3.1 Measurement of Length, Mass & Volume

Measurement of Length, Mass & Volume

Quick Summary: Measurement allows us to quantify physical attributes of objects using standardised metric units. Length measures distance between points, mass measures the quantity of matter, and volume measures three-dimensional space or liquid capacity. Converting between metric units involves multiplying or dividing by powers of 10 (10, 100, 1,000).

The metric system is an international decimal system of measurement used across the Caribbean and globally. Because it operates on powers of 10, converting between larger and smaller units requires moving the decimal point to the right (when converting to smaller units) or to the left (when converting to larger units).

Metric Units & Unit Conversions

Understanding the standard base units and prefixes is essential for accurate calculations in length, mass, and volume.

Standard Metric Standard Units

Conversion Equivalencies

• Length Equivalencies:

  • $1 \text{ kilometre (km)} = 1,000 \text{ metres (m)}$
  • $1 \text{ metre (m)} = 100 \text{ centimetres (cm)}$
  • $1 \text{ centimetre (cm)} = 10 \text{ millimetres (mm)}$
  • $1 \text{ metre (m)} = 1,000 \text{ millimetres (mm)}$

• Mass Equivalencies:

  • $1 \text{ kilogram (kg)} = 1,000 \text{ grams (g)}$
  • $1 \text{ gram (g)} = 1,000 \text{ milligrams (mg)}$
  • $1 \text{ metric ton (t)} = 1,000 \text{ kilograms (kg)}$

• Volume & Capacity Equivalencies:

  • $1 \text{ litre (L)} = 1,000 \text{ millilitres (mL)}$
  • $1 \text{ cubic centimetre (cm}^3\text{)} = 1 \text{ millilitre (mL)}$
  • $1,000 \text{ cubic centimetres (cm}^3\text{)} = 1 \text{ litre (L)}$

Rules for Unit Conversion

1. Converting Larger Units to Smaller Units: Multiply by the conversion factor (move decimal point to the right).
   • Example: To convert $3.4 \text{ km}$ to metres: $3.4 \times 1,000 = 3,400 \text{ m}$.
2. Converting Smaller Units to Larger Units: Divide by the conversion factor (move decimal point to the left).
   • Example: To convert $2,750 \text{ grams}$ to kilograms: $2,750 \div 1,000 = 2.75 \text{ kg}$.

Step-by-Step Conversion Example

Problem: A ribbon is $4.85 \text{ metres}$ long. A student cuts off $165 \text{ centimetres}$ to wrap a gift. How many metres of ribbon remain?

• Step 1: Convert $165 \text{ cm}$ to metres: $165 \div 100 = 1.65 \text{ m}$.
• Step 2: Subtract the cut ribbon length from the initial length: $4.85 \text{ m} - 1.65 \text{ m} = 3.20 \text{ m}$.
• Answer: $3.2 \text{ metres}$ of ribbon remain.

Perimeter of Polygons

The perimeter is the total continuous length around the outer edge or boundary of a closed two-dimensional shape. It is measured in linear metric units such as $\text{mm}$, $\text{cm}$, $\text{m}$, or $\text{km}$.

Formulas for Common Polygons

• General Polygon: Sum of all exterior side lengths. $$\text{Perimeter} = s_1 + s_2 + s_3 + \dots + s_n$$
• Rectangle: $$\text{Perimeter} = 2 \times (\text{length} + \text{width}) = 2l + 2w$$
• Square: $$\text{Perimeter} = 4 \times \text{side length} = 4s$$
• Regular Polygon with $n$ sides: $$\text{Perimeter} = n \times \text{side length}$$

Worked Example: Fencing a Rectangular Garden

Problem: A rectangular agricultural plot in Grenada measures $24.5 \text{ metres}$ in length and $12.0 \text{ metres}$ in width. A farmer wishes to place wire fencing around the complete perimeter, leaving a $1.5 \text{ metre}$ opening for a wooden gate. Calculate the total length of fencing wire required.

• Step 1: Calculate the total perimeter of the rectangle. $$\text{Perimeter} = 2 \times (24.5 \text{ m} + 12.0 \text{ m}) = 2 \times 36.5 \text{ m} = 73.0 \text{ m}$$
• Step 2: Subtract the width of the gate opening. $$\text{Fencing Wire Needed} = 73.0 \text{ m} - 1.5 \text{ m} = 71.5 \text{ m}$$
• Answer: The farmer needs $71.5 \text{ metres}$ of fencing wire.

