Quadratics Rectangle Worksheet: 20 Problems About Boxes That Fight Back

Part of our complete collection of free quadratic worksheets and answer keys .

This quadratics rectangle worksheet takes patios, picture frames, walkways and one very stubborn field and turns them all into the same move: name a side x, write the other side in terms of x, multiply. Twenty problems in four parts, ten with a diagram, full answer key.

By Berke Sahbazoglu · 6,000+ hours & 200+ students tutored · 10+ years teaching Algebra I & II · BS Biochemistry (Washington University in St. Louis), MS Bioinformatics (UMGC) · Burke Tutoring in Fremont

Published July 31, 2026 · Last updated Aug 01, 2026 · Every answer checked for accuracy before publishing · Free to print for classroom & home use

Level: Algebra I and Algebra II, grades 8–11
Standards: CCSS.MATH.CONTENT.HSA.CED.A.1 and HSA.REI.B.4.b
Includes: 20 problems in four parts, 10 with a diagram, three color-coded figures, and a full answer key
Assumes: Factoring and the quadratic formula already work — the picture is the new part
Skills: Naming an unknown, writing a second dimension in terms of it, expanding a product, rejecting an impossible root
Time: About 50 minutes, or one part at a time
Format: Read on this page or print the PDF, which has room to show work — no sign-up

In short

Rectangle problems are the friendliest quadratics in the book, right up until somebody puts a frame around something. Then the width sneaks in twice per dimension, the student adds it once, and a perfectly good page of algebra goes in the bin. Draw the picture. Add the x to both ends. Everything after that is factoring you can already do.

The whole method, four lines

Draw the rectangle. Ugly is fine. Ugly is honestly preferred.
Call one side x. Write every other side in terms of x.
Length × width = area. Write it, then factor or use the quadratic formula.
Delete the root that would give you a negative fence. Put a unit on the survivor.

Want the printable quadratics rectangle worksheet?

Eight pages: twenty problems with real space to show work, ten simple diagrams with the labels in a key beside them, and the full answer key. No sign-up, no email.

Download the PDF worksheet

PDF · 8 pages · prints on letter paper

If you are handing this to someone

Parts A and B are one homework night. Part C is the one worth doing together, out loud, with a pencil on the picture. Part D is review the week before a test. Teachers: the parts split cleanly into four ten-minute stations, and the answer key prints on its own page so you can hold it back.

Four Setups in This Quadratics Rectangle Worksheet

Every rectangle problem you will ever be assigned is one of these. Textbooks just keep changing the nouns.

ONE RECTANGLE

x(x + k) = A

"Three more than the width." "Twice as long, less six."

Name the short side, build the long one, multiply.

BEFORE AND AFTER

(x + a)(y + a) = old ± change

Both dimensions grow or shrink by the same amount.

Two areas, one equation between them.

FRAMES & BORDERS

(L + 2x)(W + 2x) = total

Frames, walkways, margins, mowed strips.

Width x on the left and the right. Hence 2x.

AREA + PERIMETER

x(P/2 − x) = A

They hand you a fence length and an area.

Half the perimeter minus one side is the other side.

Where the marks actually go

Two places. First, the border problem where somebody adds x instead of 2x — the algebra runs beautifully and the answer is wrong, which is the worst kind of wrong. Second, the last line, where a quadratic hands over two roots and the student boxes both. A frame cannot be −13 inches wide. Cross it out and write one short sentence saying why.

One Rectangle, Two Sides, One Equation

Start here. Every harder problem on this page is this one wearing a hat.

Whichever side looks simplest in the sentence is the one you call x. Usually that is the width, because textbooks describe the length in terms of it.

Color key for the single rectangle figure
Color	Part	What you write down
Red	Short sides	The one you name. Call it x
Blue	Long sides	x + 5, or 2x − 3, or whatever the sentence says
Pale fill	Inside	Area = red × blue. That is the equation
Gold	Corner	Where the two labeled sides meet. Start here

If the factoring stalls rather than the setup, back up one step to solving quadratics by factoring, and keep completing the square in your pocket for the ones that refuse to factor.

The Frame Problem, Which Eats Everyone Once

A picture inside a frame. A garden inside a walkway. Print inside a margin. Same figure every time, and the same mistake every time.

Two purple arrows, one equation. The band is width x, but it shows up on the left and on the right, so the outer length is the inner length plus 2x.

