7 th grade algebra - 13 perimeter of plane shapes

7th grade Algebra - 13

Perimeter of Plane Shapes

7th grade Algebra - 13

Perimeter of Plane Shapes

The chapters:

Rectangle’ perimeter Square’s perimeter Triangle’s perimeter Circle’s perimeter Challenge exercises

Question 1:

What is the perimeter of the following shape?

A. 8 cm

B. 160 cm

C. 20 cm

D. 56 cm

8 cm

20 cm

A D

CB

Solution:

What is the perimeter of the following shape?

A. 8 cm

B. 160 cm

C. 20 cm

D. 56 cm

8 cm

20 cm

A D

CB

Let us calculate:

In the following rectangle, the brown line represents the shape’s perimeter.The perimeter of a shape is the distance around the shape.We indicate this distance with the letter P (Perimeter).

The rectangle’s perimeter is 56 cm.

P = 2a + 2b = 2(a + b) is the mathematic expression representing the rectangle’s perimeter.

8 cm

20 cm

A D

CB

)ba(2b2a2P

)BCAB(2BC2AB2P

56208208

ADCDBCABP

Question 2:

The length of the rectangle is bigger than its width by 5 cm.

b

a

A D

CB

Write a mathematic expression which

will represent the rectangle’s perimeter:

A. P = 2b+5

B. P = 2(2b+5)

C. P = 2a+2b+10

D. P = a+b+10

Solution:

The length of the rectangle is bigger than its width by 5 cm.

b

a

A D

CB

Write a mathematic expression which

will represent the rectangle’s perimeter:

A. P = 2b+5

B. P = 2(2b+5)

C. P = 2a+2b+10

D. P = a+b+10

Question 3:

What is the perimeter of the following shape?

A. 8 cm

B. 16 cm

C. 32 cm

D. 64 cm

A D

CB

a = 8 cm

a = 8 cm

Solution:

What is the perimeter of the following shape?

A. 8 cm

B. 16 cm

C. 32 cm

D. 64 cm

A D

CB

a = 8 cm

a = 8 cm

Let us calculate:

The perimeter of a this square is the distance around it and it is equal to the sum of the square’s sides.A square is a rectangle with equal sides.

The square’s perimeter is 32 cm.

P = 4a is the mathematic expression representing the square’s perimeter.

a4P

aaaaP

328888

ADCDBCABP

A D

CB

a = 8 cm

a = 8 cm

Question 4:

Write a mathematic expression whichwill represent the square’s side if it’s

perimeter P is known:

A.

B.

C.

D.

a

A D

CB

a

P4

4P2P

4P

Solution:

Write a mathematic expression whichwill represent the square’s side if it’s

perimeter P is known:

A.

B.

C.

D.

a

A D

CB

a

P4

4P2P

4P

Question 5:

What is the perimeter of the following shape?

A. 8 cm

B. 12 cm

C. 20 cm

D. 17 cm

b = 4 cm

A

CB

a = 3 cmc = 5 cm

Solution:

What is the perimeter of the following shape?

A. 8 cm

B. 12 cm

C. 20 cm

D. 17 cm

b = 4 cm

A

CB

a = 3 cmc = 5 cm

Let us calculate:

The perimeter of a this triangle is the distance around it and it is equal to the sum of the triangle’s sides.

The triangle’s perimeter is 12 cm.

The triangle is this example is a right triangle. In this triangle, the hypotenuse is bigger than each of the catheti (legs) and

is smaller than their sum.

cbaP

12543

ACBCABP

b = 4 cm

A

CB

a = 3 cmc = 5 cm

b < c < a + b

Question 6:

b is bigger than a by 1, and c is bigger than b by 1.

Write a mathematic expression whichwill represent the triangle’s perimeter

according to a:

b

A

CB

ac

A. P = 2a+3

B. P = 9a

C. P = 3a+2

D. P = 3a+3

Solution:

b is bigger than a by 1, and c is bigger than b by 1.

