7th Grade · 7.G.B.4

Area of a circle in 7th grade

The area of a circle is the amount of flat space inside it, measured in square units such as square feet. In 7th grade, your child learns two circle formulas: one for area and one for circumference, the distance around the edge. Your child also learns why the two formulas are linked and uses them to solve real problems.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

What it means, in plain words

Every circle has a center. The radius, $r$ , is the distance from the center to the edge. The diameter, $d$ , goes all the way across through the center, so it is twice the radius. Pi, written $\pi$ , is the number you get when you divide any circle's circumference by its diameter. It is close to 3.14.

The circumference is $C = \pi d$ , which is the same as $C = 2\pi r$ . The area of a circle is $A = \pi r^2$ , where $r^2$ means $r \times r$ . Circumference is a length, so it is in plain units such as feet. Area covers a surface, so it is in square units such as square feet. Teachers often ask for answers in terms of $\pi$ , such as $16\pi$ , which keeps $\pi$ as a symbol. Other problems want a decimal, using 3.14 for $\pi$ .

In 7th grade, children also see why the area and circumference of a circle are linked. Cut a circle into many thin wedges, like pizza slices, and lay them side by side, alternately pointing up and down. Together they make a shape close to a rectangle. Its height is the radius, and its length is half the circumference, $\pi r$ . So the area is $\pi r \times r = \pi r^2$ .

Your child should be able to:

Tell the radius from the diameter and switch between them
Find the circumference and the area from the radius or the diameter, in terms of $\pi$ or with 3.14
Decide whether a problem asks for the distance around the edge or the space inside
Explain with the wedge picture why the area is $\pi r^2$

Example

A round rug is 8 feet across. What are its circumference and its area?

8 feet across is the diameter, so the radius is 4 feet.
Circumference: $C = \pi \times 8 = 8\pi$ , about $8 \times 3.14 = 25.12$ feet.
Area: $A = \pi \times 4^2 = 16\pi$ , about $16 \times 3.14 = 50.24$ square feet.
The edge of the rug is about 25 feet long, and the rug covers about 50 square feet.

Example

A wheel rolls exactly one full turn and travels 94.2 inches. What is the radius of the wheel?

One full turn covers the circumference, so $C = 94.2$ inches.
$C = \pi d$ , so with 3.14 for $\pi$ , $d = 94.2 \div 3.14 = 30$ inches.
The radius is half the diameter: $30 \div 2 = 15$ .
The radius of the wheel is about 15 inches.

Where children usually get stuck

A plate is 12 inches across, and your child works out its area as $\pi \times 12^2 = 144\pi$ .
Real objects and pictures usually give the full width, the diameter. The area formula needs the radius, half of that width. Using the diameter makes the area four times too big: the right answer is $36\pi$ square inches.
What helps. Before any numbers, ask your child to say whether the length is the radius or the diameter. If it goes all the way across, cut it in half first.
Your child uses $2\pi r$ when the problem asks for the area, or $\pi r^2$ for the distance around.
Both formulas have $\pi$ and $r$ in them. Learned as symbols with no picture behind them, they blur together.
What helps. Ask one question first: are we going around the edge or covering the inside? Going around is circumference, in plain units. Covering is area, in square units.
For a radius of 7, your child writes $7^2 = 14$ and gives an area of $14\pi$ .
Squaring gets mixed up with doubling. $7^2$ means $7 \times 7 = 49$ , so the area is $49\pi$ .
What helps. Draw a 7 by 7 square on graph paper and count the small squares together. There are 49, not 14.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Aisha wanted to find the exact area of a circle with diameter $10$ units. Select the line in her work below that contains the mistake, or select 'None' if the work is correct.
Formula
$\displaystyle A = \pi r^2$
Substitute
$\displaystyle = \pi(10)^2$
Student result
$\displaystyle = 100\pi$
A
Formula
B
Substitute
C
Student result
D
None — the work is correct
Show the answer and how to solve it
The mistake is on the Substitute line. Aisha plugged in the diameter $10$ where the radius belongs; using $r = 5$ gives $A = \pi(5)^{2} = 25\pi$ .
1. 1Re-read Aisha's three lines. The stem says the circle has diameter $10$ units.
2. 2Line 1 (Formula) is correct: the area of a circle is $A = \pi r^{2}$ , where $r$ is the radius.
3. 3The stem gives diameter $10$ , so the radius is $5$ . Line 2 should plug in $5$ , not $10$ .
4. 4With the correct radius, the area is $25\pi$ . Aisha's line 2, $= \pi(10)^{2}$ , used the diameter where the radius belongs — the mistake is on the Substitute line.
5. 5Line 3, $= 100\pi$ , correctly follows from $\pi(10)^{2}$ , so the error didn't originate there.
Common mistake. Plugging the diameter $10$ directly into $A = \pi r^{2}$ instead of the radius $5$ is a Substitute-line error.
The idea underneath. When the stem gives a diameter, halve it before substituting into $A = \pi r^{2}$ .

