Surface area of prisms and cylinders

1. LA = 92 sq. m.,  SA = 132 sq. m.

2. LA = 460 sq. in.,  SA = 700 sq. in.

3. LA = 118 sq. in.,  SA = 168 in. sq.

4. LA = 468 sq. cm.,  SA = 828 sq. cm.

Step-by-step explanation:

The lateral area is the area of the non-base faces. The surface area is the area of all the faces (so lateral area + area of base).

1.

Lateral area:

There are 2 same triangles each with height and base of 3 and 4, respectively. So

Area of triangle = 1/2 * base * height = 1/2 * 4 * 3 = 6

Since there are 2, so 6 * 2 = 12

Now, the slant height is gotten by using pythagorean theorem on the triangular face.

If you look at the side facing away, it is a rectangle with length 10 and width 5, so area of rectangle is length * width = 10 * 5  = 50

The side facing towards us is also a rectangle with dimensions 3 and 10. So area is 3 * 10 = 30

The lateral area is 30 + 12 + 50 = 92

Surface Area:

This is the lateral area we got before (92) + the area of the base.

The base is a rectangle with dimensions 4 and 10. So area = 4 * 10 = 40

The surface area is 92 + 40 = 132

2.

Lateral area:

all the faces are rectangles.

The dimensions of two side faces are 10 and 8. So area would be 2 * (10 * 8) = 2 * 80 = 160

The dimensions of back side and front side is 15 and 10. So area would be 2 * (15 * 10) = 2 * 150 = 300

The lateral area would be 160 + 300 = 460

Surface area:

the surface area is the lateral area (580) + the area of the bases (which is also a rectangle). The dimensions of the bottom and top are  15 and 8. So area of bottom and top  is  2(15 * 8) = 240

Thus, surface area = 460 + 240 = 700

3.

Lateral Area:

There are two rectangles (left and right) with dimensions 8 x 5 and 6 x 5. So area would be 8 * 5 = 40 and 6 * 5 = 30. Area = 40 + 30 = 70

There are two triangles (back and front) with base of 6 and height of 8. The area of both are:

(1/2 * 6 * 8 ) * 2 = 48

Lateral area is 70 + 48 = 118

Surface area:

The surface are would be lateral area (118) PLUS the base, which is a rectangle with dimensions 5 and 10. So area = 5 * 10 = 50.

Total surface area = 118 + 50 = 168

4.

lateral area:

there are 2 rectangles (left and right) with dimensions 10 x 12, and 9 x 12. So the total area would be:

(10*12) + (9*12) = 228

The front and back are trapezoids. The area of trapezoid is 1/2 (base1 + base2) * height.

The height is 8 and both the bases are 10 and 20. Plugging it into the formula, we get are of 2 of the trapezoids to be:

2 * (1/2(base1+base2)*height)

2 * (1/2(10+20)*8)

= 240

The lateral area = 228 + 240 = 468

surface area:

it will be 468 (lateral area) PLUS the area of the bases. The bottom base is a rectangle with dimensions 20 and 12. So the area of the base is 20 * 12 = 240

The top is a rectangle with dimensions 12 and 10. So area = 12 * 10 = 120

Surface area = 468 + 240 + 120  = 828

c. (3,2,6) is the answer.