Color key for the frame and border figure
Color	Part	What you write down
Green	Inner	The picture, garden or print block. Sizes are given
Shaded	The band	The border. Uniform width x, all four sides
Gold	Outer	(L + 2x) by (W + 2x)
Purple	Arrows	x twice per dimension. This is the whole trap

Cutting inward instead of building outward? Same figure, one sign flips: a strip mowed around the edge leaves a middle of (L − 2x) by (W − 2x). Problems 15 and 16 on this worksheet do exactly that, and both hand you a second root that is physically impossible — the sort of thing collected on the quadratics error analysis worksheet.

Before and After: Two Rectangles, One Sentence

"Each dimension is increased by 3 and the area goes up by 33." That is two rectangles and one equation linking them. Draw both. Nobody has ever solved one of these from memory.

Same corner, same orientation, one shape sitting inside the other. Write the old area, write the new area, and let the sentence tell you how they are related.

Color key for the before and after figure
Color	Part	What you write down
Green	Solid	Before: x by (x + k)
Blue dashed	Outline	After: (x + a) by (x + k + a)
Pale fill	The gap	The change in area the problem quotes
Gold	Bracket	The amount added. Same on both dimensions

Sentence In, Algebra Out

Five phrases cover almost everything. Learn the phrases and you have learned the chapter.

Common rectangle phrases and the algebra each one means
The sentence says	You write	Why
"7 more than the width"	w and w + 7	One unknown, one built from it
"six less than twice the width"	x and 2x − 6	Double first, subtract second
"a border of uniform width"	L + 2x and W + 2x	The band is on both ends
"48 m of fencing"	x and 24 − x	Half the perimeter, minus one side
"25% more area"	new = 1.25 × old	Percent means multiply, not add

✗ Frame, done fast

(12 + x)(16 + x) = 396

x = 6

Added the border to one end only. It even factors cleanly, which is exactly why nobody notices.

✓ Frame, done right

(12 + 2x)(16 + 2x) = 396

x = 3 (x = −17 rejected)

Thirty seconds on a sketch buys you the 2. Then it factors.

What actually goes wrong in sessions

Ten years of watching this unit and it is always the same shortlist. Nobody drew the picture. The border got added once instead of twice. Two different lengths both got called x. Or the algebra was perfect and the student answered with the width when the question asked for the area. Parts A through D below are ordered to walk into those in roughly that order, on purpose.

Quadratics Rectangle Worksheet: Twenty Practice Problems

Same twenty as the printable, same order, getting meaner as you go. Ten come with a diagram in the PDF — those are flagged below. Sketch the other ten yourself.

Quadratics rectangle worksheet — twenty problems in four parts: single rectangles, before-and-after comparisons, frames and borders, and mixed hard problems.
Part	Problems
A · ONE RECTANGLE	The length of a rectangular rug is 5 ft more than its width, w. Write a function A(w) for the area. · diagram A vegetable patch is 6 m longer than it is wide and has an area of 91 m². Find both dimensions. The length of a whiteboard is 3 ft less than twice its width, x. The area is 54 ft². Write an equation and solve it. · diagram A flower bed's width is half its length and the area is 60 ft². Find the width to the nearest tenth. A soccer field has an area of 7,500 yd² and must be 25 yd longer than it is wide. Find the dimensions.
B · BEFORE AND AFTER	Each side of a square grows by 4 in and the area becomes 9 times bigger. Find the original side. · diagram A room is 6 ft longer than it is wide. Each dimension grows by 2 ft and the area grows by 48 ft². Find the original dimensions. A rectangle is 10 cm longer than it is wide. Each dimension shrinks by 2 cm and the area drops by 76 cm². Find the original dimensions. Joe's patio is 10 ft by 14 ft. He adds the same length x to both dimensions and ends up with 192 ft². Find x. · diagram A rectangular sheet is twice as long as a square sheet and 4 in wider. Its area is 120 in². Find the side of the square.
C · FRAMES AND BORDERS	A 10 in by 14 in photo gets a frame of uniform width x. The framed picture covers 192 in². How wide is the frame? · diagram A garden 12 m by 18 m has a walkway of uniform width x around it. Together they cover 315 m². Find x. · diagram A mirror 10 in by 15 in has a uniform frame whose area equals the mirror's area. Find the frame's width. A page has a 1 in margin all round and holds 40 in² of print. The page is 3 in longer than it is wide. How big is the page? · diagram A lawn is 30 ft by 40 ft. You mow a uniform strip around the outside and stop when half the lawn is cut. How wide is the strip? · diagram
D · THE ONES THAT BITE	A field is 100 ft by 120 ft. A uniform ring is cut around the outside, leaving two-thirds uncut. Find the ring's width and say why the other root is nonsense. · diagram A field is 150 ft by 200 ft. The farmer wants 50% more area by cultivating a uniform band around the outside. How wide, to the nearest tenth of a foot? A contractor has 44 m of fencing for the perimeter of a rectangular garden of area 105 m². Find the dimensions algebraically. A square garden is redesigned: one side doubled, the other cut by 4 m. The new garden has 50% more area. Find the original side and the new area. A picture is 8 in by 10 in. Simon frames it with wood strips of equal width so the framed picture takes at most 143 in² of wall. Write an inequality and find the widest strip, to the nearest tenth. · diagram

