Practice on calculating the total and surrounding area of a cube

### 1.1. Solving exercises in the textbook Practice page 112

**Lesson 1 Textbook page 112**

Calculate the perimeter and total area of a cube with sides 2m 5cm.

__Solution guide:__

Exchange: 2m 5cm = 2.05m

The perimeter of the given cube is:

2.05 × 2.05

$\times $ 4 = 16.81 (m^{2})

The total area of the given cube is:

2.05

$\times $2.05 × 6 = 25,215 (m2)

Answer: Surrounding area: 16.81(m .)2)

Total area: 25,215(m .)2)

Lesson 2 Textbook page 112

Which of the following pieces of cardboard can fold a cube?

__Solution guide:__

– It is easy to see that shape 1 cannot be folded into a cube.

– With figure 2, when we fold the row of 4 squares below into 4 surrounding faces, the 2 squares above will overlap, not forming an upper bottom surface and a bottom bottom surface. Therefore figure 2 cannot fold figure 1 into a cube.

– Figure 3 and figure 4 can both be folded into a cube because when we fold the series of 4 squares in the middle into 4 surrounding faces, the 2 upper and lower squares will form two upper and lower bottom faces.

So each piece of cardboard in Figures 3 and 4 can be folded into a cube.

Lesson 3 Textbook page 112

Correct write D, false write S:

a) The area around cube A is twice the area around cube B

b) The area around cube A is four times the area around cube B

c) The total area of cube A is twice the total area of cube B

d) The total area of cube A is four times the total area of cube B

__Solution guide:__

The perimeter of cube A is:

10 × 10 × 4 = 400(cm^{2})

The total area of cube A is:

10 × 10 × 6 = 600(cm2)

The perimeter of cube B is:

5 × 5 × 4 = 100(cm2)

The total area of cube B is:

5 × 5 × 6 = 150(cm2)

The perimeter of cube A is the number of times the surrounding area of cube B is:

400 : 100 = 4 (times)

The total area of cube A is times more than the total area of cube B:

600 : 150 = 4 (times)

So the surrounding (total) area of figure A is 4 times the surrounding (total) area of figure B.

We have the result:

a) S b) Yes c) S d) RED

### 1.2. Solving exercises in textbooks General practice page 113, 114

Lesson 1 Textbook page 112

Calculate the perimeter and total area of a rectangular box with:

a) Length 2.5m, width 1.1m and height 0.5m.

b) Length 3m, width 15dm and height 9dm.

__Solution guide:__

a) The area × perimeter of the rectangular box is:

(2.5 + 1.1) × 2 × 0.5 = 3.6 (m .)^{2})

The area of the base of the rectangular box is:

2.5 × 1.1 = 2.75 (m2)

The total area of the rectangular box is:

3.6 + 2.75 × 2 = 9.1 (m2)

b) Change: 3m = 30dm

The area × perimeter of the rectangular box is:

(30 + 15) × 2 × 9 = 810 (dm2)

The area of the base of the rectangular box is:

30 × 15 = 450 (dm2)

The total area of the rectangular box is:

810 + 450 × 2 = 1710 (dm2)

Answer: a) 3.6m2 ; 9.1m2

b) 810dm2; 1710dm2

Lesson 2 Textbook page 113

Write the appropriate measurement in the blank:

__Solution guide:__

+) Column (1):

The perimeter of the rectangular box is:

(4 + 3) × 2 × 5 = 70(m^{2})

The area of the base of the rectangular box is:

4 × 3= 12 (m2)

$$The total area of the rectangular box is:

70 + 12 × 2 = 94(m2)

$$+) Column (2):

The bottom half circumference is: 2 : 2 = 1(cm)

The width of the rectangular box is:

\(1 – \frac{3}{5} = \frac{2}{5}(cm)\)

The perimeter of the rectangular box is:

\(2 \times \frac{1}{3} = \frac{2}{3}(c{m^2})\)

The area of the base of the rectangular box is:

\(\frac{3}{5} \times \frac{2}{5} = \frac{6}{{25}}(c{m^2})\)

The total area of the rectangular box is:

\(\frac{2}{3} + \frac{6}{{25}} \times 2 = \frac{{86}}{{75}}(c{m^2})\)

+) Column (3)

We see that the rectangular box here has three equal dimensions, so it is a cube.

The perimeter of the bottom surface is:

0.4 × 4 = 1.6(dm)

The perimeter of the given cube is:

0.4 × 0.4 × 4 = 0.64(dm^{2})

The total area of the given cube is:

0.4 × 0.4 × 6 = 0.96(dm^{2})

We have the following result:

Lesson 3 Textbook page 114

A cube has side 4cm, if the side of the cube is folded 3 times, how many times will its surrounding area and total area increase? Why ?

__Solution guide:__

The new cube has sides of:

4 x 3 = 12 (cm)

The perimeter of the new cube is:

12 x 12 x 4 = 576 (cm^{2})

The perimeter of the old cube is:

4 x 4 x 4 = 64 (cm2)

The surrounding area of the new cube is multiplied by the number of times the surrounding area of the old cube is:

576 : 64 = 9 (times)

The total area of the new cube is:

12 x 12 x 6 = 864 (cm2)

The total area of the old cube is:

4 x 4 x 6 = 96 (cm2)

The total area of the new cube is many times greater than the total area of the old cube:

864 : 96 = 9 (times)

If the side of the cube is tripled, both the surrounding area and the total area increase 9 times (because then the area of a cube face increases 9 times).

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