FIND THE DIAGONAL OF A RECTANGULAR PRISM

Find the length of the diagonal of a rectangular prism :

The following picture shows, how to find the diagonal of a rectangular prism.

Problem 1 :

Find the length of the diagonal D of the following rectangular prism.

Solution :

Considering the base of the rectangular prism, it is in the shape of rectangle.

l = 12 in, w = 5 in

In the base, diagonal length is,

l² + w² = d²

12² + 5² = d²

144 + 25 = d²

169 = d²

13 = d

So, the diagonal length of the base is13 in.

Length of the diagonal of the rectangular prism be D.

6² + 13² = D²

36 + 169 = D²

205 = D²

√205 = D

So, the diagonal length of the prism is√205 in.

Problem 2 :

Find the length of the diagonal D of the following rectangular prism.

Solution :

Length = 3 ft, width = 2 ft and height = 10 ft

Diagonal = √3² + 2² + 10²

= √9+4+100

= √113 inches

Problem 3 :

Find the length of the diagonal D of the following rectangular prism.

Solution :

Length = 7 ft, width = 3 ft and height = 4 ft

Diagonal = √7² + 3² + 4²

= √49 + 9 + 16

= √74 ft

Problem 4 :

Find the length of the diagonal D of the following rectangular prism.

Solution :

Length = 10 ft, width = 6 ft and height = 15 ft

Diagonal = √10² + 6² + 15²

= √100 + 36 + 225

= √361

= 19 ft

Problem 5 :

what is the longest object that Simone can put in a rectangular box that is 10 inches wide, 12 inches long and 20 inches tall? round to the nearest tenth of an inch.

Solution :

Length = 12 inches, width = 10 inches and height = 20 inches

Diagonal = √12² + 10² + 20²

= √(144 + 100 + 400)

= √644

= 25.37

The longest object that can fit in the box is about 25.4 inches.

Problem 6 :

A rectangular prism is 12 inches wide, 5 inches long and 6 inches tall. Tamara’s work to find the length of its longest diagonal is to the right. Unfortunately, she made a mistake. Identify the mistake and find the length of the longest diagonal in the rectangular prism

Solution :

Length = 12 inches, width = 5 inches and height = 6 inches

Diagonal = √12² + 5² + 6²

= √(144 + 25 + 36)

= √205

= 14.31

Approximately 14.3 inches

Problem 7 :

A rectangular prism is 3 feet long, 4 feet wide and 2 feet tall. What is the length of its longest diagonal?

Solution :

Length = 3 feet, width = 4 feet and height = 2 feet

Diagonal = √3² + 4² + 2²

= √(9 + 16 + 4)

= √29

= 5.385

Approximately 5.39 feet.

Problem 8 :

Petra is sending her brother a giant candy cane stick for a gift. She will use a box that measures 10 inches by 6 inches by 2 inches. What is the maximum length the candy cane stick can be to fit in the box?

Solution :

Length = 10 inches, width = 6 inches and height = 2 inches

Diagonal = √10² + 6² + 2²

= √(100 + 36 + 4)

= √140

= 11.83

= 11.9

Approximately 11.9 inches

Problem 9 :

Elena needs to ship a 61 cm concert flute to a customer. She has two rectangular boxes. One is 25 cm by 25 cm by 50 cm. The other box is 10 cm by 12 cm by 58 cm.

a. What is the longest object that will fit in the 25 cm × 25 cm × 50 cm rectangular box?

b. What is the longest object that will fit in the 10 cm × 12 cm × 58 cm rectangular box?

c. In which box will the flute best fit?

Solution :

The longest object will fit in the first box,

Length = 25 cm, width = 25 cm and height = 50 cm

Diagonal = √25² + 25² + 50²

= √(625 + 625 + 2500)

= √3750

= 61.23 cm

The longest object will fit in the second box,

Length = 10 cm, width = 12 cm and height = 58 cm

Diagonal = √10² + 12² + 58²

= √(100 + 144 + 3364)

= √3608

= 60.06 cm

The flute will fix in the first box.

Problem 10 :

A cube has a surface area of 600 square meters.

a. How many faces does a cube have?

b. What is the area of one face of the cube?

c. Find the length of one edge of the cube.

d. What is the length of the cube’s longest diagonal? Use mathematics to justify your answer.

Solution :

a) Surface area of cube = 600 square meter

6a² = 600

a² = 600/6

a² = 100

a = √100

a = 10 m

b) Area of one face of cube = a² = 100

c) Side length of cube = 10 m

d) Diagonal = 10√3