Question
what is the SA of a right square pyramid the length, width, and height are all 5
Answers
GPT 3.5
To find the surface area of a right square pyramid, you need to calculate the area of the base (the square) and the area of the four triangular faces.
Given that the length, width, and height of the right square pyramid are all 5, the area of the base square can be calculated as:
Area of base = length x width
Area of base = 5 x 5
Area of base = 25 square units
Next, we need to calculate the area of each of the four triangular faces. Since the base is a square, the triangles are isosceles triangles with base and height equal to the slant height of the pyramid. The slant height, which is the hypotenuse of a right triangle with legs equal to the height of the pyramid and half of the length of the base, can be calculated as:
Slant height = √((height)^2 + ((length/2)^2))
Slant height = √(5^2 + (2.5)^2)
Slant height = √(25 + 6.25)
Slant height = √31.25
Slant height ≈ 5.59
The area of each triangular face can be calculated as:
Area of one triangular face = (1/2) x base x height
Area of one triangular face = (1/2) x 5 x 5.59
Area of one triangular face ≈ 14 square units
Since there are four triangular faces, the total surface area of the right square pyramid can be calculated as:
SA = Area of base + 4 x Area of one triangular face
SA = 25 + 4 x 14
SA = 25 + 56
SA = 81 square units
Therefore, the surface area of a right square pyramid with length, width, and height all equal to 5 units is 81 square units.
Given that the length, width, and height of the right square pyramid are all 5, the area of the base square can be calculated as:
Area of base = length x width
Area of base = 5 x 5
Area of base = 25 square units
Next, we need to calculate the area of each of the four triangular faces. Since the base is a square, the triangles are isosceles triangles with base and height equal to the slant height of the pyramid. The slant height, which is the hypotenuse of a right triangle with legs equal to the height of the pyramid and half of the length of the base, can be calculated as:
Slant height = √((height)^2 + ((length/2)^2))
Slant height = √(5^2 + (2.5)^2)
Slant height = √(25 + 6.25)
Slant height = √31.25
Slant height ≈ 5.59
The area of each triangular face can be calculated as:
Area of one triangular face = (1/2) x base x height
Area of one triangular face = (1/2) x 5 x 5.59
Area of one triangular face ≈ 14 square units
Since there are four triangular faces, the total surface area of the right square pyramid can be calculated as:
SA = Area of base + 4 x Area of one triangular face
SA = 25 + 4 x 14
SA = 25 + 56
SA = 81 square units
Therefore, the surface area of a right square pyramid with length, width, and height all equal to 5 units is 81 square units.