Find thde total surface area of the net of a square pyramid given that the base is 3cm by 3cm and each slant edge is 5cm. Include the flaps.
To find the total surface area of the net of a square pyramid, we need to consider the individual faces that make up the pyramid and their dimensions.
The net of a square pyramid consists of a square base and four triangular faces that meet at the apex of the pyramid. The base has dimensions of 3cm by 3cm, and each slant edge measures 5cm.
Let's break down the surface area calculation step by step:
1. Calculate the area of the base:
The base of the square pyramid is a square, so we can find its area by simply squaring the length of one side:
Area of the base = side length² = 3cm × 3cm = 9cm²
2. Calculate the area of each triangular face:
Each triangular face can be divided into two right triangles and a rectangle. We can visualize the net of the pyramid and label the dimensions as follows:
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|h\ |
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The height (h) of each triangular face can be found using the Pythagorean theorem, as the slant edge is the hypotenuse and the half of the base length is one of the legs:
h = √(slant edge² - (base length/2)²) = √(5cm² - (1.5cm)²) = √(5cm² - 2.25cm²) = √(2.75cm²) ≈ 1.658 cm
The total surface area of all four triangular faces can then be calculated by multiplying the base length by the height and then multiplying by 2 (as there are two identical triangular faces):
Surface area of the triangular faces = 4 × (base length × height)/2 = 4 × (3cm × 1.658 cm) = 19.896 cm² ≈ 19.9 cm²
3. Calculate the total surface area:
To find the total surface area of the net, we sum the areas of the base and the four triangular faces:
Total surface area = Base area + Triangular faces area = 9cm² + 19.9 cm² = 28.9 cm² ≈ 28.9 cm²
Therefore, the total surface area of the net of the square pyramid, including the flaps, is approximately 28.9 cm².