 5.66 cm^2
d) The height (h4) of the fourth triangular face:
h4 = sqrt(h^2 + (t/2)^2) = sqrt(2^2 + (8/2)^2) = sqrt(4 + 16) = sqrt(20) ≈ 4.47 cm
Area of the fourth triangular face = 1/2 * 4 cm * 4.47 cm ≈ 8.94 cm^2
3. Surface area = Area of the base + Area of the four triangular faces
Surface area = 32 cm^2 + 4 cm^2 + 8.94 cm^2 + 5.66 cm^2 + 8.94 cm^2
Surface area ≈ 59.54 cm^2
Therefore, the surface area of the rectangular pyramid is approximately 59.54 cm^2.