Error Analysis: Circumference and Area of Circles | Grade 7 Math | LevelKite

Imagine a classmate hands you finished work on a circle, and the final answer is wrong. The answer tells you that something went wrong. It does not tell you what.

Each line of work makes a claim about the circle in the problem. A line is true when its claim is correct. An error is a mistake in the work. Error analysis means reading the work one line at a time and doing three jobs:

Find the first line that is not true.
Explain which fact about circles that line breaks.
Fix the work from that line on.

Why the first line? That is where the work stops matching the circle. Later lines may be worked correctly and still be untrue, because they are built on it.

In this lesson you will do these three jobs for work on the distance around a circle and the space inside it.

You judge each line against facts that are true for every circle. Each problem here states a radius or a diameter as a positive length with a unit, such as centimetres (cm).

The radius, $r$ , is the distance from the centre of the circle to its edge. The diameter, $d$ , is the distance straight across the circle through the centre. That path goes from edge to centre to edge, so it is two radius lengths: $d = 2r$ .
The circumference, $C$ , is the distance once around the circle, so it is a length. A little more than $3$ diameters fit around any circle. That number is called $\pi$ , so the circumference rule is $C = \pi d$ .
The area, $A$ , is the amount of flat space inside the circle. You count it in square units, such as square centimetres ( $\text{cm}^2$ ). The area rule is $A = \pi r^2$ , where $r^2$ means $r \times r$ .

Each problem states the form of $\pi$ to use: $3.14$ , or the symbol $\pi$ left in the answer.

A semicircle is half of a circle, cut along a diameter. Its perimeter is the total distance around it, so it includes the curved edge and the straight edge.

Key concept

Check each line against the circle facts

The three circle facts given here hold for any circle whose radius $r$ or diameter $d$ is stated as a positive length. Correct work agrees with them on every line and uses the form of $\pi$ the problem states. To find an error, read from the top and ask four questions:

Rule: Does the rule match what is asked, a distance around or the space inside?
Measure: Does the rule get the length it needs, the diameter for circumference and the radius for area?
Arithmetic: Is each calculation right, in the stated form of $\pi$ ?
Whole answer: Are the units right, and does the perimeter of part of a circle include every straight edge?

The first line that fails a question is the error. Redo the work from that line on.

d = 2r, \qquad C = \pi d, \qquad A = \pi r^2 \qquad (r > 0)

One line that fails each check, for a whole circle with stated radius 4 cm (diameter 8 cm), using 3.14 for π

Check	A line that fails it	Why the line is not true	Fixed line
Rule	$A = \pi d \approx 3.14 \times 8$	Area is the space inside, so the rule is $A = \pi r^2$ . The rule $\pi d$ gives the distance around.	$A \approx 3.14 \times 4^2 = 3.14 \times 16 = 50.24\ \text{cm}^2$
Measure	$C \approx 3.14 \times 4$	$C = \pi d$ needs the diameter, $8$ cm. The work used the radius, $4$ cm.	$C \approx 3.14 \times 8 = 25.12$ cm
Arithmetic	$A \approx 3.14 \times 4^2 = 12.56^2$	In $\pi r^2$ only the radius is squared: find $4^2 = 16$ first, then multiply by $3.14$ .	$A \approx 3.14 \times 16 = 50.24\ \text{cm}^2$
Whole answer	$A \approx 50.24$ cm	Area is counted in square units, so the unit is $\text{cm}^2$ , not cm.	$A \approx 50.24\ \text{cm}^2$

Worked example

Example 1: Check Kai's area

A circle has a diameter of $10$ cm. Use $3.14$ for $\pi$ . Kai finds the area of the whole circle and writes three lines. Line 1: $A \approx 3.14 \times 10^2$ . Line 2: $A \approx 3.14 \times 100$ . Line 3: $A \approx 314\ \text{cm}^2$ . Find the first line that is not true, explain the mistake, and fix the work.

