Find the surface area of a rectangular pyramid with these measurements l =15 m, w = 12m and h = 10m
The formula for the surface area of a rectangular pyramid is given by:
Surface Area = lw + lh + wh + (l * slant height)
l = length = 15 m
w = width = 12 m
h = height = 10 m
To find the slant height, we can use the Pythagorean theorem:
Slant height = √((w/2)² + h²)
Slant height = √((12/2)² + 10²)
Slant height = √(6² + 100)
Slant height = √(36 + 100)
Slant height = √136 ≈ 11.66 m
Now, we can calculate the surface area:
Surface Area = lw + lh + wh + (l * slant height)
Surface Area = (15 * 12) + (15 * 10) + (12 * 10) + (15 * 11.66)
Surface Area = 180 + 150 + 120 + 174.9
Surface Area ≈ 624.9 square meters
Therefore, the surface area of the rectangular pyramid is approximately 624.9 square meters.
To find the surface area of a rectangular pyramid, you need to calculate the area of each face and then sum them up. The rectangular pyramid has one rectangular base and four triangular faces.
Step 1: Calculate the area of the rectangular base.
The formula for the area of a rectangle is length multiplied by width.
Area of rectangular base = l * w = 15m * 12m = 180m²
Step 2: Calculate the area of each triangular face.
The formula for the area of a triangle is 1/2 multiplied by base multiplied by height.
The base of each triangle is the same as the length of the rectangular base.
The height of each triangle is the slant height, and it can be calculated using the Pythagorean theorem.
The slant height (s) can be found using the formula:
s = √(l/2)^2 + h^2
Let's calculate the slant height (s):
s = √(15/2)^2 + 10^2
s = √(225/4) + 100
s ≈ √(56.25) + 100
s ≈ 7.5 + 100
s ≈ 107.5
Now we can calculate the area of each triangular face:
Area of first triangular face = 1/2 * l * s = 1/2 * 15m * 107.5m = 806.25m²
Area of second triangular face = 1/2 * w * s = 1/2 * 12m * 107.5m = 645m²
Area of third triangular face = 806.25m² (same as first triangular face because they share the same base)
Area of fourth triangular face = 645m² (same as second triangular face because they share the same base)
Step 3: Sum up the areas of all faces.
Surface area of the rectangular pyramid = area of rectangular base + 2 * (area of each triangular face)
Surface area = 180m² + 2 * (806.25m² + 645m²)
Surface area = 180m² + 2 * 1451.25m²
Surface area = 180m² + 2902.5m²
Surface area ≈ 3082.5m²
Therefore, the surface area of the rectangular pyramid is approximately 3082.5 square meters.