To find the total surface area of the pyramid, we need to calculate the area of the rectangular base and the area of the triangular faces.
-
Area of the base: The base of the pyramid is a rectangle with dimensions 6 cm by 4 cm.
\[ \text{Area of the base} = \text{length} \times \text{width} = 6 , \text{cm} \times 4 , \text{cm} = 24 , \text{cm}^2 \]
-
Area of the triangular faces: The pyramid has four triangular faces, two of which are formed from the longer side (6 cm) and two from the shorter side (4 cm).
For the triangular faces on the 6 cm side:
- The base of each triangle is 6 cm.
- To find the height of the triangular face, we can use the Pythagorean theorem. The height of the pyramid is the perpendicular height (8 cm), and the midpoint of the 6 cm base is 3 cm from the apex vertically.
\[ \text{Height (h)} = \sqrt{(8 \text{ cm})^2 + (3 \text{ cm})^2} = \sqrt{64 + 9} = \sqrt{73} \text{ cm} \]
Therefore, the area of one triangular face is:
\[ \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 , \text{cm} \times \sqrt{73} \text{ cm} \]
The area of the two triangles on this side is:
\[ 2 \times \frac{1}{2} \times 6 \times \sqrt{73} = 6\sqrt{73} \text{ cm}^2 \]
For the triangular faces on the 4 cm side:
- The base of each triangle is 4 cm.
Again applying Pythagorean theorem, the height remains the same but now extends half the 4 cm base, which is 2 cm horizontally.
\[ \text{Height} = \sqrt{(8 \text{ cm})^2 + (2 \text{ cm})^2} = \sqrt{64 + 4} = \sqrt{68} \text{ cm} \]
Therefore, the area of one triangular face is:
\[ \text{Area of one triangular face} = \frac{1}{2} \times 4 , \text{cm} \times \sqrt{68} \text{ cm} \]
The area of the two triangles on this side is:
\[ 2 \times \frac{1}{2} \times 4 \times \sqrt{68} = 4\sqrt{68} \text{ cm}^2 \]
-
Adding together: The total area of the triangular faces is:
\[ 6\sqrt{73} + 4\sqrt{68} \text{ cm}^2 \]
-
Total surface area of the pyramid: Now, we add the area of the base to the area of the triangular faces:
\[ \text{Total Surface Area} = \text{Area of base} + \text{Area of triangular faces} \]
Therefore:
\[ \text{Total Surface Area} = 24 + 6\sqrt{73} + 4\sqrt{68} \text{ cm}^2 \]
This is the total surface area of the pyramid.