Glossary

9.2 Measuring Area

3 min read•june 18, 2024

Measuring is a key skill in and everyday life. We'll learn how to calculate the size of using and , and explore the metric units used to express area.

Converting between metric area units is crucial for practical applications. We'll practice using to switch between , centimeters, and millimeters, and tackle real-world problems involving area calculations.

Measuring Area

Area of rectangular surfaces

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  • Calculate area by multiplying length and width ()
  • Metric units for length and width (meters, centimeters, millimeters)
    • Meter (m) is the base unit of length in the
    • Centimeter (cm) is 1/100th of a meter
    • Millimeter (mm) is 1/1000th of a meter
  • Express area in (square meters m2\text{m}^2, , mm2\text{mm}^2)
  • Example: Rectangle with length 5 m and width 3 m has area 5 m×3 m=15 m25 \text{ m} \times 3 \text{ m} = 15 \text{ m}^2
  • Example: Rectangle with length 120 cm and width 80 cm has area 120 cm×80 cm=9,600 cm2120 \text{ cm} \times 80 \text{ cm} = 9,600 \text{ cm}^2
  • Area is a fundamental concept in geometry, used to measure the size of

Conversion of metric area units

  • Use conversion factors to change between metric area units
    • 1 m2=10,000 cm21 \text{ m}^2 = 10,000 \text{ cm}^2 (move decimal point 4 places to the right)
    • 1 cm2=100 mm21 \text{ cm}^2 = 100 \text{ mm}^2 (move decimal point 2 places to the right)
  • Larger unit to smaller unit: multiply by conversion factor
    • Example: 15 m2×10,000 cm21 m2=150,000 cm215 \text{ m}^2 \times \frac{10,000 \text{ cm}^2}{1 \text{ m}^2} = 150,000 \text{ cm}^2
  • Smaller unit to larger unit: divide by conversion factor
    • Example: 9,600 cm2÷10,000 cm21 m2=0.96 m29,600 \text{ cm}^2 \div \frac{10,000 \text{ cm}^2}{1 \text{ m}^2} = 0.96 \text{ m}^2
  • Perform to ensure conversion results in desired unit
    • Conversion factors are fractions equal to 1, allowing for
    • Example: 15m2×10,000 cm21m2=150,000 cm2\frac{15 \cancel{\text{m}^2} \times 10,000 \text{ cm}^2}{1 \cancel{\text{m}^2}} = 150,000 \text{ cm}^2

Problem-solving with area calculations

  1. Identify shape and dimensions of surface
    • Determine if surface is rectangular or can be divided into rectangles
    • Measure or estimate length and width of each rectangular section (room dimensions, rug size)
  2. Calculate area of each rectangle and sum for total area
    • Use formula A=l×wA = l \times w for each rectangle
    • Add areas of all rectangles to find total area
    • Example: Room with main section 5 m × 4 m and closet 2 m × 1 m has total area 5 m×4 m+2 m×1 m=22 m25\text{ m} \times 4\text{ m} + 2\text{ m} \times 1\text{ m} = 22 \text{ m}^2
  3. Subtract areas of excluded portions from total
    • Identify areas not part of surface or that need to be removed (fireplace, built-in shelves)
    • Calculate area of excluded portions and subtract from total
    • Example: 22 m2\text{m}^2 room with 1.5 m2\text{m}^2 fireplace has actual flooring area 22 m2−1.5 m2=20.5 m222 \text{ m}^2 - 1.5 \text{ m}^2 = 20.5 \text{ m}^2
  4. Check reasonableness of calculated area
    • Consider problem context and expected surface size
    • Compare calculated area to estimation or real-world references (typical room sizes)
    • Example: Calculation of 205 m2\text{m}^2 for a bedroom is likely an error, as this is much larger than a typical bedroom
  • : The distance around the edge of a plane figure
  • : The total area of all surfaces of a three-dimensional object
  • Plane figures: Two-dimensional shapes with length and width, but no depth

Key Terms to Review (21)

