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Length

Length Units


We can measure length in many ways, for example with our footsteps or with paper clips. But everyone's footsteps and paper clips are different sizes! How can we share our measurements if we all use different units?
To solve this, people all around the world agreed to use the same units. We call these standard units. One very common standard unit for length is the meter.

Definition Units of Length
We use different units for measuring small and large things.
  • Millimeter \(\left(\mathrm{mm}\right)\): A very small unit of length, about the thickness of a coin.
  • Centimeter \(\left(\mathrm{cm}\right)\): A small unit of length, about the width of your finger.
  • Meter \(\left(\mathrm{m}\right)\): A longer unit of length, about the height of a 6-year-old girl.
  • Kilometer \(\left(\mathrm{km}\right)\): A very large unit of length, used for long distances, like the distance between towns. It is about the height of the Burj Khalifa in Dubai, United Arab Emirates.

Conversion of Length Units

Definition Conversion of Length Units
Here are some useful metric conversions:
  • \(1~\text{km}=1\,000~\text{m}\)
  • \(1~\text{m}=100~\text{cm}\)
  • \(1~\text{cm}=10~\text{mm}\)
Method Converting with Multiplication or Division
  • Use multiplication when you go from a bigger unit to a smaller one (e.g., \(\mathrm{m} \to \mathrm{cm}\)).
  • Use division when you go from a smaller unit to a bigger one (e.g., \(\mathrm{cm} \to \mathrm{m}\)).
Method Converting Using a Table
To convert between units of length, we can use a metric place value table. This table shows the main metric units from kilometers to millimeters. Each column represents one step of 10 or 100 or 1 000 between units. Let's convert 1.2 meters to centimeters.
  1. Draw the full metric place value table.
    \(\mathrm{km}\) \(\quad\;\) \(\quad\;\) \(\;\mathrm{m}\) \(\quad\;\) \(\mathrm{cm}\) \(\mathrm{mm}\)
  2. Place the number in the table.
    The rule is: the digit in the ones place goes in the starting unit's column.
    For \(1.2\) m, the ones digit is \(1\), so it goes in the m column. The digit \(2\) (the tenths) goes in the next column to the right.
    \(\mathrm{km}\) \(\quad\;\) \(\quad\;\) \(\;\mathrm{m}\) \(\quad\;\) \(\mathrm{cm}\) \(\mathrm{mm}\)
    1 2
  3. Fill any empty spaces with zeros until you reach your target unit.
    Our target unit is cm, so we put a \(0\) in the cm column.
    \(\mathrm{km}\) \(\quad\;\) \(\quad\;\) \(\;\mathrm{m}\) \(\quad\;\) \(\mathrm{cm}\) \(\mathrm{mm}\)
    1 2 0
  4. Read the final number.
    Now read the digits as a number in centimeters:
    \(1.2 \ \text{m} = 120 \ \text{cm}.\)
    This matches the fact that we multiply by \(100\) when converting m to cm.

Perimeter

Definition Perimeter
The perimeter of a shape is the total distance all the way around its outside edge.
Method Finding the perimeter
To find the perimeter of any shape, add the lengths of all its sides together.
Example
Find the perimeter of the red shape. Each square on the grid is 1 unit long.

  • Step 1: Find the length of each side by counting the units on the grid.
    The sides are 2 units, 2 units, 2 units, and 2 units long.
  • Step 2: Add the lengths of all the sides.$$\begin{aligned} \text{Perimeter} &= 2 + 2 + 2 + 2\\ &= 8\end{aligned} $$The perimeter of the shape is 8 units.

Perimeter of Common Shapes

Method Finding a Polygon’s Perimeter
To find the perimeter of any polygon (a closed shape with straight sides), add up the lengths of all its sides.
Proposition Perimeter Formulas
Shape Diagram Perimeter Formula
Triangle \(P = a + b + c\)
Rectangle \(\begin{aligned}P &= l + w + l + w\\ &= 2l + 2w \\ &= 2(l + w)\end{aligned}\)
Square \(P = s + s + s + s = 4s\)
Circle \(P = 2\pi r\)
Example
Find the perimeter of the rectangle:

The rectangle has a length \(l = 14\) m and a width \(w = 7\) m. We can use either perimeter formula.
  • Method 1 (add both pairs of equal sides):$$\begin{aligned}[t]P &= 2 \times l + 2 \times w \\ &= 2 \times 14 + 2 \times 7 \\ &= 28 + 14 \\ &= 42 \,\mathrm{m}\end{aligned}$$
  • Method 2 (factorise):$$\begin{aligned}[t]P &= 2 \times (l + w) \\ &= 2 \times (14 + 7) \\ &= 2 \times 21 \\ &= 42 \,\mathrm{m}\end{aligned}$$
The perimeter of the rectangle is \(42\) m.

Length of an Arc

Definition Arc of a Circle
An arc is a part of the circumference of a circle, between two points on the circle, defined by its central angle \(\theta\) (theta).
Example
A semicircle is an arc with a central angle of \(180^\circ\).
Method Finding the Length of an Arc
To find the length of an arc, you take a fraction of the full circumference.
  1. Find the fraction of the circle. This is the arc’s central angle (in degrees) divided by \(360^\circ\):$$ \text{Fraction} = \frac{\text{central angle } (\theta)}{360^\circ}. $$
  2. Multiply the fraction by the full circumference. Remember, the circumference of a circle with radius \(r\) is \(C = 2 \pi r\):$$ \text{Arc Length} = \text{Fraction} \times (2 \pi r). $$
So, if the central angle is \(\theta\) (in degrees) and the radius is \(r\),$$ \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r. $$
Example
Find the length of the arc in the figure below.

We will follow the two-step method. The given values are \(\theta = 90^\circ\) and \(r = 4\) cm.
  1. Step 1: Find the fraction of the circle.$$ \text{Fraction} = \frac{90^\circ}{360^\circ} = \frac{1}{4}. $$
  2. Step 2: Multiply the fraction by the full circumference.$$ \begin{aligned}\text{Arc Length} &= \frac{1}{4} \times (2 \times \pi \times r) \\ &= \frac{1}{4} \times (2 \times \pi \times 4) \\ &= \frac{1}{4} \times 8\pi \\ &= 2\pi \,\text{cm}.\end{aligned} $$
The exact length of the arc is \(2\pi\) cm. For an approximate value, we use a calculator:$$ \text{Arc Length} \approx 6.28 \,\text{cm}. $$

Perimeter of Composite Figures

Definition Composite Figure
A composite figure is a shape made by joining two or more simple shapes (like rectangles, squares, and triangles).
Method Finding the Perimeter of a Composite Figure
To find the perimeter of a composite figure:
  1. Identify all the outer sides of the figure. Be careful not to include any lines inside the shape.
  2. Find the lengths of any unknown sides. You may need to use information from the other sides to figure these out.
  3. Add the lengths of all the outer sides together.
Example
Find the perimeter of this composite figure, which is made of a square and a triangle.

\(\quad\)
\(P=\)+
\(\begin{aligned}[t]P &= \textcolor{colordef}{3\times 4 \mathrm{~cm}} + \textcolor{colorprop}{2\times 5 \mathrm{~cm}}\\P &= 22 \mathrm{~cm}\\\end{aligned}\)