Space Inside the Figure
Area measures the amount of surface covered by a two-dimensional shape.
Free Geometry Guide
Area and Perimeter
Learn how to find area and perimeter of rectangles and squares, solve for missing dimensions, choose the correct units, work through word problems, and handle composite L-shaped figures.
Area and perimeter both describe measurements of two-dimensional figures, but they measure very different things. Area measures the space inside a figure, while perimeter measures the distance around the outside.
Many mistakes happen because students begin calculating before deciding which measurement the problem actually asks for. A reliable solution starts by identifying whether you need area or perimeter, choosing the correct formula, substituting the dimensions, and using the correct type of units.
What Is the Difference Between Area and Perimeter?
Before choosing a formula, ask what the problem is measuring. If it asks how much space is inside a shape, you need area. If it asks for the total distance around the outside edge, you need perimeter.
Distance Around the Figure
Perimeter is the sum of the lengths of all the outside sides.
| Measurement | What It Measures | Typical Question | Units |
|---|---|---|---|
| Area | Space inside a figure | How much flooring covers the room? | Square units, such as cm² or ft² |
| Perimeter | Distance around the outside | How much fencing surrounds the yard? | Linear units, such as cm or ft |
A useful question to ask yourself:
Am I measuring what is inside the figure, or am I traveling around its boundary?
Area and Perimeter Formulas for a Rectangle
A rectangle has two pairs of equal opposite sides. We usually call the longer dimension the length and the shorter dimension the width.
Rectangle Example
Length = 8 cm and width = 5 cm
5 cm
8 cm
Length = 8 cm | Width = 5 cm
A = l × w
A = 8 × 5
A = 40 cm²
Multiply the length by the width to find the number of square units inside the rectangle.
P = 2l + 2w
P = 2(8) + 2(5)
P = 26 cm
A rectangle has two lengths and two widths, so perimeter adds both pairs of sides.
Why does P = 2l + 2w work?
Walking around a rectangle gives length + width + length + width. Combining the two equal lengths and two equal widths gives 2l + 2w.
Area and Perimeter Formulas for a Square
A square is a special rectangle in which all four sides have the same length. Because every side is equal, we only need one variable, usually written as s.
Square Example
Side length = 6 m
6 m
6 m
Every side of the square is 6 m.
A = s²
A = 6²
A = 36 m²
Squaring the side length is the same as multiplying the side by itself.
P = 4s
P = 4(6)
P = 24 m
Since all four sides are equal, multiply one side length by 4.
Linear Units vs. Square Units
The number alone is not a complete measurement. Area and perimeter use different types of units because they describe different quantities.
26 cm
Perimeter measures one-dimensional distance, so use ordinary linear units.
40 cm²
Area measures two-dimensional space, so the unit must be squared.
| Original Unit | Perimeter Unit | Area Unit |
|---|---|---|
| centimeters | cm | cm² |
| meters | m | m² |
| feet | ft | ft² |
| inches | in | in² |
Do not square perimeter units.
A perimeter of 26 centimeters is written 26 cm, not 26 cm². Square units are reserved for area.
How to Find a Missing Dimension
Sometimes the area or perimeter is given, but one side length is missing. In that case, use the same formula you normally would, substitute the values you know, and solve for the unknown dimension.
The formula does not change.
The difference is that instead of calculating area or perimeter, you are using the known measurement to work backward and find a missing side.
Area = 48 cm² and width = 6 cm
Find the missing length.
- Start with the rectangle area formula: A = l × w.
- Substitute the known values: 48 = l × 6.
- Divide both sides by 6.
- l = 8.
- The missing length is 8 cm.
Perimeter = 34 m and length = 12 m
Find the missing width.
- Start with: P = 2l + 2w.
- Substitute: 34 = 2(12) + 2w.
- Simplify: 34 = 24 + 2w.
- Subtract 24: 10 = 2w.
- Divide by 2: w = 5 m.
Area:
A = l × w
Missing side:
l = A ÷ w
A missing side is a length.
Even if the given measurement is area in square units, the missing dimension itself uses linear units.
Area and Perimeter of L-Shaped Composite Figures
A composite figure is made from more than one simple shape. An L-shaped figure can often be viewed as a large rectangle with a smaller rectangular corner removed.
The strategy depends on whether the problem asks for area or perimeter. For area, subtract the missing rectangle. For perimeter, trace the full outside boundary and add every outside side.
L-Shaped Figure Example
Overall width = 12 m, overall height = 9 m
Missing horizontal = 12 − 7 = 5 m |
Missing vertical = 9 − 5 = 4 m
Outer rectangle − missing rectangle
A = (12 × 9) − (5 × 4)
A = 108 − 20
A = 88 m²
Find the area of the complete outer rectangle, then subtract the rectangular section that is missing.
P = 7 + 4 + 5 + 5 + 12 + 9
P = 42 m
Trace the outside boundary and add each of the six edge lengths.
Do not include interior lines in perimeter.
Perimeter only counts the outside boundary. Any line used to split the shape into smaller rectangles is not automatically part of the perimeter.
