What is the surface area of a square pyramid with base length of 10 cm and slant height of 20 cm
A square pyramid has a square base and four triangular faces. To find the surface area, we need to find the area of the base and the combined areas of the four triangular faces.
The area of the base can be found by squaring the length of one side of the square base:
Area of base = (side length)^2
Area of base = (10 cm)^2
Area of base = 100 cm^2
To find the area of one triangular face, we can use the formula:
Area of triangle = (base length * height) / 2
In this case, the base length is the same as the side length of the square base, which is 10 cm. The height of the triangular face can be found using the Pythagorean theorem:
height = sqrt((slant height)^2 - (base length / 2)^2)
height = sqrt((20 cm)^2 - (10 cm / 2)^2)
height = sqrt(400 cm^2 - 25 cm^2)
height = sqrt(375 cm^2)
height ≈ 19.36 cm
Area of one triangular face = (10 cm * 19.36 cm) / 2
Area of one triangular face ≈ 96.80 cm^2
Since there are four triangular faces, the combined area of the triangular faces is:
Combined area of triangular faces = 4 * 96.80 cm^2
Combined area of triangular faces ≈ 387.20 cm^2
Finally, to find the surface area of the square pyramid, we add the area of the base to the combined area of the triangular faces:
Surface area = Area of base + Combined area of triangular faces
Surface area = 100 cm^2 + 387.20 cm^2
Surface area ≈ 487.20 cm^2
Therefore, the surface area of the square pyramid is approximately 487.20 square cm.
To find the surface area of a square pyramid, we need to find the area of the base and the lateral faces.
1. To find the area of the base, we can use the formula for the area of a square:
Area of base = side length * side length
Given that the base length is 10 cm:
Area of base = 10 cm * 10 cm = 100 cm^2
2. To find the area of the lateral faces, we can use the formula:
Area of each lateral face = (1/2) * perimeter of base * slant height
The perimeter of the base can be calculated by multiplying the side length by 4, since it is a square:
Perimeter of base = 4 * side length = 4 * 10 cm = 40 cm
Given that the slant height is 20 cm:
Area of each lateral face = (1/2) * 40 cm * 20 cm = 400 cm^2
3. Since a square pyramid has four identical lateral faces, we can calculate the total area of the lateral faces by multiplying the area of one lateral face by 4:
Total area of lateral faces = 4 * 400 cm^2 = 1600 cm^2
4. The surface area of the square pyramid is the sum of the area of the base and the total area of the lateral faces:
Surface area = Area of base + Total area of lateral faces
Surface area = 100 cm^2 + 1600 cm^2 = 1700 cm^2
Therefore, the surface area of the square pyramid is 1700 cm^2.