E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Finding the Best Dimensions: Maximum Area and Minimum Surface Area

E1.4 A1About 70 minutes

Learning goals

I can show, with a table or graph, that a square gives the greatest area for a fixed perimeter and the least perimeter for a fixed area.
I can solve fencing problems, including fences that use a wall for one side.
I can show that a cube gives the least surface area for a fixed volume of a rectangular prism.
I can use a spreadsheet or a short program to test many dimensions quickly.

Words to know

optimal
The best possible, such as the greatest area or the least material.

maximum
The greatest value.

minimum
The least value.

constraint
A condition that must stay fixed, such as "the perimeter is 24 m".

table of values
A list of inputs and outputs used to look for a pattern or the best value.

A hook: the dog run

You have 24 m of fencing to build a rectangular run for your dog. You could make it long and thin, or short and wide. Which shape gives your dog the most space?

In the last two lessons you changed dimensions and watched what happened. Now you will choose the dimensions to get the best result. This is called optimization, and engineers, farmers and packaging designers do it every day.

Three rectangles with the same 24 m perimeter, drawn to scale. As the rectangle gets closer to a square, its area grows.

Strategy: use the constraint, make a table

Write the constraint. For a perimeter of 24 m: 2w + 2L = 24, so L = 12 − w.
Write what you want to make best: area A = w × L = w(12 − w).
Make a table (or a spreadsheet) of many widths, or graph A against w.
Find the best value and describe the shape.

Width w (m)	Length 12 − w (m)	Area (m²)
1	11	11
2	10	20
3	9	27
4	8	32
5	7	35
6	6	36
7	5	35
8	4	32

For a fixed perimeter of 24 m, the area is greatest when w = L = 6 m.

# Python: test every whole-number width for a 24 m perimeter

for w in range(1, 12):

L = 12 - w

print(w, L, w * L)

# Output (width, length, area):

# 1 11 11

# 2 10 20

# 3 9 27

# 4 8 32

# 5 7 35

# 6 6 36 <- greatest area

# 7 5 35

# 8 4 32

# 9 3 27

# 10 2 20

# 11 1 11

A short program does the same job as the table. Change 12 to half of any perimeter to test other fences.

The graph of area against width for a 24 m perimeter. The points rise, peak at w = 6 m, then fall symmetrically. Any curve this shape tells you where the maximum is.

The rules you discover

Fixed perimeter → the square has the greatest area.
Fixed area → the square has the least perimeter (least fencing).
Fixed volume of a rectangular prism → the cube has the least surface area (least cardboard).
If one side is a wall and needs no fence, the best rectangle is twice as long as it is wide (the length runs along the wall).

For a closed cylinder with a fixed volume, the least surface area happens when the height equals the diameter. Many real cans are not that shape, because designers also think about cost of the ends, how cans stack, and how they fit in a hand.

Fencing three sides against a barn. The two widths and the length use all 40 m: 2w + L = 40.

Worked example

Example 1: Fixed area, least fencing

A rectangular garden must have an area of 36 m². Which whole-number dimensions use the least fencing?

List the factor pairs and perimeters: 1 × 36 → 74 m; 2 × 18 → 40 m; 3 × 12 → 30 m; 4 × 9 → 26 m; 6 × 6 → 24 m.
The perimeter is smallest for the square.

6 m × 6 m, with 24 m of fencing.

Worked example

Example 2: Against a barn

A farmer has 40 m of fencing for three sides of a pen against a barn wall. Find the dimensions with the greatest area.

Constraint: 2w + L = 40, so L = 40 − 2w.
Area: A = w(40 − 2w).
Table: w = 8 → 8 × 24 = 192; w = 9 → 9 × 22 = 198; w = 10 → 10 × 20 = 200; w = 11 → 11 × 18 = 198; w = 12 → 12 × 16 = 192.
The greatest area is at w = 10 m, L = 20 m. The length is twice the width.

10 m by 20 m, area 200 m².

Worked example

Example 3: The best box

A box must hold 1000 cm³. Compare the surface area of a cube with two other boxes.

