Geometric Optimization Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Geometric Optimization.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Finding the best geometric configuration — the shape that maximizes area, minimizes perimeter, uses the least material, or achieves some other optimal outcome — subject to given constraints.

What rectangle with fixed perimeter has the most area? A square!

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Geometric optimization finds the configuration that maximizes or minimizes a measurement while a constraint stays fixed.

Common stuck point: The procedure for geometric optimization is the easy part; the trap is confusing what is fixed with what is optimized. Asking "Is one quantity held fixed while I search for the shape that makes another biggest or smallest?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is one quantity held fixed while I search for the shape that makes another biggest or smallest?

Worked Examples

Example 1

medium
A farmer has P=60P = 60 m of fence to enclose a rectangular paddock against a straight wall (so only 33 sides need fencing). Find the dimensions that maximise the area.

Answer

Width =15= 15 m, Length =30= 30 m, Maximum area =450= 450 m2^2.

First step

1
Step 1: Let width =x= x (the two sides perpendicular to the wall) and length =y= y (parallel to wall). Constraint: 2x+y=602x + y = 60, so y=60−2xy = 60 - 2x.

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Example 2

easy
A rectangle has perimeter P=40P = 40 cm. Using the formula maximum area =P2/16= P^2/16, compute the maximum area and the dimensions of the optimal rectangle.

Example 3

medium
A farmer has 100100 m of fencing to enclose a rectangular field against a straight river (no fence on the river side). What dimensions maximize the area?

Example 4

medium
A cylindrical can must hold 10001000 cm3^3. What radius and height minimize the surface area (closed top and bottom)?

Example 5

medium
A rectangle is inscribed under the curve y=4−x2y = 4 - x^2 on the xx-axis, with its base on the axis. What is the maximum area?

Example 6

medium
A rectangular sheet 2424 cm by 99 cm has equal squares of side xx cut from each corner. The flaps fold up into an open-top box. Which xx maximizes volume?

Example 7

hard
Find the largest area of an isoceles triangle inscribed in a circle of radius 11.

Example 8

hard
Find the rectangle of maximum area inscribed in the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 with sides parallel to the axes.

Example 9

challenge
Among all triangles inscribed in a circle of radius 11, find the one with maximum area and give that area.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Two rectangles have the same perimeter of 2424 cm: one is 8×48 \times 4 cm and one is 6×66 \times 6 cm. Which has greater area? Does this match the P2/16P^2/16 maximum?

Example 2

hard
A rectangular box with a square base and no lid must have volume V=32V = 32 cm3^3. Find the dimensions that minimise the total surface area (base + 4 sides).

Example 3

easy
Of all rectangles with a fixed perimeter, which shape has the most area?

Example 4

easy
Of all shapes with a fixed perimeter, which encloses the most area?

Example 5

easy
A rectangle has perimeter 20. What side length gives the maximum area?

Example 6

easy
Of all shapes enclosing a fixed area, which has the smallest perimeter?

Example 7

easy
Why do soap bubbles form spheres?

Example 8

easy
A rectangle has fixed perimeter PP. Its maximum area is P216\frac{P^2}{16}. For P=16P = 16, find the max area.

Example 9

easy
True or false: a long, thin rectangle has more area than a square with the same perimeter.

Example 10

easy
To use the least fencing for a fixed rectangular area, what proportions should the rectangle have?

Example 11

medium
A farmer has 40 m of fence for a rectangular pen. What dimensions maximize the area, and what is that area?

Example 12

medium
A farmer builds a rectangular pen against a straight wall (no fence needed on the wall side) with 40 m of fence. What dimensions maximize area?

Example 13

medium
Compare the area of a circle and a square that both have perimeter (circumference) 12. Which is larger?

Example 14

medium
Why does a balanced shape (like a square) maximize area, while extreme shapes do worse?

Example 15

medium
A can must hold a fixed volume. Roughly what proportions minimize the metal used (surface area)?

Example 16

medium
Honeybees build hexagonal cells. Why is the hexagon a good optimization choice for tiling with the least wax?

Example 17

medium
Among all triangles with a fixed perimeter, which has the largest area?

Example 18

medium
A rectangle has length twice its width and perimeter 36. Find its area, and compare to the max possible for that perimeter.

Example 19

challenge
A rectangular pen against a wall uses fence for two widths and one length, with 100 m of fence. Maximize the area.

Example 20

challenge
Why does the shortest path of light reflecting off a mirror equal the straight-line path to the mirror image of the target?

Example 21

challenge
Among all rectangles inscribed in a semicircle of radius 1 (base on the diameter), what is the maximum area?

Example 22

challenge
Explain why nature's optimization (bubbles round, honeycombs hexagonal, bones hollow) reflects the same underlying principle, and name it.

Example 23

easy
A rectangle has perimeter P=32P = 32 cm. What dimensions maximize its area?

Example 24

easy
Of all rectangles with area 3636 cm2^2, which has the smallest perimeter?

Example 25

easy
Of all triangles with fixed perimeter PP, which shape has the largest area?

Example 26

easy
A rectangle has perimeter P=48P = 48 m. Find the dimensions and the maximum area.

Example 27

medium
Among all rectangles inscribed in a circle of radius rr, which has the largest area?

Example 28

medium
A box with a square base and open top has volume 108108 cm3^3. What base side and height minimize surface area?

Example 29

medium
A rancher has 200200 m of fence to make two equal rectangular pens that share a common side. What dimensions of the combined enclosure maximize the total area?

Example 30

medium
Two positive numbers sum to 2020. What is the maximum product?

Example 31

medium
A field with perimeter 8080 m must be rectangular. If one side is constrained to be at least 2525 m, find the dimensions maximizing area.

Example 32

hard
A right triangle has hypotenuse 1010. Find the legs that maximize the area.

Example 33

hard
A cylindrical can is open at the top and must hold V=500V = 500 cm3^3. What radius minimizes the material (lateral + base)?

Example 34

hard
A poster has total area 384384 cm2^2 with 44 cm margins on top and bottom and 22 cm margins on the sides. What dimensions of the poster maximize the printable area?

Example 35

hard
Of all isoperimetric quadrilaterals with perimeter PP, what is the maximum area?
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Related Concepts

AreaPerimeter

Background Knowledge

These ideas may be useful before you work through the harder examples.

areaperimeter