Part D borrows its shape from Regents-style modeling questions — if that is the exam in your future, the New York question bank at JMAP has years of them sorted by standard. Problem 17 will not factor, so it is quadratic-formula territory: see the quadratic formula worksheet and the discriminant and number of roots page. For projectiles, revenue and consecutive integers instead of rectangles, the quadratics word problems worksheet is the companion to this one.

Quadratics Rectangle Worksheet Answer Key

Part A · One rectangle

1: A(w) = w(w + 5) = w² + 5w
2: w(w + 6) = 91 → w² + 6w − 91 = 0 → (w + 13)(w − 7) = 0 → 7 m by 13 m
3: x(2x − 3) = 54 → 2x² − 3x − 54 = 0 → (2x + 9)(x − 6) = 0 → x = 6 → 6 ft by 9 ft
4: (2w)(w) = 60 → w² = 30 → w ≈ 5.5 ft, length ≈ 11.0 ft
5: w(w + 25) = 7500 → w² + 25w − 7500 = 0 → (w − 75)(w + 100) = 0 → 75 yd by 100 yd

↑ Back to Part A

Part B · Before and after

6: (x + 4)² = 9x² → 8x² − 8x − 16 = 0 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → side 2 in
7: (x + 2)(x + 8) = x(x + 6) + 48 → x² + 10x + 16 = x² + 6x + 48 → 4x = 32 → 8 ft by 14 ft
8: (x − 2)(x + 8) = x(x + 10) − 76 → x² + 6x − 16 = x² + 10x − 76 → 15 cm by 25 cm
9: (10 + x)(14 + x) = 192 → x² + 24x − 52 = 0 → (x + 26)(x − 2) = 0 → x = 2 ft
10: 2s(s + 4) = 120 → s² + 4s − 60 = 0 → (s + 10)(s − 6) = 0 → square side 6 in

↑ Back to Part B

Part C · Frames and borders

11: (10 + 2x)(14 + 2x) = 192 → x² + 12x − 13 = 0 → (x + 13)(x − 1) = 0 → 1 in
12: (12 + 2x)(18 + 2x) = 315 → 4x² + 60x − 99 = 0 → (2x + 33)(2x − 3) = 0 → x = 1.5 m
13: Mirror is 150 in², so the total is 300 in². (10 + 2x)(15 + 2x) = 300 → 2x² + 25x − 75 = 0 → (2x − 5)(x + 15) = 0 → 2.5 in
14: (w − 2)(w + 1) = 40 → w² − w − 42 = 0 → (w − 7)(w + 6) = 0 → page is 7 in by 10 in
15: (30 − 2x)(40 − 2x) = 600 → x² − 35x + 150 = 0 → (x − 5)(x − 30) = 0 → 5 ft. x = 30 would mow the lawn twice over

↑ Back to Part C

Part D · The ones that bite

16: (100 − 2x)(120 − 2x) = 8000 → x² − 110x + 1000 = 0 → (x − 10)(x − 100) = 0 → 10 ft. x = 100 is rejected: the field is only 100 ft across, so a 100 ft ring on each side is impossible
17: (150 + 2x)(200 + 2x) = 45,000 → x² + 175x − 3750 = 0 → x = (−175 + √45625) / 2 = (25√73 − 175) / 2 ≈ 19.3 ft
18: l + w = 22, so w(22 − w) = 105 → w² − 22w + 105 = 0 → (w − 7)(w − 15) = 0 → 7 m by 15 m
19: 50% more means multiply by 1.5, so 2x(x − 4) = 1.5x² → 0.5x² − 8x = 0 → x(x − 16) = 0 → original side 16 m, new garden 32 m by 12 m = 384 m²
20: (8 + 2x)(10 + 2x) ≤ 143 → 4x² + 36x − 63 ≤ 0 → (2x + 21)(2x − 3) ≤ 0 → 0 < x ≤ 1.5 → widest strip 1.5 in

↑ Back to Part D

Every answer above was checked for accuracy before publishing, most recently on July 31, 2026. If you spot an error, email burketutoringinfremont@outlook.com or text (510) 453-0350 and it will be corrected.

Common Questions

What is a quadratics rectangle problem?

