Question

The composite figures below are made of right prisms.

Note: figures not drawn to scale.

Place the figures in order from least surface area to greatest surface area.

Answer 1

Least surface area to greatest surface area:

Blue - Red - Green

Explanation:

All the surfaces of the shapes are rectangles, then their surface is computed as:

Surface = Length*Width

Blue figure

Area of the top:

5*5 + 5*5 + 5*5 + 5*5 + 5*5 = 125 ft^2

Area of the sides:

2*(5*10) = 100 ft^2

Area of the bottom:

10*5 = 50 ft^2

Area of the front and the back

2*(5*5 + 15*5) = 200 ft^2

Total: 125 + 100 + 50 + 200 = 475 ft^2

Red figure

Area of the top:

5*5 + 13*5 = 90 ft^2

Area of the left side:

5*10 = 50 ft^2

Area of the right side:

5*5 = 25 ft^2

Area of the bottom:

17*5 = 85 ft^2

Area of the front and the back:

2*(17*5 + 12*5/2 + 5*5) = 280 ft^2

Total: 90 + 50 + 25 + 85 + 280 = 530 ft^2

Green figure

Area of the top:

12.5*11 + 13*11 = 280.5 ft^2

Area of the left side:

11*5 = 55 ft^2

Area of the bottom:

(12.5+12)*11 = 269.5 ft^2

Area of the front and the back:

2*(12.5*5 + 12*5/2) = 185 ft^2

Total: 280.5 + 55 + 269.5 + 185 = 790 ft^2

Answer 2

Figure 1's surface area: 550
`feet^2`

Figure 2's surface area: 530
`feet^2`

Figure 3's surface area: 790
`feet^2`

So, the order of the figures' surface areas is: Figure 2 < Figure 1 < Figure 3.

Explanation:

Bearing in mind that the surface area of a 3D object is the total area of its surface (that is the sum of the areas of its faces), we can count the number of distinct shapes in each figure, calculate their areas and add them up. Be careful not to add the faces that are "inside".

Figure 1

Its surface area consists of 13 squares of 5x5
`feet^2`and 3 rectangles of 15x5
`feet^2`. So, its surface area is (13x25 + 3x75)
`feet^2 = 550feet^2`

Figure 2

Its surface area consists of 6 squares of 5x5
`feet^2`, 3 rectangles of 17x5
`feet^2`, 1 rectangle of 13x5
`feet^2` and 2 triangles of base 12 feet and height 5 feet. So, its surface area is (17x5x3+6x5x5+13x5+12x5x2:2)
`feet^2 = 530feet^2`

Figure 3

Its surface area consists of 2 rectangles of 12.5x5
`feet^2`, 1 rectangle of 11x5
`feet^2`, 2 triangles of base 12 feet and height 5 feet, 1 rectangle of 11x13
`feet^2`, 2 rectangles of 12.5x11
`feet^2` and 1 rectangle of 12x11
`feet^2`.

So, its surface area is (2x12.5x5+11x5+2x12x5:2+11x13+2x12.5x11+12x11)
`feet^2 = 790feet^2`