The base of a regular pyramid is a square whose sides have a length 8 cm. If a lateral edge of this pyramid has length 20 cm, what is the surface area of the pyramid?
Look at one of the triangles, which is isosceles with a base of 8 and each of the equal sides of length 20
Height of triangle: let it be h
h^2 + 4^2 = 20^2
h^2 = 384
h = √384
Area of one triangle = (1/2)(8)(√384) = 4√384
Assuming the surface area consists only of the 4 triangles,
surface area = 4(4√384) or 16(8√6) = 128√6
Thanks, but I need the base too - that I can do by myself!
One edge of a right square pyramid is 6cm long. If the length of the lateral edge is 8cm, then find: its total surface area
To find the surface area of the pyramid, we need to calculate the area of each individual face and then sum them up.
First, let's find the area of the base. Since the base is a square, we can use the formula for the area of a square: A = side^2. In this case, the side length is 8 cm, so the area of the base is 8^2 = 64 sq cm.
Next, let's calculate the area of each triangular face. We know that the lateral edge length is 20 cm, and the height of each triangle can be found using the Pythagorean theorem. The height is the perpendicular distance from the apex to the base of the pyramid. Since the base is a square, the height is also the height of the triangular face.
Using the Pythagorean theorem, we can find the height. The hypotenuse is the lateral edge length (20 cm), and one of the legs is half the length of the base (8 cm / 2 = 4 cm). Let's call the other leg x. Using the theorem, we have:
20^2 = 4^2 + x^2
400 = 16 + x^2
x^2 = 400 - 16
x^2 = 384
x ≈ √384
x ≈ 19.60 cm
Now we can find the area of each triangular face using the formula for the area of a triangle: A = 1/2 * base * height. Since the base of each face is 8 cm (the side length of the base square), and the height is approximately 19.60 cm, the area of each triangular face is approximately 1/2 * 8 cm * 19.60 cm = 78.4 sq cm.
Since the pyramid has 4 triangular faces, the total surface area of the pyramid is 4 * 78.4 sq cm = 313.6 sq cm.
Therefore, the surface area of the pyramid is approximately 313.6 square cm.