$\text{cm}^2$: $\text{cm}^2$ is a unit of area measurement that stands for square centimeters. This unit is commonly used to quantify the size of two-dimensional surfaces, such as those found in geometry and real-world applications. It helps in comparing areas, determining the size of objects, and calculating space in various contexts like home design, landscaping, and more.
$A = l \times w$: The equation $A = l \times w$ represents the formula for calculating the area of a rectangle, where $A$ is the area, $l$ is the length, and $w$ is the width. This formula is fundamental in understanding how to quantify two-dimensional space and is applicable in various real-world scenarios, such as in architecture, landscaping, and design. By multiplying the length by the width, one can determine how much surface is covered within the rectangle.
Area: Area is a measure of the amount of space contained within a two-dimensional shape, typically expressed in square units. Understanding area is crucial for calculating how much surface space a shape covers, which connects directly to determining quantities like land size, material needs, or even visual representations. It serves as a foundation for more complex mathematical concepts and is essential in practical applications such as architecture, engineering, and landscaping.
Conversion factors: Conversion factors are ratios used to convert a quantity expressed in one unit to another unit, maintaining the same quantity. They are essential for ensuring accurate measurements across different systems, especially in area measurement where different units may be utilized, such as square feet and square meters. Understanding conversion factors allows for seamless calculations and comparisons between various area measurements.
Decomposition Method: The decomposition method is a mathematical technique used to simplify complex geometric shapes into smaller, manageable components to measure area accurately. By breaking down a larger figure into familiar shapes, like rectangles, triangles, or circles, one can calculate the area of each part separately and then sum those areas to find the total area of the original shape. This method is especially useful for irregular shapes that do not have straightforward formulas for area calculation.
Geometry: Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. It encompasses various concepts such as measuring area, understanding the nature of points and lines, and calculating the dimensions of different shapes. Geometry plays a crucial role in real-world applications, from architecture to engineering, by providing the tools to quantify and analyze space.
Length: Length is a measure of distance or extent in one dimension, often represented as the measurement of an object from one end to another. In mathematical contexts, length is crucial for understanding more complex concepts such as area and perimeter, as it provides the foundation upon which these measurements are built. It is typically measured in standard units such as meters, feet, or inches and is a fundamental aspect of geometry that facilitates the calculation of various properties of shapes and figures.
M²: The term m², or square meter, is the SI unit of area measurement, representing the area of a square with sides that are one meter in length. This unit is widely used in various fields, including construction, real estate, and agriculture, to quantify land and surface areas. Understanding m² is essential for calculating space, determining property sizes, and working with dimensions in practical applications.
Metric system: The metric system is an international decimal-based system of measurement that uses units such as meters for length, liters for volume, and grams for mass. It provides a standardized method for measuring various quantities, making it easier to communicate and compare measurements across different fields. The metric system is widely used in scientific, educational, and everyday contexts, as it simplifies calculations and conversions through its consistent base of ten.
Mm²: The term mm², or square millimeters, is a unit of area measurement in the metric system. It represents the area of a square whose sides each measure one millimeter in length. This unit is commonly used in various fields such as engineering, architecture, and biology to quantify small areas accurately.
Perimeter: Perimeter is the total distance around the boundary of a two-dimensional shape. It plays a crucial role in various geometric calculations, linking to the area, volume, and surface area of shapes by providing a foundational measure of length that helps in understanding the dimensions and characteristics of polygons and circles.
Plane figures: Plane figures are two-dimensional shapes that lie flat on a single plane and are defined by their boundaries. They include various geometric shapes such as circles, triangles, squares, and rectangles, each characterized by specific properties such as sides, angles, and area. Understanding plane figures is essential when calculating area, as the measurement of space within these figures is fundamental to many mathematical applications.
Rectangular surfaces: Rectangular surfaces refer to flat, two-dimensional shapes that have four sides, with opposite sides being equal in length and all angles measuring 90 degrees. These surfaces are fundamental in measuring area since they provide a straightforward way to calculate space within defined boundaries using simple geometric formulas.
Square centimeters: Square centimeters are a unit of area measurement that represent the size of a two-dimensional surface. They are often used in various fields such as geometry, architecture, and design to quantify the extent of flat surfaces. When measuring area, square centimeters provide a practical way to express how much space an object occupies, making them essential for calculations involving length and width.
Square meters: Square meters is a unit of area measurement in the metric system, representing the area of a square whose sides each measure one meter in length. This term is crucial for quantifying space in a variety of contexts, from real estate to landscaping, and it helps in understanding dimensions and scaling in various applications such as construction and design.
Square millimeters: Square millimeters are a unit of area measurement equal to the area of a square with each side measuring one millimeter. This term is crucial in understanding how to quantify small areas, particularly in fields like engineering, architecture, and biology where precise measurements are essential. Square millimeters allow for a standardized way to compare and calculate area on a very fine scale.
Square units: Square units are a measurement used to quantify area, representing the size of a two-dimensional surface. This term is significant in calculating and expressing the area of various shapes, such as rectangles, squares, and circles. By using square units, we can standardize the measurement of area, making it easier to compare and analyze different surfaces.
Surface area: Surface area is the total area that the surface of a three-dimensional object occupies. It plays a crucial role in understanding properties like volume, material use, and heat transfer. Knowing how to calculate surface area helps in practical applications, such as determining the amount of paint needed for a wall or the capacity of containers.
Unit analysis: Unit analysis is a method used to convert measurements from one unit to another, ensuring that the calculations are consistent and accurate. This technique helps in verifying that the units in a mathematical expression are appropriately aligned, preventing errors in calculations, especially when dealing with area measurements. By systematically applying conversion factors, one can transition seamlessly between different unit systems, like converting square meters to square feet.
Unit Cancellation: Unit cancellation is a mathematical process used to simplify expressions and calculations by removing units from the numerator and denominator in a fraction. This technique ensures that the final result is expressed in the desired units and helps verify that calculations are consistent and correct, particularly when dealing with measurements such as area.
Width: Width refers to the measurement of an object's extent from one side to the other, typically representing the shorter dimension in the context of geometric shapes. It plays a vital role in calculating area, as understanding width helps determine how much space a shape occupies on a flat surface. In practical applications, width is essential for designing and organizing spaces, ensuring that objects fit appropriately within designated areas.
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