The bases of a right prism are parallelograms with sides a=10 cm, b=6 cm, and altitude of the parallelogram towards side a, ha = 3 cm. Find the surface area of the parallelepiped, if the height of the prism is h=12 cm.
444 cm^2
To find the surface area of the right prism, we need to find the areas of the two bases and the area of the lateral faces.
First, let's find the area of each base. We have a parallelogram with sides a = 10 cm, b = 6 cm, and altitude ha = 3 cm.
The area of a parallelogram is given by the formula: A = base * height.
For the first base, base1 = a = 10 cm and height1 = ha = 3 cm.
So, the area of the first base is A1 = base1 * height1 = 10 cm * 3 cm = 30 cm².
For the second base, the base2 is also equal to a = 10 cm, and the height2 is the same as height1.
So, the area of the second base is A2 = base2 * height2 = 10 cm * 3 cm = 30 cm².
To find the lateral faces' area, we need to find the perimeter of the base and multiply by the height of the prism.
The perimeter of a parallelogram is given by the formula: P = 2 * (side a + side b).
For our base, the perimeter is P = 2 * (10 cm + 6 cm) = 2 * 16 cm = 32 cm.
Now, we can find the surface area of the lateral faces.
The area of one lateral face is given by: A = perimeter of the base * height of the prism.
So, the area of one lateral face is A = 32 cm * 12 cm = 384 cm².
Since we have two identical lateral faces, the total area of the lateral faces is 2 * A = 2 * 384 cm² = 768 cm².
Finally, to find the surface area of the parallelepiped, we sum the areas of the two bases and the total area of the lateral faces.
Surface Area = 2 * (A1 + A2) + Total lateral area
Surface Area = 2 * (30 cm² + 30 cm²) + 768 cm²
Surface Area = 2 * 60 cm² + 768 cm²
Surface Area = 120 cm² + 768 cm²
Surface Area = 888 cm²
Therefore, the surface area of the parallelepiped is 888 cm².
To find the surface area of the parallelepiped, we need to consider the lateral surface area and the surface area of the two bases.
1. Lateral Surface Area:
The lateral surface area of a right prism can be calculated by multiplying the perimeter of the base by the height of the prism. In this case, the base is a parallelogram, so we need to find its perimeter.
The perimeter of a parallelogram is given by the formula: P = 2(a+b), where a and b are the lengths of the sides. In this case, a = 10 cm and b = 6 cm, so the perimeter of the parallelogram is:
P = 2(10 cm + 6 cm) = 2(16 cm) = 32 cm
Now, we can calculate the lateral surface area using the formula: L = P * h, where L is the lateral surface area and h is the height of the prism.
L = 32 cm * 12 cm = 384 cm²
2. Base Surface Area:
The surface area of a parallelogram is given by the formula: A = a * ha, where a is the length of the base and ha is the altitude of the base. In this case, a = 10 cm and ha = 3 cm. Since there are two bases in a prism, we need to calculate the surface area of the bases separately and then multiply by 2.
Surface area of one base = 10 cm * 3 cm = 30 cm²
Surface area of both bases = 2 * 30 cm² = 60 cm²
3. Total Surface Area:
Finally, we can calculate the total surface area of the parallelepiped by adding the lateral surface area and the surface area of the bases.
Total surface area = L + 2 * A
Total surface area = 384 cm² + 60 cm²
Total surface area = 444 cm²
So, the surface area of the parallelepiped is 444 cm².
the area of each base is just 10*3=30
Then you have four rectangles. Two that are 10x12 and two that are 6x12
Now just add up all those six areas.