A regular square pyramid has a height of 2 and lateral edges of length 10. What is the lateral area and total surface area of the pyramid?
To find the lateral area and total surface area of a regular square pyramid, we need to understand the different components of the pyramid and their formulas.
First, let's discuss the components of the pyramid:
1. Base: In a regular square pyramid, the base is a square.
2. Slant height: The slant height is the distance from any corner of the base to the apex (top) of the pyramid.
3. Height: The height is the vertical distance from the center of the base to the apex.
Now, let's calculate the lateral area and total surface area step by step:
Step 1: Find the slant height.
In a square pyramid, the slant height is calculated using the formula:
slant height = √(height^2 + (side length / 2)^2)
Given that the height of the pyramid is 2 and the side length is 10, we can substitute these values into the formula:
slant height = √(2^2 + (10 / 2)^2) = √(4 + 25) = √29
Step 2: Calculate the lateral area.
The lateral area of a pyramid is the sum of the areas of the lateral faces. In a regular square pyramid, all the lateral faces are triangles.
The formula to calculate the area of a triangle is:
area = (base * height) / 2
Since all the triangles have the same base and height, we can calculate the area of one triangle and then multiply it by 4 to get the lateral area of the pyramid.
The base of the triangle is equal to the side length of the square base. In this case, the side length is 10.
The height of the triangle is the slant height of the pyramid, which we found to be √29.
Using these values, we can calculate the area of one triangle:
area of one triangle = (10 * √29) / 2 = (10√29) / 2 = 5√29
Finally, we multiply the area of one triangle by 4 to obtain the lateral area of the pyramid:
lateral area = 4 * (5√29) = 20√29
Step 3: Calculate the total surface area.
The total surface area of a pyramid is the sum of the lateral area and the area of the base (which is a square in this case).
The area of the base can be found by squaring the side length. In this case, it is 10 * 10 = 100.
We add the lateral area and the area of the base to get the total surface area:
total surface area = lateral area + area of the base = 20√29 + 100 = 100 + 20√29
Therefore, the lateral area of the pyramid is 20√29 square units, and the total surface area is 100 + 20√29 square units.
If the diagonal of the base has length 2d, then
d^2 + 2^2 = 10^2
The sides of the base have length 2b, where
b^2 + b^2 = d^2
Each triangular face has slant height h, where
b^2 + 2^2 = h^2
Now you can find the area of the base, and the area of each face.