Circumference and Area Word Problems | Grade 7 Math | LevelKite

A circle story rarely tells you which measurement to find. It describes something real, such as edging around a pond or seed spread over a lawn, and you decide what the story measures.

You choose between two measurements:

The circumference of a circle is the distance around it. It is a length, so its unit is a length unit such as meters.
The area of a circle is the amount of flat surface inside it. Its unit is a square unit such as square meters.

So your first question is always the same: does this story go around the circle, or does it cover the inside?

Both measurements come from one stated length. The radius is the distance from the center of a circle to its edge. The diameter is the distance straight across the circle through its center.

Let $r$ stand for the radius and $d$ for the diameter, in the same length unit. Both are positive. Let $C$ stand for the circumference in that unit and $A$ for the area in the matching square unit, such as meters and square meters. You have already used three relationships between them:

$d = 2r$ , because a diameter runs from the edge to the center and on to the opposite edge, so it is two radius lengths.
$C = \pi d$ , because $\pi$ is the name of the number you get when you divide any circle's circumference by its diameter. That number is the same for every circle and is a little more than $3$ . Each problem states which form of $\pi$ to use.
$A = \pi r^2$ , where $r^2$ means $r \times r$ .

The next table sorts four stories. Notice that one object can appear in two stories that measure different things.

Stories about whole circles, where r is the positive radius and d = 2r is the diameter

The story says	What it measures	Relationship	Unit of the answer
Trim goes once around the whole edge of a circular tabletop	Distance around: circumference	$C = \pi d$	A length unit, such as feet
Paint covers the whole top surface of that tabletop	Surface inside: area	$A = \pi r^2$	A square unit, such as square feet
A fence follows the whole edge of a circular pen	Distance around: circumference	$C = \pi d$	A length unit, such as meters
Straw covers the whole floor inside that pen	Surface inside: area	$A = \pi r^2$	A square unit, such as square meters

Key concept

Decide what the story measures before you calculate

Start from the stated radius $r$ or diameter $d$ of the circle, its unit, and the stated form of $\pi$ . The radius and the diameter are positive, and $d = 2r$ .

If the story goes around the whole circle, find the circumference, $C = \pi d$ . The answer is a length.
If the story covers the whole inside, find the area, $A = \pi r^2$ . The answer is in square units.
If the shape is only part of a circle and the story goes around its whole boundary, add every straight edge to the curved edge.

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

Earlier you worked backward from a circumference or an area to a radius or a diameter. Here you work forward. Every story states a radius or a diameter, and you use four moves:

Find what is stated. Write down the radius or the diameter, which is a positive length, with its unit. Note the form of $\pi$ the problem asks for.
Decide what the story measures. Check exactly which edges or which region the story names.
Get the length you need. $C = \pi d$ uses the diameter and $A = \pi r^2$ uses the radius. Move between them with $d = 2r$ .
Calculate and label. Show the arithmetic, then answer in a sentence with the correct unit.

When a problem says to use $3.14$ for $\pi$ , you are using a rounded value, so your result is a close estimate. Write the symbol $\approx$ , which means is about equal to, in place of $=$ . When a problem says to leave the answer in terms of $\pi$ , keep the symbol, and your result is exact.

Worked example

Edging around a pond

You place stone edging once around the whole outer edge of a circular pond. The pond has a diameter of $6$ meters. Use $3.14$ for $\pi$ . About how many meters of edging do you need?

Decide what the story measures. The edging goes around the edge of the circle, so you need the distance around, which is the circumference. Let d be the diameter in meters and C be the circumference in meters.
$d = 6$
The diameter is stated and positive, so the circumference is π times the diameter. Replace π with 3.14, the form the problem states. The result is an estimate because 3.14 is a rounded value of π.
$C = \pi d \approx 3.14 \times 6$
Split 3.14 into 3 + 0.14. Multiplying each part by 6 and adding the two products gives the same result as multiplying 3.14 by 6.
$3.14 \times 6 = (3 \times 6) + (0.14 \times 6) = 18 + 0.84 = 18.84$

Answer

You need about $18.84$ meters of edging to go once around the pond.

Worked example

Seed over a lawn

You spread grass seed over a whole circular lawn, so the seed covers all of the surface inside the circle. The lawn has a diameter of $8$ meters. First give the area the seed covers in terms of $\pi$ . Then use $3.14$ for $\pi$ to estimate that area.

