Question

A house has an area of \(215 \mathrm{~m}^{2}\). What is its area in each unit? (a) \(\mathrm{km}^{2}\) (b) \(\mathrm{dm}^{2}\) (c) \(\mathrm{cm}^{2}\)

Short answer

The area of the house is 0.000215 km^2, 21,500 dm^2, and 2,150,000 cm^2.

Step-by-step solution

Understanding the conversion factors

To convert the area from square meters to square kilometers, square decimeters, and square centimeters, one must know the conversion factors. There are 1,000 meters in a kilometer, 10 decimeters in a meter, and 100 centimeters in a meter. Since these are area measurements, we square the conversion factors for linear measurements: 1 km^2 = (1,000 m)^2 = 1,000,000 m^2, 1 dm^2 = (1/10 m)^2 = 1/100 m^2, and 1 cm^2 = (1/100 m)^2 = 1/10,000 m^2.

Converting to square kilometers (km^2)

To find the area in square kilometers, divide the given area in square meters by the number of square meters in a square kilometer. Area in km^2 = area in m^2 / 1,000,000.

Calculating area in square kilometers

Area in km^2 = 215 m^2 / 1,000,000 = 0.000215 km^2.

Converting to square decimeters (dm^2)

To find the area in square decimeters, multiply the given area in square meters by the number of square decimeters in a square meter. Area in dm^2 = area in m^2 * 100.

Calculating area in square decimeters

Area in dm^2 = 215 m^2 * 100 = 21,500 dm^2.

Converting to square centimeters (cm^2)

To find the area in square centimeters, multiply the given area in square meters by the number of square centimeters in a square meter. Area in cm^2 = area in m^2 * 10,000.

Calculating area in square centimeters

Area in cm^2 = 215 m^2 * 10,000 = 2,150,000 cm^2.

Key concepts

Units of Area

Understanding different units of area is crucial for interpreting land size, room dimensions, or any space measurement. The most common units of area include square meters (m²), square kilometers (km²), square centimeters (cm²), and square decimeters (dm²).

It's important to recognize that when we talk about area, we are referring to the size of a two-dimensional space. The 'square' in these units signifies that we are dealing with an area rather than a linear measure. For example, a square meter represents the area covered by a square that is one meter long on each side.

Square Meters

Square meters (m²) are the SI (International System of Units) unit used to measure area, commonly used worldwide. When visualizing a square meter, think of a square with each side measuring exactly one meter. It's about the size of a large door and is a convenient measure for rooms, houses, and localized land areas.

Visualizing Square Meters

In the context of everyday life, understanding the physical size of a square meter can help in visualizing the space being measured. For a concrete example, an average car parking space is about 12 square meters.

Conversion Factors

Conversion factors are essential tools that allow us to convert between different units of measurement. They act as a multiplier that transforms one unit into another, ensuring that the same quantity is expressed in a different way.

To convert between areas, you should square the conversion factor of the corresponding linear measure. For instance, to go from square meters to square centimeters, since 1 m = 100 cm, you square this to find that 1 m² = 10,000 cm². Always remember to use the square of the conversion factor to translate units of area.

Metric System

The metric system is an international decimal-based system of measurement. It is used in almost every country and is the standard for scientific measurement worldwide. The system is based on multiples of ten, making it especially straightforward to understand and use.

When dealing with area in the metric system, we use square meters as the base unit. Understanding and utilizing the metric system's prefixes, such as 'kilo-', 'deci-', and 'centi-', allow for easy conversions between larger and smaller units of area just by moving the decimal point.