find the lateral area and the total surface area of the right prism if the base of the prism is a right triangle whole legs meausure 5in and 12 in . and the altitude of the prism measure 8 in.

To find the lateral area and total surface area of a right prism, you need to know the dimensions of the prism.

In this case, the base of the prism is a right triangle with legs measuring 5 inches and 12 inches, and the altitude of the prism measures 8 inches.

The lateral area of a prism is the sum of the areas of all the lateral faces (excluding the top and bottom faces). To find it:

1. Calculate the area of the right triangle base using the formula for the area of a triangle: Area = (1/2) * base * height.
Given that the legs of the right triangle measure 5 inches and 12 inches, the area of the base is (1/2) * 5 * 12 = 30 square inches.

2. Multiply the base area by the height of the prism to get the lateral area:
Lateral Area = base area * height = 30 square inches * 8 inches = 240 square inches.

So, the lateral area of the prism is 240 square inches.

The total surface area of a prism includes the sum of the areas of all the faces, including the top and bottom faces. To find it:

1. Calculate the area of the right triangle base again, which is 30 square inches.

2. Calculate the area of the lateral faces, which we already found to be 240 square inches.

3. Calculate the area of the top and bottom faces. The top face is a right triangle with sides measuring 5 inches and 12 inches, so its area is (1/2) * 5 * 12 = 30 square inches. Similarly, the bottom face has the same area since the base is identical.

4. Add up the areas of all the faces to find the total surface area:
Total Surface Area = base area + lateral area + top face area + bottom face area
= 30 square inches + 240 square inches + 30 square inches + 30 square inches
= 330 square inches.

Therefore, the total surface area of the prism is 330 square inches.

To find the lateral area and total surface area of the right prism, we need to first calculate the area of the base triangle.

Step 1: Calculate the area of the base triangle.
The base triangle is a right triangle with legs measuring 5 inches and 12 inches, and the altitude measures 8 inches. The area of a right triangle is given by the formula A = (1/2) * base * height.

Let's substitute the values into the formula:
A = (1/2) * 5in * 12in
A = (1/2) * 60in^2
A = 30in^2

Step 2: Calculate the lateral area.
The lateral area of a prism is the sum of the areas of the lateral faces. In this case, the lateral faces are rectangles with sides matching the legs of the base triangle and the height of the prism.

The lateral area is given by the formula A = perimeter * height.

The perimeter of a rectangle is given by the formula P = 2 * (length + width).

In this case, the length and width of the rectangle are 5 inches and 12 inches, respectively. The height of the prism is 8 inches.

Substituting the values into the formula:
P = 2 * (5in + 12in)
P = 2 * 17in
P = 34in

Now, let's calculate the lateral area:
A = 34in * 8in
A = 272in^2

Step 3: Calculate the total surface area.
The total surface area of a prism is the sum of the areas of all faces.

In this case, we have two identical triangular bases and three rectangular lateral faces.

The area of each triangular base is 30in^2 (from Step 1).

The area of all three rectangular lateral faces is 272in^2 (from Step 2).

Therefore, the total surface area is:
A = (2 * 30in^2) + (3 * 272in^2)
A = 60in^2 + 816in^2
A = 876in^2

So, the lateral area of the right prism is 272 square inches and the total surface area is 876 square inches.