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Volume Equals Surface Area

Find the cuboids with integer side lengths where the volume is numerically equal to the surface area.
Volume Surface Area Challenge Help More

There are ten cuboids with edge lengths one or two digit integers where the numerical value of the surface area in square centimetres is equal to the value of the volume in cubic centimetres.

Use technology to find the dimensions of all ten cuboids ( width ≤ length ≤ height ) in centimetres.

1

Volume: 882cm3, Surface Area: 882cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

2

Volume: 576cm3, Surface Area: 576cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

3

Volume: 486cm3, Surface Area: 486cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

4

Volume: 450cm3, Surface Area: 450cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

5

Volume: 432cm3, Surface Area: 432cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

6

Volume: 400cm3, Surface Area: 400cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

7

Volume: 288cm3, Surface Area: 288cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

8

Volume: 256cm3, Surface Area: 256cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

9

Volume: 250cm3, Surface Area: 250cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

10

Volume: 216cm3, Surface Area: 216cm2

Width = cm☐ ✓ ✗ Length = cm☐ ✓ ✗ Height = cm☐ ✓ ✗

Check

Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.
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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

Comment recorded on the 19 October 'Starter of the Day' page by E Pollard, Huddersfield:

"I used this with my bottom set in year 9. To engage them I used their name and favorite football team (or pop group) instead of the school name. For homework, I asked each student to find a definition for the key words they had been given (once they had fun trying to guess the answer) and they presented their findings to the rest of the class the following day. They felt really special because the key words came from their own personal information."

Comment recorded on the 28 May 'Starter of the Day' page by L Smith, Colwyn Bay:

"An absolutely brilliant resource. Only recently been discovered but is used daily with all my classes. It is particularly useful when things can be saved for further use. Thank you!"

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Suggestions

This activity is not intended to be a trial-and-error activity for novice mathematicians. It was designed to encourage higher attaining students to investigate new problem solving methods using technology. It could be an individual task or a collaborative effort with everyone in class having a part to play.

here are some thoughts I have had for how technology could be used. Please let me know if you have any other suggestions.

Spreadsheet

Set up a two-way table for length and height with row and column headings from one to one hundred. In a different cell type in the value 1 for the width. In the table set up conditional formatting to flag up surface area = volume with a formula. Change the value of the width to 2,3,4 etc. noting the dimensions of the successful cuboids as you go. You will have found all ten when you have reached width = 6.

Program

Write a program on your graphing calculator or computer with nested loops to check all possible combinations of width, length and height then print out the details of the successful tests. For example, in the Visual Basic programming language the code might look like this:

code example

Investigate how to create loops in Online Logo to perform this task.

Calculator

Use the calculator the find the prime factors of the given volumes then show all the ways those prime factors could be combined to produce three factors (width (W), length (L) and height (H) where W ≤ L ≤ H).

FACT button

Define a graphic display calculator (GDC) function to work out surface area from the dimensions of the cuboid.

Graphic Display Calculator

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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