TACHS: Perimeter and area: solve for the sides first
Use a perimeter or area to find unknown side lengths, then the other measure, and count perimeter on a grid figure.
When a rectangle's sides are written with x, how do you get from its perimeter to its area?
Short answer
Write the formula you are given information about, solve for x, then find the real side lengths and use them in the other formula. Perimeter is 2 × (length + width); area is length × width.
Know first: Solving a two-step equation, Combining like terms such as x + x
Step 1 of 6
Perimeter goes around, area fills in
Perimeter is the distance around a shape. For a rectangle, it is 2 × (length + width), measured in plain units such as cm. Area is the space inside, length × width, measured in square units such as square cm.
When sides are written as expressions like (x + 5) and (x − 1), the measurement you are given becomes an equation for x. Solve it, find the actual sides, then use them in the other formula.
| You know | Write | Then |
|---|---|---|
| Perimeter, sides with x | 2 × (length + width) = perimeter | Solve for x, find sides, multiply for area |
| Area, one side a multiple of the other | length × width = area | Find the sides, then 2 × (length + width) |
| Both sides grow by the same amount | New area − old area | Do not just square the growth |
| A figure on a grid | Trace the outline | Count each unit edge once |
x is almost never the final answer. Find the sides.
Step 2 of 6
Solve for x, then build the sides
Worked example: Sides written with x
A rectangle has length (x + 5) cm and width (x − 1) cm. Its perimeter is 36 cm. What is its area, in square centimeters?
- A7
- B18
- C36
- D72
- 1
Write the perimeter equation
2 × ((x + 5) + (x − 1)) = 36, so 2 × (2x + 4) = 36.
- 2
Solve
Halve both sides: 2x + 4 = 18. Subtract 4: 2x = 14. Divide: x = 7.
- 3
Find the sides
Length: 7 + 5 = 12 cm. Width: 7 − 1 = 6 cm. Check: 2 × (12 + 6) = 36.
- 4
Multiply for area
12 × 6 = 72 square centimeters.
- 5
See why the other choices are there
7 is x, not an area. 18 is half the perimeter, length + width. 36 is the perimeter itself. Each one stops too early.
Answer
The area is 72 square centimeters.
From area to perimeter
A rectangle's length is twice its width, and its area is 50 square centimeters. What is its perimeter?
- 1
Let the width be W, so the length is 2W and W × 2W = 50.
- 2
2W² = 50, so W² = 25 and W = 5. The length is 10.
- 3
Perimeter: 5 + 10 = 15 cm.
What went wrong
Length + width covers only half of the way around. A rectangle has two lengths and two widths.
The fix
Perimeter = 2 × (5 + 10) = 30 cm.
Step 3 of 6
Grid figures and growing sides
On a grid of 1 cm squares, find perimeter by tracing the outline and counting each unit edge once. Take a rectangle 4 squares long and 3 squares tall: its perimeter is 14 cm. Remove the corner square and the perimeter is still 14, because the outline just steps in. Remove a square from the middle of the top edge instead, and the notch adds 2 edges, making 16. Both shapes have an area of 11 square cm.
When both sides grow, compute the new area and subtract. An 8 by 5 rectangle has area 40. Add 3 to each side: 11 × 8 = 88, an increase of 48, not 3 × 3 = 9.
Step 4 of 6
Find the sides before you answer
Check yourself · Question 1
A rectangle has width x cm and length (2x + 1) cm. Its perimeter is 50 cm. What is its area, in square centimeters?
Answer: D
2 × (x + 2x + 1) = 50, so 3x + 1 = 25 and x = 8. Width 8 cm, length 17 cm. Check: 2 × (8 + 17) = 50. Area: 8 × 17 = 136.
Trap answer: 25
Stopping at half the perimeter
- Why it looks right
- 25 shows up in the middle of the work, right after you divide by 2.
- Why it's wrong
- 25 is length + width, a distance in centimeters. The question asks for the space inside, in square centimeters.
- The right answer
- 136, because the sides are 8 cm and 17 cm.
Check yourself · Question 2
A rectangle's length is 3 cm more than its width, and its area is 54 square centimeters. What is its perimeter, in centimeters?
Answer: C
Try widths: 6 × 9 = 54, and 9 is 3 more than 6. Perimeter: 2 × (6 + 9) = 30 cm.
Check yourself · Question 3
On a grid of 1 cm squares, a figure is a rectangle 5 squares long and 2 squares tall, with one more square attached on top of its left end. What is the perimeter of the figure, in centimeters?
Answer: C
The rectangle alone has perimeter 2 × (5 + 2) = 14. The added square covers 1 edge of the rectangle and adds 3 new edges, a net gain of 2. Perimeter: 16 cm.
Common mistake
"The perimeter is length + width."
- Why it's tempting
- Length + width is the first sum you write, and it is half of the right answer.
- Do this instead
- Perimeter goes all the way around: 2 × (length + width). Sketch the rectangle and label all four sides if you are unsure.
Step 5 of 6
What to remember
Remember
- Perimeter is 2 × (length + width); area is length × width, in square units.
- Solve for x, then find the real side lengths before using the other formula.
- On a grid, count each outside edge once; shared edges are inside the figure.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what are the two side lengths, and which wrong choice is the perimeter or half of it?