Find the lateral area of the square pyramid 22 height 8 base 8 width.
each face of the pyramid is an isosceles triangle with base 8 and height √(22^2+4^2)
To see this, view the pyramid from the side. Drop an altitude to the center of the base, and you have a right triangle whose hypotenuse is the height of the triangular face.
To find the lateral area of a square pyramid, you need to calculate the sum of the areas of all the lateral faces.
A square pyramid has four lateral faces that are congruent triangles. The formula to find the area of a triangle is:
Area = (1/2) * base * height
In this case, the base of each triangle is one side of the square base of the pyramid, and the height is the slant height of the pyramid. The slant height can be found using the Pythagorean theorem, as the hypotenuse of a right triangle with the height and one side of the base as its legs.
Using the given measurements:
Height (h) = 22
Base side length (b) = 8
To find the slant height (s), calculate:
s = sqrt(h^2 + (b/2)^2)
s = sqrt(22^2 + (8/2)^2)
s = sqrt(484 + 16)
s = sqrt(500)
s = 10*sqrt(5)
Now that we have the slant height, we can calculate the area of one lateral face:
Area of a lateral face = (1/2) * b * s
= (1/2) * 8 * 10*sqrt(5)
= 40 * sqrt(5)
Since there are four lateral faces, we multiply the area of one face by four to find the total lateral area:
Total lateral area = 4 * 40 * sqrt(5)
= 160 * sqrt(5)
Therefore, the lateral area of the given square pyramid with a height of 22, base side of 8, and width of 8 is 160 * sqrt(5).