Cube: 10 × 10 × 10. SA = 6 × 100 = 600 cm².
5 × 10 × 20: SA = 2(50 + 100 + 200) = 700 cm².
4 × 10 × 25: SA = 2(40 + 100 + 250) = 780 cm².
The cube uses the least cardboard.

The 10 cm cube (600 cm²).

Watch out!

Always use the constraint to write one dimension in terms of the other before you make the table.
With a wall, the fence covers only 3 sides: 2w + L, not 2w + 2L.
If dimensions don't have to be whole numbers, check decimals near the best value (try 6.5 or 7.5).

Hands-on activity

Toothpick and cracker experiments

You need: 24 toothpicks (or 24 paper clips in a chain), 36 square crackers or paper squares, and paper.

Use all 24 toothpicks to make rectangles (each toothpick is 1 unit). Make every possible rectangle. Record width, length and area in a table. Which has the biggest area?
Use all 36 squares to make solid rectangles. Record the perimeter of each. Which has the smallest perimeter?
If you have a computer, put your first table in a spreadsheet: A2 = width, B2 = 12-A2, **C2 = A2B2*, and make a scatter graph.

Model answer: the 6 × 6 square has the largest area (36 units²) with 24 toothpicks, and the 6 × 6 square of crackers has the smallest perimeter (24 units).

Just for fun

Why did the rectangle start working out?

It wanted to get into square shape: the biggest area for its perimeter.

Your turn

A rectangle has a perimeter of 40 m. What dimensions give the greatest area, and what is that area?

A rectangle has an area of 64 m². What dimensions give the least perimeter?

You have 60 m of fence for three sides of a pen along a wall. Find the best dimensions and the area.

A rectangular prism has a volume of 27 cm³. What dimensions give the least surface area, and what is it?

Both have a perimeter of 20 cm. Which has more area: 4 cm × 6 cm or 5 cm × 5 cm?

With 100 m of fencing (all four sides), what is the greatest rectangular area?

Compare the surface areas of a 4 cm cube and a 2 × 4 × 8 box. They have the same volume.

More practice: warm-up

Perimeter 16 m: side of the best square?

Area 49 m²: side of the best square?

Which rectangle shape gives the most area for a fixed perimeter?

What is the perimeter of a 5 m × 5 m square?

What is the area of a 3 m × 9 m rectangle?

More practice: on your own

You have 36 m of fence. List the whole-number rectangles from width 6 m to 9 m and find the greatest area.

A rectangular patio must be 100 m². Edging costs $12/m. What is the cheapest edging?

A farmer has 80 m of fence for three sides along a river. Find the best dimensions and area.

A box must hold 125 cm³. Compare the SA of the 5 cm cube with a 1 × 5 × 25 box.

A rectangle has a perimeter of 30 m. What is the greatest area with whole-number sides? With any sides?

Challenge

Two side-by-side pens share a middle fence. There are 120 m of fence in total: 3 widths and 2 lengths. Find the dimensions with the greatest total area.

An open-top box is made from a 20 cm × 20 cm sheet by cutting an x cm square from each corner and folding up the sides. V = x(20 − 2x)². Which whole-number x gives the greatest volume?

A closed cylinder must hold 1000 cm³. The least surface area happens when h = 2r. Find r, h and the SA, and compare with a 10 cm cube.

Check your understanding

Quiz

For a fixed perimeter, which rectangle has the greatest area?

A rectangle must have an area of 81 m². What is the least possible perimeter?

With 48 m of fence along a wall (3 sides), what is the best width w?

Which rectangular box with volume 216 cm³ uses the least cardboard?

For P = 24 m, what is the area when w = 4 m?

Why do people make tables or spreadsheets for these problems?

For parents and teachers

Big idea: This lesson completes E1.4 by using changes in dimensions to find optimal shapes: the square for rectangles and the cube for rectangular prisms. Tables, graphs, spreadsheets and a short Python program show the pattern (technology is named in the expectation). The graph shape here is a parabola, which your teen will study more in Grade 10.

Common difficulties:

Forgetting the constraint. Ask: "What has to stay the same?"
Wall problems: counting four sides of fence instead of three.
Stopping at the first big number instead of checking the values on both sides.

How to help: Plan a real project: a garden bed or a pet enclosure with a fixed amount of fencing. Let your teen build the table and recommend dimensions.