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Chapter 9: Problem 77

Describe how area is measured. Explain why linear units cannot be used.

Short Answer

Expert verified
Area, a two-dimensional measure, is measured in square units like square meters, square inches, etc. It represents the amount of space within a shape or surface. Linear units, which measure one-dimensional length, cannot be used because they don't accurately represent the size of a two-dimensional space.

Step by step solution

01

Introduction to Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. It can be visualized as the amount of paint that would be necessary to cover a surface, and is the two-dimensional counterpart of the one-dimensional length of a curve, and three-dimensional volume of a solid.
02

Measurement of Area

Area is measured in square units. These can be squares of any size, and they do not even need to be actual squares. Any shape that covers a space perfectly without gaps or overlaps can be used as a 'unit' for measuring area. Common units for measuring area include square millimeters (mm虏), square centimeters (cm虏), square meters (m虏), square kilometers (km虏), etc. in the metric system; square inches (in虏), square feet (ft虏), square yards (yd虏), square miles (mi虏), etc. in the Imperial system.
03

Linear Units vs Square Units

Linear units measure length, which is a one-dimensional measurement. They're used to measure the distance from one point to another. On the other hand, the area is a two-dimensional measurement; it's about measuring the amount of space a shape covers. While you can measure the length of a boundary with a ruler (linear measure), it won't tell you how much space is inside that boundary. That's why area is measured in square units, not linear units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Units
Linear units are essential for measuring the distances of lines or boundaries. Examples include meters, feet, and inches. These are one-dimensional measurements, which indicate length or distance.
  • Used to describe how far one point is from another.
  • Examples: kilometers, meters, miles, yards.
However, when it comes to measuring spaces inside borders, linear units fall short. Imagine measuring just the edges of a park. Knowing the perimeter won't tell you how much grass is inside. That's the limitation of linear units. They're fantastic for understanding boundaries or lengths but not the space enclosed by them.
Square Units
Square units come into play when measuring area, which is a two-dimensional space. Think of covering a floor with tiles. Each tile represents a square unit. The total area of a space tells us how many of these tiles we need.
  • Indicators of area include square meters, square kilometers, square feet, and so on.
  • Each square unit represents a small square that fits into the space.
Using square units provides a more accurate depiction of the size of a surface. It measures the entire area within a boundary, ensuring a full grasp of how much space is being utilized or needed.
Two-dimensional Space
Two-dimensional spaces refer to surfaces that have length and width, but no depth. This is the realm where area measurement is crucial, as it tells us how much flat space exists within boundaries.
  • Commonly seen in fields of architecture, design, and even art.
  • Includes spaces like floors, walls, and tabletops.
Two-dimensional measurements help us in practical applications, such as real estate or planning. When buying carpet, for instance, knowing the two-dimensional space of a room tells you how much material you'll need. This highlights that measuring these dimensions goes beyond simple lines; it's about fully grasping how much usable space is present.

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