Practice Geometric Optimization in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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!

Showing a random 20 of 50 problems.

Example 1

medium

A field with perimeter

80

m must be rectangular. If one side is constrained to be at least

25

m, find the dimensions maximizing area.

Example 2

easy

Two rectangles have the same perimeter of

24

cm: one is

8 \times 4

cm and one is

6 \times 6

cm. Which has greater area? Does this match the

P^2/16

maximum?

Example 3

hard

A poster has total area

384

^2

with

4

cm margins on top and bottom and

2

cm margins on the sides. What dimensions of the poster maximize the printable area?

Example 4

medium

A rectangle is inscribed under the curve

y = 4 - x^2

on the

x

-axis, with its base on the axis. What is the maximum area?

Example 5

easy

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

Example 6

challenge

Among all triangles inscribed in a circle of radius

1

, find the one with maximum area and give that area.

Example 7

easy

Of all rectangles with perimeter

40

inches, what is the smallest possible area (assuming positive sides)?

Example 8

easy

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

Example 9

easy

True or false: among all closed curves of length

L

, the circle encloses the largest area.

Example 10

easy

A rectangle has perimeter

P = 40

cm. Using the formula maximum area

= P^2/16

, compute the maximum area and the dimensions of the optimal rectangle.

Example 11

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 12

hard

A cylindrical can is open at the top and must hold

V = 500

^3

. What radius minimizes the material (lateral + base)?

Example 13

hard

An open-top box has a square base of side

x

and height

h

. If the total surface area equals

48

^2

, what is the maximum volume?

Example 14

easy

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

Example 15

medium

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

Example 16

easy

Using

A_{\max} = P^2/16

, find the maximum area of a rectangle with perimeter

P = 24

Example 17

hard

Find the rectangle of maximum area inscribed in the ellipse

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

with sides parallel to the axes.

Example 18

hard

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

1

Example 19

medium

A square and a circle have the same perimeter

P

. Which has the larger area, and by what ratio?

Example 20

challenge

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

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Related Concepts

Area Perimeter