How to Find the Area of Squares, Rectangles, and Parallelograms

How to Find the Area of Squares, Rectangles, and Parallelograms

 Read,5 minutes

Squares, rectangles, and parallelograms all use base-height area relationships. For rectangles and squares, adjacent sides are perpendicular. For parallelograms, the height must be perpendicular to the chosen base, not the slanted side unless it is also perpendicular.

Core Formulas

  • Square: \(A=s^2\).
  • Rectangle: \(A=lw\).
  • Parallelogram: \(A=bh\).
  • Missing dimension: divide area by the known dimension.

Worked Example

A parallelogram with base \(15\) and height \(8\) has area \(15(8)=120\) square units.

Reference Diagram

Parallelograms

Square

Rectangles

Original Practice Figures

These saved figures are kept with the lesson for continuity. The exercise text below gives all needed measurements.

Parallelograms Squares Rectangles Rectangles Rectangles Parallelograms Squares Squares Squares Rectangles Parallelograms Squares Rectangles Rectangles Rectangles Parallelograms Squares Squares Squares Rectangles

Area of Squares, Rectangles, and Parallelograms

Think of this lesson as more than a rule to memorize. Area of Squares, Rectangles, and Parallelograms is about shape relationships, formulas, and units. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Sketch or label the shape.
  • Decide whether the question asks for length, area, volume, or surface area.
  • Substitute values into the matching formula.
  • Keep units squared for area and cubed for volume.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) Find the area of a square with side length 7 cm.

2) Find the area of a rectangle 12 ft by 5 ft.

3) Find the area of a parallelogram with base 10 in and height 6 in.

4) A rectangle has area 84 square m and length 12 m. Find its width.

5) A square has area 121 square inches. Find its side length.

6) Find the area of a rectangle 8.5 yd by 4 yd.

7) A parallelogram has area 96 square cm and base 16 cm. Find its height.

8) A square garden has side 13 ft. Find its area.

9) A rectangle is 18 m long and 11 m wide. Find its area.

10) A parallelogram has base 14 and height 14. Find its area.

11) A rectangle has perimeter 54 and length 17. Find its area.

12) A square has diagonal \(10\\sqrt{2}\). Find its area.

13) A parallelogram has base \(3x\), height 8, and area 120. Find \(x\).

14) A rectangle has length \(x+5\), width 9, and area 126. Find \(x\).

15) A square and rectangle have equal areas. The square side is 12, and the rectangle length is 18. Find the rectangle width.

16) A parallelogram has base 25 ft and height 18 ft. Find the area.

17) A rectangular floor is 24 ft by 15 ft. Tile costs $3 per square foot. Find the total cost.

18) A square has perimeter 64 cm. Find its area.

19) A parallelogram has area 210 square yd and height 14 yd. Find its base.

20) A rectangle has length twice its width and area 200. Find the width.

 
1) Use \(A=s^2\).
\(A=7^2=49\).
The area is \(49\) square cm.
2) Use \(A=lw\).
\(A=(12)(5)=60\).
The area is \(60\) square ft.
3) Use \(A=bh\).
\(A=(10)(6)=60\).
The area is \(60\) square in.
4) Divide the area by the known dimension.
\(w=84/12=7\).
The width is \(7\) m.
5) For a square, \(A=s^2\).
\(121=s^2\), so \(s=\\sqrt{121}=11\) in.
6) Use \(A=lw\).
\(A=(8.5)(4)=34\).
The area is \(34\) square yd.
7) Use \(A=bh\).
\(96=16h\).
So \(h=6\) cm.
8) Use \(A=s^2\).
\(A=13^2=169\).
The area is \(169\) square ft.
9) Use \(A=lw\).
\(A=(18)(11)=198\).
The area is \(198\) square m.
10) Use \(A=bh\).
\(A=14(14)=196\).
The area is \(196\) square units.
11) First find width from \(P=2L+2w\): \(54=2(17)+2w\).
\(2w=20\), so \(w=10\).
Area \(=17(10)=170\).
12) For a square, diagonal \(=s\\sqrt{2}\).
If the diagonal is \(10\\sqrt{2}\), then \(s=10\).
Area \(=10^2=100\).
13) Use \(A=bh\).
\(120=(3x)(8)=24x\).
\(x=5\).
14) Use \(A=lw\).
\(126=9(x+5)\). Divide by \(9\): \(14=x+5\).
So \(x=9\).
15) The square area is \(12^2=144\).
Set rectangle area equal: \(18w=144\).
\(w=8\).
16) Use \(A=bh\).
\(A=(25)(18)=450\).
The area is \(450\) square ft.
17) Area \(=(24)(15)=360\) square ft.
Cost \(=360(3)=\$1080\).
18) A square has \(4\) equal sides, so \(s=64/4=16\).
Area \(=16^2=256\) square cm.
19) Use \(A=bh\).
\(210=14b\).
\(b=15\) yd.
20) Let the width be \(w\), so the length is \(2w\).
Area: \(200=(2w)(w)=2w^2\).
\(w^2=100\), so \(w=10\) units.

More Practice

Free printable Worksheets

Related Lessons