Area in the Metric System
A classroom floor covers about $60\ \text{m}^2$. A soccer field covers about $7{,}000\ \text{m}^2$. The whole United States covers about $9{,}800{,}000\ \text{km}^2$. All of these are area: how much flat space a shape covers. But how do you compare numbers written in such different units? That's what this page is about.
The Square Meter
Length tells you how far it is from one point to another. Area tells you how much surface a figure covers. To measure it, we count squares.
Area units are always "square" units: $\text{cm}^2$, $\text{m}^2$, $\text{km}^2$. The little $^2$ reminds you that you are counting squares, not lengths.
Select the most appropriate unit of measurement to express the area of the following objects: a) The screen of a smartphone. b) A professional soccer field. c) The area…
Select the most appropriate unit of measurement to express the area of the following objects:
- a) The screen of a smartphone.
- b) A professional soccer field.
- c) The area of the United States.
- d) The floor of your bedroom.
(Options: $km^2$, $hm^2$, $m^2$, $cm^2$)
Calculate the area of the following shaded figure by counting the squares on the grid. Note that each small square represents an area of $1 \text{ m}^2$. Express the…
Calculate the area of the following shaded figure by counting the squares on the grid. Note that each small square represents an area of $1 \text{ m}^2$. Express the final result in square decameters ($\text{dam}^2$).
The Metric Area Staircase
Metric units have prefixes. For length, each unit is $10$ times the next one:
| Unit | Symbol | In meters |
|---|---|---|
| kilometer | km | $1{,}000\ \text{m}$ |
| hectometer | hm | $100\ \text{m}$ |
| decameter | dam | $10\ \text{m}$ |
| meter | m | $1\ \text{m}$ |
| decimeter | dm | $0.1\ \text{m}$ |
| centimeter | cm | $0.01\ \text{m}$ |
| millimeter | mm | $0.001\ \text{m}$ |
For area, each step is not $\times 10$. It is $\times 100$.
Moving several steps at once:
| Steps | Multiply or divide by |
|---|---|
| 1 | $100$ |
| 2 | $100 \times 100 = 10{,}000$ |
| 3 | $1{,}000{,}000$ |
Each step moves the decimal point two places.
For an Art project, Lucas needs a piece of card that measures exactly $2.400 \text{ cm}^2$. When he goes to the stationery store, the shop assistant tells him that the…
For an Art project, Lucas needs a piece of card that measures exactly $2.400 \text{ cm}^2$. When he goes to the stationery store, the shop assistant tells him that the measurements are expressed in square decimeters. How many $\text{dm}^2$ should Lucas ask for for his project?
Express the following area measurement, given in mixed units, as a single unit in square meters: $3 \text{ dam}^2$, $25 \text{ m}^2$ and $75 \text{ dm}^2$
Express the following area measurement, given in mixed units, as a single unit in square meters:
$3 \text{ dam}^2$, $25 \text{ m}^2$ and $75 \text{ dm}^2$
Converting Between Units
To convert, count how many steps you move on the staircase. Then multiply by $100$ for every step down, or divide by $100$ for every step up.
When you compare or order areas, first convert them all to the same unit. A bigger number does not always mean a bigger area. For example, $9{,}000\ \text{m}^2$ is bigger than $50\ \text{dam}^2$, because $50\ \text{dam}^2 = 5{,}000\ \text{m}^2$.
A farmer has three plots with the following areas: Plot A: $0,45 \text{ hm}^2$ Plot B: $5.200 \text{ m}^2$ Plot C: $48 \text{ dam}^2$ Order the plots from smallest to…
A farmer has three plots with the following areas:
-
Plot A: $0,45 \text{ hm}^2$
-
Plot B: $5.200 \text{ m}^2$
-
Plot C: $48 \text{ dam}^2$
Order the plots from smallest to largest area by comparing their measurements in $\text{m}^2$.
A city council is planning the creation of a new suburban park. They have bought three adjacent plots whose areas are expressed in different units: Plot A:…
A city council is planning the creation of a new suburban park. They have bought three adjacent plots whose areas are expressed in different units:
-
Plot A: $0,052\text{ km}^2$
-
Plot B: $140\text{ dam}^2$
-
Plot C: $8.500\text{ m}^2$
Express the total area of the future park in hectares ($ha$).
On a scale plan of a shopping center, the central aisle is represented by a rectangle with an area of $450\text{ cm}^2$. If we know that in reality this aisle occupies…
On a scale plan of a shopping center, the central aisle is represented by a rectangle with an area of $450\text{ cm}^2$. If we know that in reality this aisle occupies $0,18\text{ dam}^2$:
- a) Express both quantities in $\text{dm}^2$.
- b) How many times larger is the real area than the area on the plan?
Land Units: Hectare, Are, Centiare
For fields, parks, and forests, $\text{m}^2$ gives numbers with too many zeros. Land has its own units, which match the staircase.
Hectares are used in science and around the world. In the United States, land is usually sold in acres, and $1\ \text{ha}$ is about $2.5$ acres. The are and the centiare are rarely used in the US, but some problems use them, so you should know them.
