Question

Determine the number of a. square meters \(\left(\mathrm{m}^{2}\right)\) in 1 square kilometer \(\left(\mathrm{km}^{2}\right)\) b. cubic centimeters \(\left(\mathrm{cm}^{3}\right)\) in 1 cubic foot \(\left(\mathrm{ft}^{3}\right)\) c. square feet \(\left(\mathrm{ft}^{2}\right)\) in 1 square yard \(\left(\mathrm{yd}^{2}\right)\)

Short answer

a. 1,000,000 square meters in 1 square kilometer, b. approximately 28,316.8 cubic centimeters in 1 cubic foot, c. 9 square feet in 1 square yard.

Step-by-step solution

Convert square kilometers to square meters

1 square kilometer is equivalent to the area of a square with sides of 1 kilometer each. Since 1 kilometer is equal to 1,000 meters, the area in square meters is found by squaring the sides in meters, so it becomes \(1 \mathrm{km} \times 1 \mathrm{km} = 1,000 \mathrm{m} \times 1,000 \mathrm{m} = 1,000,000 \mathrm{m}^{2}\). Therefore, 1 square kilometer is equal to 1,000,000 square meters.

Convert cubic feet to cubic centimeters

1 cubic foot is the volume of a cube with sides of 1 foot each. There are 12 inches in a foot and 2.54 centimeters in an inch. Thus, the volume in cubic centimeters is found by converting each side to centimeters and then finding the volume: \(1 \mathrm{ft} = 12 \mathrm{in} = 12 \times 2.54 \mathrm{cm} = 30.48 \mathrm{cm}\). So, \(1 \mathrm{ft}^{3} = 30.48 \mathrm{cm} \times 30.48 \mathrm{cm} \times 30.48 \mathrm{cm} \approx 28,316.8 \mathrm{cm}^{3}\).

Convert square yards to square feet

1 square yard is the area of a square with sides of 1 yard each. Given that 1 yard is equal to 3 feet, the area in square feet is found by squaring the sides in feet, \(1 \mathrm{yd} \times 1 \mathrm{yd} = 3 \mathrm{ft} \times 3 \mathrm{ft} = 9\mathrm{ft}^{2}\). Therefore, 1 square yard is equal to 9 square feet.

Key concepts

Understanding Unit Conversion from Square Meters to Square Kilometers

When it comes to understanding large areas, converting from square meters to square kilometers is essential. Imagine a plot of land that is one kilometer by one kilometer in size. It's a vast space, right? Now, we know that one kilometer is equivalent to 1,000 meters. To calculate the area of this large square, we square the side lengths,

which means we multiply 1,000 meters by itself:
\(1,000 \mathrm{m} \times 1,000 \mathrm{m} = 1,000,000 \mathrm{m}^2\).

This calculation shows us that one square kilometer is actually made up of 1,000,000 square meters! That's a lot of ground cover. When observing geographical maps or discussing the size of cities, this unit conversion becomes particularly useful for better understanding and visualization.

Converting Cubic Centimeters to Cubic Feet: A Volumetric Perspective

In everyday situations, like measuring the volume of packages or aquariums, we frequently switch between cubic centimeters to cubic feet depending on which unit is more convenient. To illustrate this conversion, let's look at a cubic foot. It’s a cube whose sides are all one foot long. Now, since there are 12 inches in one foot and exactly 2.54 centimeters in one inch, calculating the conversion is all about multiplying these lengths threefold (for the three dimensions of the cube).

Let's do this step by step:
\(1 \mathrm{ft} = 12 \mathrm{in} = 12 \times 2.54 \mathrm{cm} = 30.48 \mathrm{cm}\).

Now, to find the volume in cubic centimeters, simply raise that number to the third power:
\(30.48 \mathrm{cm} \times 30.48 \mathrm{cm} \times 30.48 \mathrm{cm} \approx 28,316.8 \mathrm{cm}^3\).

This conversion factor allows you to understand how much smaller a cubic centimeter is as compared to a cubic foot, with roughly 28,316.8 cubic centimeters fitting into a cubic foot.

From Square Feet to Square Yards: Calculating Area with Different Units

Calculating floor space for carpeting or measuring lawns, you might need to work in either square feet or square yards. Knowing how to convert between these two is quite handy. One square yard can be visualized as a square-shaped area where each side is one yard (3 feet) long. To find how many square feet are in a square yard, you'd square the number of feet per yard:

\(1 \mathrm{yd} \times 1 \mathrm{yd} = 3 \mathrm{ft} \times 3 \mathrm{ft} = 9\mathrm{ft}^2\).

Thus, a single square yard equals nine square feet. Remembering this conversion factor of 9 can simplify purchasing the right amount of material for your next home project, whether it’s laying down new tiles or sowing grass seeds in your garden.