Any situation where two sides of a rectangle depend on each other and you are given the area. Naming one side x forces the other side to be written in terms of x, and multiplying them gives a quadratic. Patios, gardens, picture frames, mowed lawns and printed pages are all the same problem.

Why is it 2x and not x for a frame?

Because the frame is on both sides. A 2 in frame around a 10 in photo makes the outside 14 in, not 12. Draw the picture and count the bands: two per dimension, every time. This single line is worth more marks than any other sentence on the page.

Which side should I call x?

Whichever one the sentence describes the other in terms of. "The length is 7 more than the width" means call the width x and the length x + 7. Choosing the other way round works too, it is just uglier for no reward.

Both answers are positive. Now what?

Check them against the picture, not the algebra. Problem 15 gives 5 and 30, and a 30 ft strip taken off each side of a 30 ft lawn removes 60 ft from a 30 ft width. Positive is not the same as possible.

Why do some problems give a perimeter instead of a second side?

Same information, wrapped differently. If the perimeter is 44, then length plus width is 22, so the sides are x and 22 − x. Halve first, then subtract. Problem 18 is exactly this.

What grade level is this quadratics rectangle worksheet?

Parts A and B sit squarely in Algebra I. Parts C and D show up in Algebra I too, then return in Algebra II and in the SAT's harder math module wearing slightly different clothes.

Can I use this in my classroom?

Print it, copy it, project it, hand it out. No sign-up and no attribution required, though a link back is always welcome.

More Quadratics Practice

Rectangle problems assume the solving already works. When a part keeps stalling, the trouble is almost always upstream of the picture.

Solving: Solving by factoring · the quadratic formula · completing the square · the discriminant
Maximum area: Largest-pen questions are vertex questions: vertex and axis of symmetry · graphing quadratic functions · writing the equation from a graph · quadratic transformations
Other stories: Projectiles, revenue and consecutive integers: quadratics word problems. Curved reflectors instead of flat rectangles: focus and directrix word problems. Not sure the model is even quadratic? Compare the function families
Keeping it warm: Quadratics spiral review · quadratics error analysis
Checking work: Sketch the area function in GeoGebra or push the roots through MathPapa. Neither will read the sentence for you, which is the actual difficulty

Standards and References

These problems were written against the following standards and checked against the following references.

Building an equation from a described geometric situation is CCSS.MATH.CONTENT.HSA.CED.A.1. Standard text: Common Core, High School Algebra: Creating Equations.
Solving the resulting equation by factoring, completing the square or the formula is HSA.REI.B.4.b.
Exam-style parallels for Parts C and D: JMAP, which sorts released New York Regents questions by standard, including the geometric applications of quadratics set.
Classic rectangle, frame and spiral-mowing problems in an openly licensed text: LibreTexts Mathematics, which hosts Tyler Wallace's Beginning and Intermediate Algebra under CC BY 3.0.
Position on modeling and problem representation in algebra: National Council of Teachers of Mathematics.
Plain-English refreshers for students: MathBits Notebook, Cuemath on area of a rectangle, and eMathHelp for a stepped second opinion.
Reuse: the problems on this page are original to Burke Tutoring and free to print, copy or project. The classic problem types they follow — frames, borders and spiral mowing — appear in openly licensed algebra texts distributed under CC BY 3.0.
Answer verification: every solution was substituted back into its own equation, checked against the picture for impossible roots, and cross-checked with an independent area calculator and GeoGebra.

About the Author

Berke Sahbazoglu

Berke has taught K–12 math and science for more than 10 years, logging over 6,000 tutoring hours with 200+ students. He holds a BS in Biochemistry from Washington University in St. Louis and an MS in Bioinformatics from UMGC.

The four parts on this page are the order the mistakes actually arrive in. Students rarely fail at factoring here; they fail at the picture, and then the factoring never had a chance.

How these are made: problems are prepared from previous class notes and assigned homework, then checked for accuracy before publishing. Corrections come in by email or text and get fixed the same week.

More: tutor profile · LinkedIn · burketutoringinfremont@outlook.com

Page Updates

July 31, 2026 — quadratics rectangle worksheet published with twenty problems in four parts, three color-coded figures, the eight-page printable PDF and the full answer key.

Aug 01, 2026

Need one-on-one help?

Frame and border problems turn up in Algebra I at every high school around here — Irvington, American, Mission San Jose, Washington and Kennedy in Fremont, Newark Memorial, and James Logan in Union City. They are quick to fix in person, because most of the fix is sitting next to a student while they draw the picture. Burke Tutoring offers in-home algebra tutoring in Fremont, Newark, and Union City. Call (510) 453-0350.

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