Decide what the story measures. The seed covers the surface inside the circle, so you need the area, in square meters. Let d be the diameter in meters, r be the radius in meters, and A be the area in square meters.
$d = 8$
The area relationship uses the radius, but the story states the diameter. A diameter is two radius lengths, so d = 2r. Divide both sides by 2; the two sides stay equal. A positive diameter gives a positive radius.
$r = d \div 2 = 8 \div 2 = 4$
Put the radius into the area relationship. Squaring the radius means multiplying it by itself. Keep the symbol π, the first form the problem asks for, so the area is exact.
$A = \pi r^2 = \pi \times 4 \times 4 = \pi \times 16 = 16\pi$
Now replace π with 3.14, the second form the problem asks for, to estimate the area. Split 3.14 into 3 + 0.14, multiply each part by 16, and add the two products.
$A \approx 3.14 \times 16 = (3 \times 16) + (0.14 \times 16) = 48 + 2.24 = 50.24$

Answer

The seed covers exactly $16\pi$ square meters, which is about $50.24$ square meters.

Some stories use only part of a circle. A half circle, also called a semicircle, is one of the two equal pieces you get when a straight cut passes through the center of a circle. The perimeter of a flat shape is the total distance around its boundary.

The boundary of a half circle has two parts:

a curved edge, which is half of the full circle's circumference, and
a straight edge, which is a diameter of the full circle.

Let $P$ stand for the perimeter of a half circle, in the same length unit as the positive diameter $d$ of the full circle. The perimeter counts both parts, because both belong to the boundary. The curved edge is half of $\pi d$ , and the straight edge is $d$ .

P = \frac{1}{2}\pi d + d

Worked example

A fence around a half-circle garden

A garden bed is shaped like half of a circle that has a radius of $5$ meters. Its boundary has two parts: a curved edge, which is half of that circle's circumference, and a straight edge, which is a diameter of that circle. You put a fence along the whole boundary, both the curved edge and the straight edge. Use $3.14$ for $\pi$ . About how many meters of fence do you need?

A fence follows the boundary, so you need the perimeter, a length in meters. The story names both edges, so both count. Let r be the radius in meters, d be the diameter in meters, and P be the perimeter in meters.
$P = \text{curved edge} + \text{straight edge}$
The straight edge is a diameter. The radius is stated and positive, so doubling it gives a positive diameter.
$d = 2r = 2 \times 5 = 10$
Let C be the circumference of the full circle in meters. Find it with 3.14 for π, as the problem states. Multiplying by 10 moves each digit one place to the left.
$C = \pi d \approx 3.14 \times 10 = 31.4$
The curved edge is exactly half of the full circle's edge, so divide the circumference by 2.
$31.4 \div 2 = 15.7$
Add the straight edge, because the fence runs along it too.
$P \approx 15.7 + 10 = 25.7$

Answer

You need about $25.7$ meters of fence: about $15.7$ meters for the curved edge and $10$ meters for the straight edge.

Check each answer with a quick estimate. These checks work for any circle with a positive radius $r$ and diameter $d = 2r$ , because $\pi$ is a little more than $3$ :

The circumference is a little more than $3 \times d$ . For the pond, $3 \times 6 = 18$ , and the answer was about $18.84$ meters.
The area is a little more than $3 \times r \times r$ . For the lawn, $3 \times 4 \times 4 = 48$ , and the answer was about $50.24$ square meters.

For the half-circle garden, estimate both edges. Half of $3 \times 10$ is $15$ , and the straight edge adds $10$ , so the fence should be a little more than $25$ meters. The answer was about $25.7$ meters. If an answer lands far from its estimate, reread the story and recheck your arithmetic.

Takeaway: A circle word problem is a decision before it is a calculation. Ask what the story measures. The distance around a whole circle is the circumference, $C = \pi d$ . The surface inside is the area, $A = \pi r^2$ . Start from the stated positive radius $r$ or diameter $d$ , use $d = 2r$ to get the length you need, and calculate in the stated form of $\pi$ with the correct unit. When the shape is part of a circle, its perimeter includes every straight edge.

Next, you will explain how the circumference and the area of a circle are related.