1. A hexagonal prism 6ft tall with a regular base measuring 9ft on each and an apothegm of length 7.8ft.
2. A prism 2m tall. The base is trapezoid whose parallel sides measure 7m and 3m. The other sides are each 4m. The altitude of the trapezoid measures 3.5m.
1. To find the surface area of a hexagonal prism, you need to calculate the areas of each face and then add them together.
- First, calculate the area of the hexagonal base:
The area of a regular hexagon can be found using the formula: A = (3√3 / 2) * s^2, where A is the area and s is the length of one side.
Substituting the given values, we get:
A = (3√3 / 2) * (9ft)^2
- Next, calculate the area of each triangular face:
The base of each triangle is the side length of the hexagon, which is 9ft.
The height of each triangle is the length of the apothem, which is 7.8ft.
The formula to calculate the area of a triangle is: A = (1 / 2) * b * h, where A is the area, b is the base, and h is the height.
Thus, the area of each triangle is: A = (1 / 2) * 9ft * 7.8ft
- Lastly, multiply the area of each triangular face by 6 (since there are 6 triangular faces) and add it to the area of the hexagonal base.
2. To find the surface area of this prism, you also need to calculate the areas of all its faces and then add them together.
- First, calculate the area of the trapezoidal base:
The formula to calculate the area of a trapezoid is: A = (1 / 2) * (a + b) * h, where A is the area, a and b are the lengths of the parallel sides, and h is the altitude.
Substituting the given values, we get:
A = (1 / 2) * (7m + 3m) * 3.5m
- Next, calculate the area of each rectangular face:
The height of each rectangular face is the same as the height of the prism, which is 2m.
The length of each rectangular face is the same as the sides of the trapezoid, which is 4m.
The formula to calculate the area of a rectangle is: A = l * w, where A is the area, l is the length, and w is the width.
Thus, the area of each rectangular face is: A = 4m * 2m
- Lastly, multiply the area of each rectangular face by 2 (since there are 2 rectangular faces) and add it to the area of the trapezoidal base.
1. To find the surface area of a hexagonal prism, you need to calculate the areas of all its faces and add them up.
First, let's find the area of the hexagonal base. Since the base is a regular hexagon, we can use the formula:
Area = (3 * sqrt(3) * s²) / 2
Where s is the length of a side of the hexagon. In this case, the side length is 9ft, so the area of the base would be:
Area_base = (3 * sqrt(3) * 9²) / 2
Next, let's find the area of one of the lateral faces. Each lateral face of a hexagonal prism is a rectangle, and its dimensions are the height of the prism (6ft) and the perimeter of the base. Since the base is a regular hexagon, the perimeter is equal to 6 times the side length (9ft):
Perimeter_base = 6 * 9ft
Area_lateral = 6ft * (Perimeter_base)
Finally, let's find the area of the top and bottom faces. Since they are regular hexagons, we use the same formula as we did for the base. The side length is still 9ft:
Area_top_bottom = (3 * sqrt(3) * 9²) / 2
To find the total surface area, we add up all the areas we calculated:
Total_surface_area = 2 * (Area_base) + (Area_lateral) + 2 * (Area_top_bottom)
Now, substitute the values into the formula and calculate the result.
2. To find the surface area of this prism, we need to calculate the areas of all its faces and add them up.
First, let's find the area of the trapezoidal base. The formula for the area of a trapezoid is:
Area = (1/2) * (b₁ + b₂) * h
Where b₁ and b₂ are the lengths of the parallel sides, and h is the altitude (the height of the trapezoid). In this case, b₁ is 7m, b₂ is 3m, and h is 3.5m:
Area_base = (1/2) * (7m + 3m) * 3.5m
Next, let's find the areas of the two triangular faces. Each one is a right triangle, and its base is one of the non-parallel sides of the trapezoid (which measures 4m) and its height is the height of the prism (2m):
Area_triangular = (1/2) * 4m * 2m
Finally, let's find the areas of the two rectangular faces. Each one has a length equal to the perimeter of the trapezoid's base (b₁ + b₂ + 2 × side), and a height equal to the height of the prism (2m):
Perimeter_base = 7m + 3m + 2 × 4m
Area_rectangular = (Perimeter_base) * 2m
To find the total surface area, we add up all the areas we calculated:
Total_surface_area = (Area_base) + 2 * (Area_triangular) + 2 * (Area_rectangular)
Now, substitute the values into the formula and calculate the result.