Calculate the lateral and surface area of a prism whose height is 12 cm and whose base is a rectangle with dimensions 5 cm by 7 cm.
To calculate the lateral area and surface area of a prism, we need to understand the definitions of these terms.
The lateral area of a prism refers to the total area of all its lateral faces, excluding the bases. In this case, the prism has three lateral faces.
The surface area of a prism, however, includes the lateral faces as well as the bases. To calculate the surface area, we need to consider the area of the bases as well.
Now let's calculate the lateral area:
1. Start by finding the perimeter of the base, which is a rectangle. The formula for the perimeter of a rectangle is P = 2(length + width).
In this case, the length is 7 cm and the width is 5 cm. So, the perimeter is P = 2(7 + 5) = 2(12) = 24 cm.
2. Multiply the perimeter by the height of the prism to find the lateral area. The formula for the lateral area of a prism is A = perimeter × height.
In this case, the height is 12 cm. So, the lateral area is A = 24 cm × 12 cm = 288 cm².
Now let's calculate the surface area:
1. Find the area of each base, which is a rectangle. The formula for the area of a rectangle is A = length × width.
In this case, the length is 7 cm and the width is 5 cm, so the area of each base is A = 7 cm × 5 cm = 35 cm².
2. Multiply the area of each base by 2 (since there are two bases) to get the total area of the bases.
In this case, the total area of the bases is 2 × 35 cm² = 70 cm².
3. Add the lateral area and the total area of the bases to find the surface area of the prism.
In this case, the surface area is 288 cm² (the lateral area) + 70 cm² (the total area of the bases) = 358 cm².
Therefore, the lateral area of the prism is 288 cm² and the surface area is 358 cm².