2On level

Aisha tried to find the area of a circle with radius 3 cm. Which explanation best describes the mistake in her work below?

Formula	$\displaystyle A$	$\displaystyle =$	$\displaystyle \pi r^2$
Substitute	$\displaystyle A$	$\displaystyle =$	$\displaystyle \pi \cdot 3$
Simplify	$\displaystyle A$	$\displaystyle =$	$\displaystyle 3\pi$

The student used the circumference formula instead of the area formula.

The student treated the radius as if it were the diameter.

The student forgot to square the radius when substituting the given value.

The student squared the whole product

\pi r

instead of just the radius.

Show the answer and how to solve it

The correct explanation is that Aisha forgot to square the radius, because substituting $r = 3$ into $A = \pi r^{2}$ gives $A = \pi(3)^{2}$ , not $A = \pi \cdot 3$ .

1Look at the first line. Aisha starts with the correct formula for the area of a circle: $A = \pi r^{2}$ .
2Since the radius is 3, replace r with 3 and keep the square on the radius value.
3Aisha's second line is $A = \pi \cdot 3$ . That turns $r^{2}$ into just 3, so the square was left out.
4So the best explanation is that Aisha forgot to square the radius when substituting 3.

Common mistake. A common mistake is replacing $r^{2}$ with 3 instead of $(3)^{2}$ . When you substitute a number for a variable, the exponent stays with that number.

The idea underneath. When you substitute into a formula, exponents stay attached to the value of the variable, so $r^{2}$ becomes $(3)^{2}$ .

3On level
Use A for area and k for radius. Write the formula for the area of a circle.
Loading...
Show the answer and how to solve it
Answer. A = \pi k^{2}
The formula is $A = \pi k^{2}$ because the area of a circle is π times the square of the radius, and this problem uses k for the radius.
1. 1Start with the standard area formula for a circle.
2. 2This problem uses k for the radius, so replace r with k.
3. 3That is the full formula with the symbols the prompt asked you to use.
Common mistake. A common mistake is writing $A = 2\pi k$ , which uses the circumference expression for area. Another is keeping r instead of k; the mathematical relationship is the same, but the variable letter must match the prompt.
The idea underneath. Changing the letter names does not change the relationship: whatever symbol stands for the radius is the one that gets squared.
4Stretch
A semicircle has radius 5. Which expression gives its exact perimeter?
A
$\displaystyle \frac{25\pi}{2}$
B
$\displaystyle 5\pi + {}$
$\displaystyle 10$
C
$\displaystyle 10\pi + {}$
$\displaystyle 10$
D
$\displaystyle 5\pi + {}$
$\displaystyle 5$
Show the answer and how to solve it
The exact perimeter is $5\pi + 10$ because the semicircle arc is $5\pi$ and the diameter straight side is $10$ .
1. 1A semicircle's perimeter includes $1$ curved arc and $1$ straight side equal to the diameter.
2. 2With radius $5$ , the full circumference is $2\pi(5) = 10\pi$ . The semicircle arc is half of that.
3. 3The straight side is the diameter: $2(5) = 10$ .
4. 4Add the arc and the straight side to get the exact perimeter.
Common mistake. A common mistake is using the full circumference $10\pi$ as the arc, producing $10\pi + 10$ . A semicircle uses only half the circle, so the arc is $5\pi$ .
The idea underneath. For any partial circle, perimeter means the arc length plus every straight edge on the boundary.

Formula	$\displaystyle A = \pi r^2$
Substitute	$\displaystyle = \pi(10)^2$
Student result	$\displaystyle = 100\pi$