Ask the Rule question. The problem asks for the area, the space inside the whole circle, so the rule is π times the radius squared. Line 1 multiplies 3.14 by a squared number, so the rule and the form of π are right.
$A = \pi r^2$
Ask the Measure question. The area rule needs the radius, but the 10 cm in the problem is the diameter. Line 1 squares 10, so it puts the diameter where the radius belongs. Line 1 is the first line that is not true.
Find the radius. The diameter is 2 times the radius, so dividing the diameter by 2 gives the radius.
$r = 10 \div 2 = 5\ \text{cm}$
Redo the work from Line 1 with the radius. Square 5 first, because only the radius is squared in the area rule. Then multiply by 3.14, the stated form of π.
$A \approx 3.14 \times 5^2 = 3.14 \times 25 = 78.5\ \text{cm}^2$
Check the size of the new answer. π is a little more than 3, so the area is a little more than 3 times 25, which is 75. The answer 78.5 fits. Kai's 314 does not.
$3 \times 25 = 75$

Answer

Line 1 is the first line that is not true: Kai used the diameter, $10$ cm, where the radius, $5$ cm, belongs. Using $3.14$ for $\pi$ , the area of the circle is about $78.5\ \text{cm}^2$ .

Worked example

Example 2: Check Mia's area

A circle has a radius of $9$ m. Leave the symbol $\pi$ in the answer. Mia finds the area of the whole circle and writes three lines. Line 1: $A = \pi \times 9^2$ . Line 2: $A = \pi \times 18$ . Line 3: $A = 18\pi\ \text{m}^2$ . Find the first line that is not true, explain the mistake, and fix the work.

Check Line 1 with the Rule and Measure questions. Area is the space inside the circle, so the rule is π times the radius squared. The problem states the radius, 9 m, and Line 1 squares 9. Line 1 is true.
$A = \pi \times 9^2$
Check Line 2 with the Arithmetic question. Squaring 9 means multiplying 9 by 9, which gives 81. Line 2 has 18, which is 9 times 2. Line 2 is the first line that is not true.
$9^2 = 9 \times 9 = 81, \quad \text{not } 9 \times 2 = 18$
Explain why doubling cannot give an area. Doubling the radius gives the diameter, 18 m, so π times 18 is the distance around this circle, a length. Area is counted in square units, and those come from multiplying 9 m by 9 m.
$9\ \text{m} \times 9\ \text{m} = 81\ \text{m}^2$
Redo the work from Line 2. Keep π as a symbol, because the problem says to leave the symbol π in the answer.
$A = \pi \times 81 = 81\pi\ \text{m}^2$

Answer

Line 2 is the first line that is not true: Mia doubled the radius instead of squaring it. The area of the circle is $81\pi\ \text{m}^2$ .

Worked example

Example 3: Check Omar's border

A flower bed is a semicircle whose straight edge is a diameter, $6$ m long. A border runs all the way around the bed, along the curved edge and the straight edge. Use $3.14$ for $\pi$ . Omar finds the length of the border in three lines. Line 1: for the full circle, $C \approx 3.14 \times 6 = 18.84$ m. Line 2: for the curved edge, $18.84 \div 2 = 9.42$ m. Line 3: the border is about $9.42$ m long. Find the first line that is not true, explain the mistake, and fix the work.

Check Line 1. It finds the distance around the full circle that the bed is half of. That is a circumference, so the rule is π times the diameter, and the stated diameter is 6 m. The multiplication is right, so Line 1 is true.
$C \approx 3.14 \times 6 = 18.84\ \text{m}$
Check Line 2. A semicircle is half of a circle, so its curved edge is half of the circumference. The division is right, so Line 2 is true. It gives the curved edge only.
$18.84 \div 2 = 9.42\ \text{m}$
Check Line 3 with the Whole answer question. The border also runs along the straight edge, which is 6 m long. Line 3 counts the curved edge and leaves the straight edge out, so Line 3 is the first line that is not true.
Redo the work from Line 3. The border is the perimeter of the bed, so add the straight edge to the curved edge. Both are lengths in metres, so the sum is in metres.
$9.42 + 6 = 15.42\ \text{m}$

Answer

Line 3 is the first line that is not true: Omar left out the straight edge of the bed. Using $3.14$ for $\pi$ , the border is about $15.42$ m long.

Here is the idea to keep. Error analysis is a search for the first line that is not true. When circle work starts from a stated radius or diameter that is a positive length, every line must agree with $d = 2r$ , $C = \pi d$ and $A = \pi r^2$ , in the form of $\pi$ the problem states. The perimeter of part of a circle must also include every straight edge.

Read from the top and ask the four questions: rule, measure, arithmetic, whole answer. Name the line that fails, say which fact it breaks, and redo the work from that line on. Correct arithmetic does not make a line true when the line starts from the wrong rule or the wrong length.