Perimeter: Dimensions & Units

Learn and practice calculating perimeter with length unit conversions, and understand the difference between 1D and 2D scaling.

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Dimensions & Units

Use the workspace below. Write equations like P = 24 to solve for the perimeter.

📖 Perimeter Study Guide

1. What Is a Dimension?

A dimension tells us how many independent directions we need to measure a shape. Refer to the Dimensions & Units guide on the Area Page for details on dimensions and the metric/US customary systems.
• A line or boundary is 1-dimensional (1D) — it has only length.
• A flat surface is 2-dimensional (2D) — it has width and height.
• A solid object is 3-dimensional (3D).

2. Units of Perimeter

Because perimeter measures the boundary of a shape (a 1D length), it is always measured in linear units (e.g., mm, cmcm, ftft, inin), never in square units (which are for area). For example, if a rectangle has sides of 5 m5\text{ m} and 3 m3\text{ m}, its perimeter is 16 m16\text{ m} (linear), while its area is 15 m215\text{ m}^2 (square).
Rectangle (5m × 3m)5 m3 mArea = 15 m²(5 × 3 = 15)[2D Surface]Perimeter = 16 m(5 + 3 + 5 + 3 = 16) [1D Line]
Triangle (3m - 4m - 5m)3 m4 m5 mArea = 6 m²(½ × 4 × 3)[2D Surface]Perimeter = 12 m(3 + 4 + 5 = 12) [1D Line]
Circle (r = 3m)3 mArea ≈ 28.27 m²(π × 3²)[2D Surface]Perimeter ≈ 18.85 m(2 × π × 3) [1D Line]

3. Converting Length Units

To convert perimeter, use the standard length conversion formulas from the Formula Sheet:
• Metric: 1 m=100 cm=1,000 mm1\text{ m} = 100\text{ cm} = 1{,}000\text{ mm}, 1 cm=10 mm1\text{ cm} = 10\text{ mm}, 1 dm=10 cm1\text{ dm} = 10\text{ cm}.
• US Customary: 1 mi=1,760 yd1\text{ mi} = 1{,}760\text{ yd}, 1 yd=3 ft1\text{ yd} = 3\text{ ft}, 1 ft=12 in1\text{ ft} = 12\text{ in}.
• Multiply when converting from a larger unit to a smaller one (e.g., 3 ft×12=36 in3\text{ ft} \times 12 = 36\text{ in}). Divide when going from smaller to larger.

4. Perimeter vs. Area Scaling

When you enlarge a shape by a percentage (scaling all its dimensions by a scale factor kk):
• Perimeter (1D) scales linearly by kk. The ratio of the new perimeter to the old is k:1k:1.
• Area (2D) scales quadratically by k2k^2. The ratio of the new area to the old is k2:1k^2:1.

Example: If you increase all sides of a shape by 50%50\%, the scale factor is 1.5=3/21.5 = 3/2.
• The ratio of new perimeter to old perimeter is 3:2\mathbf{3:2}.
• The ratio of new area to old area is (3/2)2=9:4(3/2)^2 = \mathbf{9:4}.

5. Curiosity: The Rope Around the Earth

Imagine wrapping a rope tightly around the Earth's equator (assuming the Earth is a perfect sphere of radius RR). Its length is the circumference C1=2πRC_1 = 2\pi R.
Now, suppose you want to add just enough rope so that the rope hovers exactly 1 meter1\text{ meter} above the ground all the way around. The new radius is R+1 mR + 1\text{ m}, and the new circumference is C2=2π(R+1)=2πR+2πC_2 = 2\pi(R + 1) = 2\pi R + 2\pi.
The amount of rope you need to add is:
C2−C1=2π(R+1)−2πR=2π≈6.28 metersC_2 - C_1 = 2\pi(R + 1) - 2\pi R = 2\pi \approx 6.28\text{ meters}

Surprisingly, it does not depend on the radius of the Earth! Whether wrapping a tennis ball or the entire Earth, you only need to add about 6.28 m6.28\text{ m} of rope. This is because the relationship is linear (1-dimensional), a concept we will explore in future topics.

Mastering SealMath: Ratio & Unit Answers

• For ratio problems: enter your answer in a:b format (e.g., type 3:2 or 9:4). Make sure the ratio is fully simplified (no common factors).
• For calculation problems: enter your numerical answer as P = value. The unit is shown in the question, so do not type it.
Learning Topics

Frequently Asked Questions

What is perimeter?

Perimeter is the total boundary of a two-dimensional shape. It includes both the outer boundary and any inner boundaries (like the edges of holes inside the shape).

How do you calculate the perimeter of a rectangle?

The perimeter of a rectangle is the sum of all its four sides: P = 2w + 2h or P = 2(w + h), where w is the width and h is the height.

Why is a square's perimeter formula P = 4s?

A square is a special case of a rectangle where width and height are equal (w = h = s). Substituting this into the rectangle formula gives P = 2(s + s) = 4s.

How does a hole affect the perimeter?

Since perimeter measures the entire boundary of a shape, a hole adds to the perimeter. The total perimeter is the outer perimeter plus the inner perimeter (the perimeter of the hole).

How do you find the perimeter of a right-angled triangle if one side is missing?

Since we've already learned the Pythagorean theorem (a² + b² = c²), we can calculate the missing side length first and then sum all three sides together to get the perimeter.

What unit is used for perimeter?

Since perimeter is a one-dimensional length (boundary), it is measured in linear units like meters (m), centimeters (cm), feet (ft), or inches (in). It is never measured in square units, which are reserved for area.

How does doubling a shape's dimensions affect its perimeter and area?

Doubling all dimensions (scale factor of 2) doubles the perimeter (ratio 2:1) because perimeter is linear (1D). However, it quadruples the area (ratio 4:1) because area is two-dimensional (2D) and scales quadratically (2² = 4).

Perimeter: Dimensions & Units | SealMath