10.6 Areas

If you are planning to paint a wall, one of the first questions to ask is how much paint will you need. This will obviously depend on the size of the wall. The paint cans usually include information about the amount of area that the paint will cover. For example, on the back of a 2.5-liter can of emulsion paint, it says that the paint will cover “up to 35 square meters.” A square meter is the area that is covered by a square whose sides are 1 m. This can also be written as one times m super two. So 35 times m super two will be the same area as the area of 35 of these 1-meter squares.

Areas can also be measured in other square units, such as square centimeters ( cm super two) or square kilometers ( km super two), depending on what is appropriate to the situation. A square centimeter is a square whose sides are 1 cm long, and a square kilometer has sides which are 1 km long.

If the tangram puzzle below is drawn on a grid in which the lines are 1 cm apart, then each grid square will have an area of one times cm super two. Since the whole puzzle measures 4 cm by 4 cm, its area is 16 times cm super two. You can confirm the area by counting the grid squares on the puzzle.

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Activity Symbol Activity: Counting Squares

All the shapes of the puzzle are made up out of whole squares or half squares, so to find the area of any of the shapes, you can count the squares. Using the 4 cm tangram above, work out the area of each shape in square centimeters. Write the area on each shape.

Solution Symbol

Answer

The areas in square centimeters are shown below.

To check, you can add the individual areas and see if they total 16 times cm super two. [ It’s always a good idea to check your answer a different way if you can! ]

The sum of the individual areas in cm super two = sum with, 7 , summands four plus four plus one plus two plus one plus two plus two equals 16.

If you do need to calculate the area of a shape, laying a square grid (marked with m, cm, or mm squares) over the shape and then counting the number of squares is one way to do it. This is particularly useful if the shape is irregular—for example, if it is a handprint or footprint.

However, many areas can be built out of basic shapes such as rectangles, triangles, or circles, and these areas can be calculated by using formulas. For example, if a rectangular room measures 4 m by 3 m and you are covering it in carpet tiles that are each 1 meter square, the tiles can be arranged in 3 rows each with 4 tiles. So the total number of tiles will be three multiplication four equals 12 and the area covered is 12 m super two.

This area could have been calculated directly by multiplying the length of the room by its width. Provided both measurements are in the same units, the following formula holds for any rectangle.

area of a rectangle equals length multiplication width

For example, if the room measured 3.24 m by 4.38 m, the floor area would be 3.24 m multiplication four .38 m equals 14 equation left hand side .1912 m super two almost equals right hand side 14.2 times m super two. Did you notice that the answer has been rounded?

Any measurement that you make is rounded—for example, lengths may be measured to the nearest mm or nearest cm depending both on the context and the instrument you are using to make the measurement. In the example above, the measurements have been made to the nearest centimeter. This means that the answer cannot be given to more than the nearest square centimeter under any circumstances, and it is likely that the precision of the answer will be considerably less than that, as the analysis below shows.

The width of the room has been measured as 324 centimeters, but all lengths from 323.5 cm up to but not including 324.5 cm, would be rounded to 324 cm.

Similarly, the length has been measured as 438 cm, and all measurements from 437.5 cm up to, but not including, 438.5 cm, would be rounded to 438 cm.

This means that the smallest possible area of the room is 4.375 m prefix multiplication of 3.235 m equals 14 .153125 m super two.

The largest possible area of the room is 4.385 m multiplication three .245 m equals 14 .229325 m super two.

So, the area of the room is approximately 14.2 times m super two.

  • Note: If a calculation involves measurements, remember to round your answer appropriately at the end of your calculation.

The answer should be no more accurate than the least accurate number you have used in your calculation. For example, let's suppose you need to perform the calculation 3.4 multiplication 5.26.

The least accurate number is 3.4 since it is only measured to the tenths place as supposed to the hundredths place in 5.26.

When you multiply these two numbers, you will initially obtain 17.884, but since your least accurate number was only to the tenths place, at best you can round and report your final answer to the tenths place as well, yielding a final answer of 17.9. To be safe, you might even round it to the units place, giving 18 as the product. However, this is not always appropriate.

10.5.2 Protecting an Old Tree

10.6.1 Areas of Rectangles