Area and Perimeter Word Problems
In a word problem, the first challenge is deciding which measurement matches the situation. Certain words and phrases often suggest area or perimeter, but the meaning of the situation matters more than memorizing keywords.
| Situation | Usually Means | Why |
|---|---|---|
| Flooring, carpet, tile, grass, paint covering a flat surface | Area | You are covering the inside surface. |
| Fencing, edging, trim, border, frame | Perimeter | You are measuring around the outside. |
| Size of a rectangular garden | Depends on the question | Soil or grass may require area; fencing requires perimeter. |
| Finding an unknown side from a known area or perimeter | Missing dimension | Use the appropriate formula and solve backward. |
New carpet for a bedroom
A rectangular bedroom is 12 ft long and 10 ft wide. How many square feet of carpet are needed?
- Carpet covers the floor, so find area.
- Use A = l × w.
- A = 12 × 10 = 120.
- Answer: 120 ft².
Fence around a garden
A rectangular garden is 15 m long and 8 m wide. How much fencing is needed to surround it?
- Fencing goes around the outside, so find perimeter.
- Use P = 2l + 2w.
- P = 2(15) + 2(8).
- P = 30 + 16 = 46.
- Answer: 46 m.
Focus on what is being measured.
Instead of relying only on keywords, picture the situation. Are you covering the interior surface or measuring around the boundary?
Worked Area and Perimeter Examples
These examples show the same decision-making process used throughout the guide: identify what the problem asks for, choose the correct formula or strategy, substitute the known values, calculate, and attach the correct units.
Length 11 cm, width 4 cm
- The problem asks for area.
- A = l × w
- A = 11 × 4
- A = 44
- 44 cm²
Side length 7 m
- The problem asks for perimeter.
- P = 4s
- P = 4(7)
- P = 28
- 28 m
Area 72 ft², width 8 ft
- A = l × w
- 72 = l × 8
- l = 72 ÷ 8
- l = 9
- 9 ft
Outer rectangle 10 × 8, cutout 3 × 2
- Outer area = 10 × 8 = 80.
- Missing area = 3 × 2 = 6.
- Composite area = 80 − 6.
- Composite area = 74.
- 74 square units
Notice what changes and what stays the same.
The figures may become more complicated, but the core process remains consistent: understand the measurement, use the correct relationship, calculate, and check the units.
Common Area and Perimeter Mistakes
Most area and perimeter mistakes come from choosing the wrong measurement, using the wrong formula, overlooking a side, or attaching the wrong units. Checking these details before finishing can catch many errors.
1. Finding area when the problem asks for perimeter
Area measures the space inside a figure. Perimeter measures the distance around it. Decide which measurement the problem asks for before choosing a formula.
2. Using square units for perimeter
Perimeter uses linear units such as cm, m, or ft. Area uses square units such as cm², m², or ft².
3. Forgetting that a rectangle has four sides
Adding only the length and width gives half of the rectangle's perimeter. Use P = 2l + 2w, or add all four sides.
4. Multiplying when finding perimeter
Multiplying length by width finds rectangle area, not perimeter. Perimeter comes from adding side lengths.
5. Giving a missing side in square units
A side length is one-dimensional. If you solve an area problem and find that the missing length is 8 centimeters, write 8 cm, not 8 cm².
6. Ignoring missing sides in a composite figure
An L-shaped figure may not label every side directly. Use the overall width or height and the corresponding shorter segment to determine the missing length before calculating.
7. Adding an interior line to the perimeter
Only count edges that are part of the outside boundary. A line drawn inside a composite figure to divide it into rectangles does not automatically belong in the perimeter.
8. Treating every word problem the same way
A garden problem could ask for area if you are covering it with grass, or perimeter if you are surrounding it with fencing. Use the context to determine what is actually being measured.
A Step-by-Step Strategy for Area and Perimeter Problems
Whether you are working with a basic rectangle, a missing dimension, or an L-shaped composite figure, use a consistent process instead of trying to calculate immediately.
Step 1: Identify the goal
Decide whether the problem asks for area,
perimeter, or a missing side length.
Step 2: Study the figure
Identify the dimensions you know and determine
whether any necessary side lengths are missing.
Step 3: Choose the relationship
Select the appropriate formula or composite-figure
strategy.
Step 4: Substitute
Replace the variables with the known dimensions
before doing the arithmetic.
Step 5: Calculate
Perform the multiplication, addition, subtraction,
or division required by the problem.
Step 6: Attach units
Use linear units for perimeter and side lengths,
and square units for area.
Step 7: Check your answer
Ask whether the result makes sense for the figure
and whether the units match the measurement.
Do not start with the formula.
Start by understanding what the problem is asking. Once you know whether you need area, perimeter, or a missing dimension, choosing the correct formula becomes much easier.
Quick check: area = inside + square units. Perimeter =
outside boundary + linear units.
Free Interactive Practice
Practice Area and Perimeter Step by Step
Use the interactive Area & Perimeter tool to practice rectangles, squares, area, perimeter, missing dimensions, word problems, and advanced composite figures with guided feedback.
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