Area of Rectangles, Squares & Triangles

Area measures the amount of flat two-dimensional surface surface enclosed inside a boundary. It is measured in square units such as $\text{cm}^2$ or $\text{m}^2$.

Primary Area Formulas

1. Square: $$\text{Area} = s \times s = s^2$$
2. Rectangle: $$\text{Area} = \text{length} \times \text{width} = l \times w$$
3. Triangle: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height} = \frac{1}{2}bh$$

Important Note on Triangles: The height $h$ used in the area formula must ALWAYS be perpendicular (forming a $90^\circ$ angle) to the chosen base $b$. Never use a slanted side length as the perpendicular height unless it forms a right-angled triangle.

Worked Example: Area of a Triangle

Problem: Find the area of a triangular sign board with a base length of $14 \text{ cm}$ and a vertical perpendicular height of $9 \text{ cm}$.

• Calculation: $$\text{Area} = \frac{1}{2} \times 14 \text{ cm} \times 9 \text{ cm} = 7 \text{ cm} \times 9 \text{ cm} = 63 \text{ cm}^2$$
• Answer: The area of the sign board is $63 \text{ cm}^2$.

Area of Composite Shapes

A composite shape is a figure constructed from two or more standard geometric shapes (such as combining rectangles, squares, or triangles).

Strategy for Finding Composite Area

1. Decompose: Split the complex figure into smaller non-overlapping standard shapes (Shape A, Shape B, etc.).
2. Determine Missing Dimensions: Use given side lengths to calculate any unknown edge lengths.
3. Calculate Individual Areas: Apply the appropriate area formula to each individual component.
4. Sum or Subtract Areas: Add the individual areas together (or subtract cut-out areas) to determine total composite area.

Worked Example: L-Shaped Floor Plan

Problem: An L-shaped room has a long bottom edge of $10 \text{ m}$, a total vertical left edge of $8 \text{ m}$, a top edge of $4 \text{ m}$, and a right-most vertical edge of $3 \text{ m}$. Calculate the total floor area.

• Step 1: Split the shape horizontally into two rectangles:
  • Top Rectangle (A): Width $= 4 \text{ m}$, Height $= 8 \text{ m} - 3 \text{ m} = 5 \text{ m}$.
  • Bottom Rectangle (B): Length $= 10 \text{ m}$, Height $= 3 \text{ m}$.
• Step 2: Calculate Area of Rectangle A: $$\text{Area}_A = 4 \text{ m} \times 5 \text{ m} = 20 \text{ m}^2$$
• Step 3: Calculate Area of Rectangle B: $$\text{Area}_B = 10 \text{ m} \times 3 \text{ m} = 30 \text{ m}^2$$
• Step 4: Sum the areas: $$\text{Total Area} = 20 \text{ m}^2 + 30 \text{ m}^2 = 50 \text{ m}^2$$
• Answer: The total floor area is $50 \text{ m}^2$.

Volume of Rectangular Prisms (Cuboids) & Capacity

Volume measures the amount of three-dimensional space occupied by a solid object, expressed in cubic units (such as $\text{cm}^3$ or $\text{m}^3$). Capacity refers to the volume of liquid a container can hold (measured in $\text{L}$ or $\text{mL}$).

Volume Formula for Rectangular Prisms

Alternatively, since $\text{length} \times \text{width}$ represents the base area:

Connecting Cubic Volume and Liquid Capacity

• $1 \text{ cm}^3 = 1 \text{ mL}$
• $1,000 \text{ cm}^3 = 1,000 \text{ mL} = 1 \text{ Litre (L)}$
• $1 \text{ m}^3 = 1,000 \text{ Litres (L)}$

Worked Example: Tank Capacity Calculation

Problem: A rectangular water storage tank in Barbados has an interior length of $150 \text{ cm}$, a width of $80 \text{ cm}$, and a height of $100 \text{ cm}$. How many litres of water can the tank hold when completely filled?

• Step 1: Calculate volume in cubic centimetres ($\text{cm}^3$): $$\text{Volume} = 150 \text{ cm} \times 80 \text{ cm} \times 100 \text{ cm} = 1,200,000 \text{ cm}^3$$
• Step 2: Convert cubic centimetres to litres ($1,000 \text{ cm}^3 = 1 \text{ L}$): $$\text{Capacity in Litres} = 1,200,000 \div 1,000 = 1,200 \text{ L}$$
• Answer: The water tank holds $1,200 \text{ litres}$ of water.