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Calculating the Area of Circles
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1. Our Training Videos
Skills Application: Calculating the Area of Circles
Step-by-Step: Calculating the Area of Circles
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2. Running with Pi ## Tagged in

A partial clog in a pipeline can be caused by mineral depositing that sticks to the side walls inside the water lines. The clog changes the pipes original circular area to a smaller circular area. It is critical to keep pipelines supplying hot water heaters and car radiators near their design flow rate. When plumbers or technicians repair clogged water or drain lines, they are essentially restoring the pipes internal cross sectional area to its original circular area for maximum efficiency and flow rate.

## Calculating the Area of Circles ## Running with Pi  . . . . . . . . .
3. Our Training Worksheets

The Area of a circle with radius r is:

A = πr²

Remember that π ≈ 3.14

Example:

Area of a circle with radius 3 inches is A = πr² = π9 ≈ 3.14 × 9 = 28.26 in²

Check out the ADVANCED tab. The exercises will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, to sketch the shapes, label the shapes with the given numbers, tracking changes to units of measure, visualizing to cross-link the concepts of area and numbers. We use the numbers from this boiler top example. If you spend 30 – 60 min working with the different ideas in the ADVANCED page, youll remember key formulas easier and youll learn new skills faster. ### A Short Review

Area of a circle can be calculated a number of ways. You might be familiar with the one your line of work uses. Other ways to calculate as just as valid, though may not be just as practical.

A = r²π    which is sometimes written as    A = πr²

or

${A = d^2}\frac{\pi}{4}$
which is sometimes written as
${A =}\frac{d^2}{4}\cdot\pi$
or      A = d² 0.7854

These formulas all produce the same result.

Note : π/4 ≈ 0.785398163 …. ### Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

### Worksheet: Level 1 Sample Questions

Example: Further example:

If the radius of a circle = 7, how long is its diameter?

### Worksheet: Level 1 Answer Key  ### Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

### Worksheet: Level 2 Sample Questions

Calculate the area of the following circles. Answer in decimals.

Example:

Radius = 35 cm,  Area = ? Question:

1.   r = 5, A = ?

Further example:

Calculate the area of a circle with a diameter of 24 ¾ inches

### Worksheet: Level 2 Sample Answer Key

r² ⋅ ? = A
radius² × ? = area of circle
5² × ? = 78.5398 ### Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

### Worksheet: Level 3 Sample Questions

Calculate the area of the following circles. Answer in decimals.

Remember:

r² ⋅ ? = A

or

${A =}\frac{d^2 \pi}{4}$

Example:

1. What area does a circular patch of sheet metal have if the radius is 2400 mm?

Further example:

Whats the cross-sectional area of a water main line whose internal height is 1.8 m?

### Worksheet: Level 3 Sample Answer Key

r² ⋅ ? = A
radius² × ? = area of circle
(2400 mm)² × ? = 18, 095, 573.68 mm² Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

One way to get the answer

Calculate the area of the top of a boiler (circle) if the diameter is 0.5 m

One way to get the answer, using the formula:  A = r²π

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this: Using a calculator, do the following:

Enter 0.25

Press x² button

Press x (multiplication) button (this could be omitted, depending on your calculators circuits)

Press π button or 2ndF button then the π button, depending on how your calculator is laid out.

Press =

You should see 0.19634954 on your display.

Get a $20-or-so dual-display scientific calculator, even if you cant take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you cant do without. ### Additional Information: Another Way To Get The Answer Another way to get the answer Calculate the area of the top of a boiler (circle) if the diameter is 0.5 m Another way to get the answer, using the formula ${A = }\frac{d^2 \pi}{4}$ Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this: Using a calculator, do the following: Enter 0.5 Press button Press x (multiplication) button (this could be omitted, depending on your calculators circuits) Press π button or 2ndF button then the π button, depending on how your calculator is laid out. Press ÷ button Enter 4 Press = You should see 0.19634954 on your display. Get a$20-or-so dual-display scientific calculator, even if you cant take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you cant do without.

Compare the 2 circle area calculations. The results are the same, 0.1963 m2. Both methods have exactly 3 calculations. The formulas are different.

The first method

A = r²π

has a division up front (d ÷ 2 = r), then a squaring (r²) and a multiplication by π. The order of multiplication is interchangeable, so the formula could also be written as A = πr². Try it.

The second method

${A =}\frac{d^2 \pi}{4}$

has squaring (d²), multiplication by π and a division (by 4) The order of multiplication is interchangeable, so the formula could also be written as
${A =}\frac{\pi d^2}{4}$
.  Try it.

Both of the fractions in the formulas for the second method can also be written as

${A = d^2}\frac{\pi}{4}$
and
${A =}\frac{d^2}{4}\pi$
.

So now we have 6 possible ways to calculate the area of circles. Same results, same number of steps, same amount of work.

A = r²π             A = πr²
${A = d^2}\frac{\pi}{4}$

${A =}\frac{d^2 \pi}{4}$

${A = }\frac{d^2}{4}\cdot\pi$

${A =}\frac{\pi d^2}{4}$

A formula is a symbolic short-hand way to write the above 3-step procedures. Use a formula thats easiest to remember for you.

$\frac{\pi}{4}$
, which you need to memorize to 4 decimal places.

1. Memorizing numbers comes from an age when pocket calculators with π button were very expensive. Not anymore.
2. In some trades, accuracy beyond 4 decimal digits is not required since laying out measurements for cutting and fitting is not practical beyond 1 or 2 decimal digits. Who has a microscope on the job to mark 5.28 mm or 3.579 inches?  Any tenth of a mm or hundredth of an inch is impossible to see on a ruler or tape.  In some trades, calculating with only a few decimal digits is fine.
3. The trades that use diameter-based circle formulas to work with are the ones where cylindrical raw materials are described with diameter: piping trades, plumbing, boilermaking, machining, electrical trades, HVAC trades.

Trades where radius-based calculations dominate are the ones where drawing and laying out of circles and arches to build or to calculate cost of materials are frequent tasks: metal fabricating, sheet metal work, carpentry, tailoring, joinery, tile setting and flooring, bricklaying, stucco and lather work, architectural glazing, painting.