To go the other way, divide. For example, $7{,}000\ \text{m}^2 \div 10{,}000 = 0.7\ \text{ha}$, and $7{,}000\ \text{m}^2 \div 100 = 70\ \text{a}$.
In the world of agriculture, special units are used. Match each International System measurement with its corresponding agricultural unit: a) $1 \text{ hm}^2$ b)…
In the world of agriculture, special units are used. Match each International System measurement with its corresponding agricultural unit:
- a) $1 \text{ hm}^2$
- b) $1 \text{ dam}^2$
- c) $1 \text{ m}^2$
I. Are ($a$)
II. Hectare ($ha$)
III. Centiare ($ca$)
A city council has decided to create a central park of $5 \text{ hectares}$. If they want to install a play area that occupies $1.500 \text{ m}^2$, how many square…
A city council has decided to create a central park of $5 \text{ hectares}$. If they want to install a play area that occupies $1.500 \text{ m}^2$, how many square meters will be available for trees and paths in the rest of the park?
A farmer has a farm of $3\text{ ha}$, $5\text{ a}$ and $25\text{ ca}$. They decide to sell a part of their land for the construction of a road. The sold part measures…
A farmer has a farm of $3\text{ ha}$, $5\text{ a}$ and $25\text{ ca}$. They decide to sell a part of their land for the construction of a road. The sold part measures exactly $4.200\text{ m}^2$. What area of the farm, expressed in mixed units (hectares, ares and centiares), does the farmer have left?
Area of Composite Figures
To find the area of an irregular shape, break it into simple pieces like rectangles and squares. Find each area and add. If a piece is cut out, like a window or a hole, subtract it.
Rectangle: area $=$ length $\times$ width. Square: area $=$ side $\times$ side.
In a residential development, a square communal area measuring $10 \text{ meters}$ per side is going to be paved. Square tiles with an area of $1 \text{ dm}^2$ each will…
In a residential development, a square communal area measuring $10 \text{ meters}$ per side is going to be paved. Square tiles with an area of $1 \text{ dm}^2$ each will be used. How many tiles will be needed to cover the whole area?
We want to change the floor of a kitchen that has a rectangular shape and measures $420\text{ cm}$ long by $3\text{ m}$ wide. If we are going to use square tiles with an…
We want to change the floor of a kitchen that has a rectangular shape and measures $420\text{ cm}$ long by $3\text{ m}$ wide. If we are going to use square tiles with an area of $20\text{ dm}^2$ each:
- a) Calculate the total area of the kitchen in $\text{m}^2$.
- b) How many tiles will we need to cover the whole floor?
Calculate the area of the following irregular figure by breaking it down into squares and rectangles. Each square on the background grid represents an area of…
Calculate the area of the following irregular figure by breaking it down into squares and rectangles. Each square on the background grid represents an area of $100\text{ cm}^2$. Express the final result in square decimeters ($\text{dm}^2$).
Multi-Step Area Problems
Harder problems add percents, costs, or scales. Take them one step at a time: convert to one unit, find the area, then use the percent, the price, or the scale.
-
Percent of an area: find the percent of the total, then subtract or add as needed.
-
Cost: $\text{cost} = \text{price per unit of area} \times \text{area}$. The price and the area must use the same unit.
-
Scale on a map: if $1\ \text{cm}^2$ on the map stands for a certain real area, multiply the map area by that number.
-
Cutting out a piece: subtract its area from the total.
For the construction of a new shopping center, a rectangular plot measuring $2,4 \text{ hm}$ long and $150 \text{ m}$ wide has been fenced off. The local council…
For the construction of a new shopping center, a rectangular plot measuring $2,4 \text{ hm}$ long and $150 \text{ m}$ wide has been fenced off. The local council requires $25 \%$ of the total area to be dedicated to green areas and another $15 \%$ to parking lots. If the rest is divided into commercial units with an average area of $120 \text{ m}^2$, how many units can be built in total?
The floor of a sports center with an area of $12,5 \text{ dam}^2$ is going to be renovated. Boxes of heavy-duty tiles have been bought, where each box contains enough…
The floor of a sports center with an area of $12,5 \text{ dam}^2$ is going to be renovated. Boxes of heavy-duty tiles have been bought, where each box contains enough material to cover $2,5 \text{ m}^2$ and costs $\$45 \text{ \$}$. In addition to the cost of the material, the installation company charges $\$18 \text{ \$}$ for each square meter laid. What is the total bill if $5 \%$ more material than needed is bought to make up for what is wasted in cuts and breakages?
A graphic designer is creating a logo on a metal sheet. The logo consists of a central square with an area of $4 \text{ dm}^2$. Around this square, they add four smaller…
A graphic designer is creating a logo on a metal sheet. The logo consists of a central square with an area of $4 \text{ dm}^2$. Around this square, they add four smaller squares whose sides are half the length of the side of the central square. Express the total area of the logo in square millimeters and in square decimeters using only a single unit.
Back to the start: a $7{,}000\ \text{m}^2$ soccer field is $0.7\ \text{ha}$, and the United States is $9{,}800{,}000\ \text{km}^2$. Put both in the same unit and you can compare them. Convert first, then compare.