Write a mathematic expression whichwill represent the triangle’s perimeter

according to a:

b

A

CB

ac

A. P = 2a+3

B. P = 9a

C. P = 3a+2

D. P = 3a+3

Circle’s perimeter

Definition:

If the radius of a circle is r, its perimeter will be 2r.

If the diameter of a circle is d, its perimeter will be d.

= 3.14

Question 6:

What is the circle’s perimeter?

A. 62.8 cm

B. 31.4 cm

C. 15.74 cm

D. 10 cm

10 cm

Perimeter formula: P = 2r = d = 3.14

Solution:

What is the circle’s perimeter?

A. 62.8 cm

B. 31.4 cm

C. 15.74 cm

D. 10 cm

10 cm

Perimeter formula: P = 2r = d = 3.14

The perimeter formula is:

14.3

dr2P

Solution:

10 cm

Given: the circle’s perimeter = 10 cmThus, r = 5 cmWe will place the data in the formula:

4.311014.3514.32P The circle’s perimeter is 31.4 cm.

B is the correct answer.

Question 7:

If we increase the circle’s diameter times 2, the circle’s perimeter is:

A. 62.8 cm

B. 31.4 cm

C. 15.74 cm

D. 10 cm

10 cm

Perimeter formula: P = 2r = d = 3.14

Solution :

If we increase the circle’s diameter times 2, the circle’s perimeter is:

A. 62.8 cm

B. 31.4 cm

C. 15.74 cm

D. 10 cm

10 cm

Perimeter formula: P = 2r = d = 3.14

Question 8:

What is the perimeter of the following shape?

A. 8 cm

B. 16 cm

C. 24 cm

D. None of the above

The length of each of the triangles’ side is 1 cm.

Solution :

What is the perimeter of the following shape?

A. 8 cm

B. 16 cm

C. 24 cm

D. None of the above

The length of each of the triangles’ side is 1 cm.

The shape’s perimeter is the blue line. We can see that:

ADCDBCAB

Solution:

Each of the shape’s sides is composed of two triangle’s sides. The length of each triangle side is 1 cm. Thus,

2ADCDBCAB The shape’s perimeter is:

A

D

C

B

cm842 A is the correct answer.

Question 9:

What is the perimeter of the following shape?

A. 48 cm

B. 16 cm

C. 32 cm

D. 96 cm

The length of each of the squares’ side is 2 cm.

Solution :

What is the perimeter of the following shape?

A. 48 cm

B. 16 cm

C. 32 cm

D. 96 cm

The length of each of the squares’ side is 2 cm.

The shape’s perimeter is the blue line. We can see that the shape’s perimeter contains 16 square sides.

Solution:

Each of the squares’ side is 2 cm. Thus, the shape’s perimeter is:

cm32216

C is the correct answer.

Question 10:

Each of the following sentences is correct?

A. The shape’s perimeter is smaller than 25

B. The shape’s perimeter is smaller than 30

C. The shape’s perimeter is larger than 34

D. The shape’s perimeter is larger than 30

12 cm

A

CB

5 cm

8 cmD

Solution :

Each of the following sentences is correct?

A. The shape’s perimeter is smaller than 25

B. The shape’s perimeter is smaller than 30

C. The shape’s perimeter is larger than 34

D. The shape’s perimeter is larger than 30

12 cm

A

CB

5 cm

8 cmD

We will calculate the trapeze known sides:

Solution:

In order to calculate the perimeter, we need to add the length of CD to the result. Thus, answer A (smaller than 25) is not correct.

251285 12 cm

A

CB

5 cm

8 cmD

To estimate the length of CD, we will add the dotted lines.

Solution:

In right-angle triangle, the hypotenuse is the longest side, but its length is smaller then the sum of the other two sides. Which means: 9CD5

D is the correct answer.

12 cm

A

CB

5 cm

8 cm D

5 cm

4 cm

Thus, the shape’s perimeter cannot be smaller than 30; therefore, answer B is not correct. The perimeter also cannot be larger than 34 based on the same reason, so answer C is also the wrong answer.

Question 11:

True ore false?

A. True

B. False

The perimeter of shape I is 12 units.

IIIIII

IVV

VI

Solution:

True ore false?

