Measurement: Length, Area, Surface Area and Study Guide

Measurement Skills in Occupations and Real-World Contexts

• Measurement skills are essential in nearly every practical field of work. Occupations requiring these skills include bakers, boilermakers, bricklayers, builders, carpenters, carpet layers, concreters, cooks, engineers, farmers, forestry workers, furniture makers, glaziers, graphic artists, hairdressers, house painters, landscapers, machinists, mechanics, pipelayers, painters, plumbers, plasterers, sheet-metal workers, surveyors, tailors, tilers, and welders.
• Specific examples of measurement in context:
  • Sports Courts and Athletics Tracks: Olympic running tracks have official global dimensions. They consist of straight sections measuring $84.39\,m$ and two semicircles with a radius of $36.50\,m$.
  • Shipping Containers: These rectangular prisms revolutionized global cargo transport. The largest container ships as of 2023 can carry more than $24,000$ containers.
  • Cylindrical Calculations: Surface area and volume are used for food cans, water storage tanks, fuel tankers, milk tankers, and underground fuel station tanks.

Learning Intentions and Syllabus Outcomes

• Conceptual Development (MAO-WM-01): Developing mathematical understanding and fluency by exploring concepts, choosing techniques, and communicating reasoning clearly.
• Length and Perimeter (MA4-LEN-C-01): Applying knowledge of the perimeter of plane shapes and the circumference of circles to solve problems.
• Area of Plane Shapes (MA4-ARE-C-01): Applying knowledge of area and composite area involving triangles, quadrilaterals, and circles.
• Surface Area and Composite Shapes (MA5-ARE-C-01): Solving problems involving the surface area of right prisms and practical problems involving composite shapes and solids.
• Volume (MA5-VOL-C-01): Solving problems involving the volume of composite solids consisting of right prisms and cylinders.

Length and Perimeter

• Measurement Branch: Mathematics branch including length, area, and volume. It encompasses units, perimeter, circumference, surface area, composite shapes, and capacity.
• Metric Units of Length:
  • Kilometre ($km$)
  • Metre ($m$)
  • Centimetre ($cm$)
  • Millimetre ($mm$)
• Metric Unit Conversions:
  • $1\,km = 1000\,m$
  • $1\,m = 100\,cm = 1000\,mm$
  • $1\,cm = 10\,mm$
  • To convert to a smaller unit, multiply by the appropriate power of $10$.
  • To convert to a larger unit, divide by the appropriate power of $10$.
  • $km \xrightarrow{\times 1000} m \xrightarrow{\times 100} cm \xrightarrow{\times 10} mm$
  • $mm \xrightarrow{\div 10} cm \xrightarrow{\div 100} m \xrightarrow{\div 1000} km$
• Perimeter Definition: Perimeter ($P$) is the distance around the outside of a closed shape. To find the perimeter, all side lengths are added together.
• Geometric Notations: Sides marked with the same symbols are of equal length.
• Composite Shape Perimeter: Finding the perimeter of a composite shape involves identifying and labeling missing side lengths before summing all outer boundaries. For shapes with right angles, missing horizontal or vertical lengths can be found by subtracting known parallel segments (e.g., $6\,cm - 5\,cm = 1\,cm$).

Circumference of Circles and Perimeter of Sectors

• Pi ($\pi$): The ratio of a circle's circumference to its diameter. It has an infinite number of decimal places with no pattern. Approximations include $3.14$, $3.14159$, or $\frac{22}{7}$.
• Circle Features:
  • Diameter ($d$): The distance across the circle through the centre.
  • Radius ($r$): The distance from the centre to the circumference. The radius is half the diameter ($d = 2r$).
  • Circumference ($C$): The distance around the circle, calculated as $C = 2\pi r$ or $C = \pi d$.
• Sectors and Portions:
  • Sector: A fraction of a circle bounded by two radii.
  • Semicircle: A half circle ($\frac{1}{2}$ of a circle).
  • Quadrant: A quarter circle ($\frac{1}{4}$ of a circle).
• Perimeter of Sectors: Finding the perimeter of a semicircle or quadrant requires calculating the fraction of the circumference and adding the lengths of the straight sides (radii or diameter).
  • Example for semicircle perimeter: $P = \frac{1}{2} \pi d + d$ or $P = \pi r + 2r$.
  • Example for quadrant perimeter: $P = \frac{1}{4}(2\pi r) + 2r$.

Area of Quadrilaterals and Triangles

• Area Definition: A measure of the space enclosed within the boundaries of a two-dimensional shape.
• Metric Units of Area:
  • Square millimetre ($mm^2$)
  • Square centimetre ($cm^{2}$): $1\,cm^2 = 10^2\,mm^2 = 100\,mm^2$
  • Square metre ($m^2$): $1\,m^2 = 100^2\,cm^2 = 10,000\,cm^2$
  • Hectare ($ha$): $1\,ha = 10,000\,m^2$
  • Square kilometre ($km^2$): $1\,km^2 = 1000^2\,m^2 = 1,000,000\,m^2$
• Area Conversion:
  • $km^2 \xrightarrow{\times 1000^2} m^2 \xrightarrow{\times 100^2} cm^2 \xrightarrow{\times 10^2} mm^2$
  • $mm^2 \xrightarrow{\div 10^2} cm^2 \xrightarrow{\div 100^2} m^2 \xrightarrow{\div 1000^2} km^2$
• Area Formulas:
  • Square: $A = l^2$
  • Rectangle: $A = lb$
  • Triangle: $A = \frac{1}{2}bh$
  • Parallelogram: $A = bh$
  • Trapezium: $A = \frac{h}{2}(a+b)$
  • Rhombus: $A = \frac{1}{2}xy$ (where $x$ and $y$ are diagonal lengths) or $A = bh$
  • Kite: $A = \frac{1}{2}xy$
• Perpendicular Height: In formulas for triangles, parallelograms, and trapeziums, the height ($h$) must be perpendicular (at $90^\circ$) to the base.

