What is the surface area of a square pyramid if the base edge = 68 in and the slant height = 125 in
To find the surface area of a square pyramid, we need to calculate the area of the base and the area of the four triangles that make up the lateral faces.
1. Calculate the area of the base:
The base of the pyramid is a square, so the area is equal to the square of the length of one side. In this case, the base edge is 68 inches, so the area of the base is 68^2 = 4624 square inches.
2. Calculate the area of the lateral faces:
Each lateral face of the pyramid is an isosceles triangle. The slant height is given as 125 inches. To calculate the base of the triangle (which is equal to the side length of the square base), we can use the Pythagorean theorem:
(base)^2 = (slant height)^2 - (height)^2
(base)^2 = 125^2 - 68^2
(base)^2 = 15625 - 4624
(base)^2 = 11001
base = √11001 ≈ 104.83 inches
Now we can calculate the area of one triangle:
Area of a triangle = (base * height) / 2
Area of one triangle = (104.83 * 68) / 2 = 3561.22 square inches
Since the pyramid has four identical triangular faces, the total area of the lateral faces is 4 * 3561.22 = 14244.88 square inches.
3. Calculate the total surface area:
The total surface area is the sum of the area of the base and the area of the lateral faces:
Total surface area = area of base + area of lateral faces
Total surface area = 4624 + 14244.88 = 18868.88 square inches
Therefore, the surface area of the square pyramid is approximately 18868.88 square inches.
To find the surface area of a square pyramid, you need to calculate the sum of the areas of its base and its lateral faces.
1. Calculate the area of the base:
The base of the square pyramid is a square, so to find its area, you need to square the length of one side.
Given that the base edge is 68 in, the area of the base is 68 in × 68 in.
Area of the base = 68 in × 68 in = 4624 in²
2. Calculate the area of the lateral faces:
The lateral faces of a square pyramid are triangular in shape. To find their areas, you need to multiply the perimeter of the base by half the slant height.
The perimeter of the base square is 4 times the length of one side, so it is 4 × 68 in.
The slant height is given as 125 in.
Area of one lateral face = (perimeter of base × slant height) / 2
Area of one lateral face = (4 × 68 in × 125 in) / 2
Area of one lateral face = 17000 in²
3. Calculate the total surface area:
To find the total surface area, add the area of the base to the sum of the areas of the lateral faces.
Total surface area = Area of the base + (Area of one lateral face × 4)
Total surface area = 4624 in² + (17000 in² × 4)
Total surface area = 4624 in² + 68000 in²
Total surface area = 72624 in²
Therefore, the surface area of the square pyramid is 72624 square inches.