A. True

B. False

The perimeter of shape I is 12 units.

IIIIII

IVV

VI

Question 12:

True ore false?

A. True

B. False

IIIIII

IVV

VI

The perimeter of shape V is equal to the perimeter of shape I + the perimeter of shape II.

Solution:

True ore false?

A. True

B. False

IIIIII

IVV

VI

The perimeter of shape V is equal to the perimeter of shape I + the perimeter of shape II.

Question 13:

True ore false?

A. True

B. False

IIIIII

IVV

VI

The perimeter of shape IV is equal to the perimeter of shape VI.

Solution:

True ore false?

A. True

B. False

IIIIII

IVV

VI

The perimeter of shape IV is equal to the perimeter of shape VI.

Question 14:

Which of the following shapes has the longest perimeter?

A B C D E

Solution:

Which of the following shapes has the longest perimeter?

A B C D E

Question 15:

Are shapes with the same perimeter always have the same area?

A B C D E

Calculate the area and perimeter of the following shapes.

A. Yes

B. No

Solution:

Are shapes with the same perimeter always have the same area?

A B C D E

Calculate the area and perimeter of the following shapes.

A. Yes

B. No

Question 16:

What is the perimeter of the following shape?

A. 27 cm

B. 32 cm

C. 16 cm

D. 19 cm

a

b

cd a = 8

b = 5

c = 3

d = 3

Solution:

What is the perimeter of the following shape?

A. 27 cm

B. 32 cm

C. 16 cm

D. 19 cm

a

b

cd a = 8

b = 5

c = 3

d = 3

Question 17:

What is the external perimeter of the following shape?

A. 20.85 cm

B. 28.85 cm

C. 26 cm

D. 18 cm

a = 8

b = 5

d = 5a

b d

Solution:

What is the external perimeter of the following shape?

A. 20.85 cm

B. 28.85 cm

C. 26 cm

D. 18 cm

a = 8

b = 5

d = 5a

b d

Homework1. In a rectangle: a = 8 cm ; P = 26 cm ; b = ?2. In a square: P = 48 cm ; a = ?3. In a circle: P = 15.7 cm ; r = ?4. In a triangle: a = 5 cm ; b = 9 cm ; P = 27 cm ; c = ?5. The size of a picture that was framed is The frame width is 4 cm. What is the external perimeter of the frame?

cm60cm40

Homework Check

Question 1:

Given: a = 8 cm ; P = 26 cm.

a

b

= ?b

A. 18 cm

B. 9 cm

C. 6 cm

D. 5 cm

Solution:

Given: a = 8 cm ; P = 26 cm.

a

b

= ?b

A. 18 cm

B. 9 cm

C. 6 cm

D. 5 cm

Question 2:

Given: P = 48 cm.

= ?a

A. 24 cm

B. 8 cm

C. 12 cm

D. 16 cm

a

Solution:

Given: P = 48 cm.

= ?a

A. 24 cm

B. 8 cm

C. 12 cm

D. 16 cm

a

Question 3:

Given: P = 15.7 cm.

= ?r

A. 2.5 cm

B. 5 cm

C. 10 cm

D. 6 cm

Solution:

Given: P = 15.7 cm.

= ?r

A. 2.5 cm

B. 5 cm

C. 10 cm

D. 6 cm

Question 4:

Given: a = 5 cm ; b = 9 cm ; P = 27 cm.

= ?c

A. 12 cm

B. 5 cm

C. 13 cm

D. 8 cm

a

b

c

Solution:

Given: a = 5 cm ; b = 9 cm ; P = 27 cm.

= ?c

A. 12 cm

B. 5 cm

C. 13 cm

D. 8 cm

a

b

c

Question 5:

What is the external perimeter of the frame?

A. 200 cm

B. 216 cm

C. 232 cm

D. 240 cm

The size of a picture that was framed is

The frame width is 4 cm.cm60cm40

Solution:

What is the external perimeter of the frame?

A. 200 cm

B. 216 cm

C. 232 cm

D. 240 cm

The size of a picture that was framed is

The frame width is 4 cm.cm60cm40