Area of Circles and Circular Sectors

• Circle Area Formula: $A = \pi r^2$. If the diameter is given, it must be halved to find the radius ($r$).
• Semicircle Area: $A = \frac{1}{2} \pi r^2$
• Quadrant Area: $A = \frac{1}{4} \pi r^2$
• Rearranging the Formula: To find the radius of a circle when the area is known, use the formula $r = \sqrt{\frac{A}{\pi}}$.

Perimeter and Area of Composite Shapes

• Composite Shape Composition: These are made of multiple simple shapes (e.g., a rectangle and a semicircle).
• Calculation Methods:
  • Addition: Summing the areas or perimeters of the component basic shapes.
  • Subtraction: Calculating the area of a larger bounding shape and subtracting the area of internal "cut-outs" or clear spaces.
• Perimeter Considerations: When finding the perimeter of composite shapes, only the external boundaries are summed; internal lines where shapes meet are excluded.

Surface Area of Prisms

• Surface Area ($SA$): The sum of the areas of all the outside surfaces of a three-dimensional solid.
• Net: A two-dimensional illustration of all the surfaces of a solid if it were unfolded.
• Right Prism: A solid with a uniform cross-section and rectangular side faces at right angles to the base and top.
• Specific Surface Area Formulas:
  • Cube: $SA = 6l^2$
  • Rectangular Prism (Cuboid): $SA = 2(lb + bh + lh)$
  • Triangular Prism: Sum of the areas of three rectangles and two identical triangles: $SA = (\text{area of 3 rectangles}) + 2 \times (\frac{1}{2}bh)$.
  • Pyramid: The sum of the area of the base and the areas of the triangular faces.

Surface Area of Cylinders

• Cylinder Definition: A solid with two parallel, congruent circular faces connected by a curved surface.
• Components of Cylinder Surface Area:
  • Two circular ends: $2 \times \pi r^2$
  • Curved surface: A rectangle with a length equal to the circumference ($2\pi r$) and a width equal to the height ($h$). Area of curved surface $= 2\pi rh$.
• Total Surface Area Formula: $SA = 2\pi r^2 + 2\pi rh$
• Surface Area of Cylindrical Portions:
  • Half Cylinder: Includes two semicircular ends (forming one full circle: $\pi r^2$), half the curved surface ($\pi rh$), and one rectangular base ($2r \times h$).

Volume of Prisms and Capacity

• Volume: The amount of space inside a three-dimensional object, measured in cubic units.
• Metric Units of Volume:
  • Cubic millimetre ($mm^3$)
  • Cubic centimetre ($cm^3$): $1\,cm^3 = 10^3\,mm^3 = 1000\,mm^3$
  • Cubic metre ($m^3$): $1\,m^3 = 100^3\,cm^3 = 1,000,000\,cm^3$
  • Cubic kilometre ($km^3$): $1\,km^3 = 1000^3\,m^3 = 1,000,000,000\,m^3$
• Capacity Units: Used for liquids and gases.
  • Millilitre ($mL$)
  • Litre ($L$): $1\,L = 1000\,mL$
  • Kilolitre ($kL$): $1\,kL = 1000\,L$
  • Megalitre ($ML$): $1\,ML = 1000\,kL$
• Volume and Capacity Equivalencies:
  • $1\,cm^3 = 1\,mL$
  • $1000\,cm^3 = 1\,L$
  • $1\,m^3 = 1000\,L = 1\,kL$
• Volume of Uniform Solids: For any solid with a constant cross-section (prism or cylinder), the volume is calculated as the area of the cross-section ($A$) times the height ($h$) perpendicular to that cross-section.
  • General Formula: $V = Ah$
  • Rectangular Prism: $V = lbh$ (length $\times$ breadth $\times$ height).
  • Triangular Prism: Identify the area of the triangular cross-section ($\frac{1}{2}bh$) and multiply by the length of the prism.

Volume of Cylinders

• Right Cylinder: A cylinder with sides at right angles to its base.
• Volume Formula: $V = \pi r^2h$.
  • $\pi r^2$ is the area of the circular cross-section.
  • $h$ is the distance or length between the circular ends.

Practical Mathematics in Farming and Industry

• Dam Volume Estimation: The volume of a farm property dam is estimated using the formula: $\text{Volume} = A \times d \times 0.4$.
  • $A$ is the surface area of the top of the dam.
  • $d$ is the maximum depth.
  • The factor $0.4$ accounts for the dam's sloping sides.
• Irrigation Calculations:
  • Plant Watering Depth: Percentage of moisture used by plant $\times$ root depth (in $mm$).
  • Water Volume for Trees: $V = \pi r^2 \times \text{Plant Watering Depth}$.
  • An area of $1\,m^2$ covered by $1\,mm$ of water equals $1\,L$ of water.
• Road Construction (Asphalt Rolling):
  • The area of ground pressed in one revolution of a roller wheel is equal to the curved surface area of the cylinder ($2\pi rh$).
  • Calculations for roadwork must account for the number of strips needed to cover the road width and the possibility of required overlaps (e.g., a $10\,cm$ overlap between pressed strips).
  • The cost of hiring equipment can be based on the total number of revolutions required to complete a project.