How to Find the Area and Circumference of a Circle

Every shape so far has had straight sides you could count and add up. A circle has none. It still has a length running around the outside and a flat space filling the inside, a perimeter and an area, just like the others. Measuring them brings in one famous number, $\pi$, which turns up in every circle. Once you know how to use it, you can find the distance around and the space inside for any circle at all.

Problem

In the diagram above is a circle whose radius is $8$. Find its diameter. Give the number of units.

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The diameter is twice the radius, $d = 2r$. Read the radius off the diagram above.
The radius is $8$, so double it.

Show the full solution

The diameter is twice the radius, $d = 2r$. The diagram above marks the radius as $8$, so $$d = 2 \times 8 = 16.$$ The diameter is $\boxed{16}$.

Problem

In the diagram above is a circle with diameter $10$. Its circumference is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show a hint

The circumference of a circle is $C = \pi d$, where $d$ is the diameter. Read the diameter off the diagram above.
The diameter is $10$, so the circumference is $10$ times $\pi$. The whole number you report is the one sitting in front of $\pi$.

Show the full solution

Circumference is $\pi$ times the diameter. The diagram above gives $d = 10$, so $$C = \pi d = 10\pi.$$ The whole number multiplying $\pi$ is $\boxed{10}$.

Problem

In the diagram above is a circle with radius $7$. Its circumference is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show a hint

Circumference is $C = 2\pi r$. Read the radius off the diagram above and put it in.
With $r = 7$ you get $2 \times 7 \times \pi$. Work out the whole number in front of $\pi$.

Show the full solution

The circumference of a circle is $C = 2\pi r$. The radius in the diagram above is $7$, so $$C = 2 \times 7 \times \pi = 14\pi.$$ That is a whole number times $\pi$, and the whole number in front is $\boxed{14}$.

Problem

In the diagram above is a circle with radius $6$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

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Area of a circle is $A = \pi r^2$. Read the radius off the diagram above and square it.
With $r = 6$ you get $6^2$. The whole number multiplying $\pi$ is that value.

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The area of a circle is $A = \pi r^2$. Read the radius $6$ off the diagram above and square it. $$A = \pi \times 6^2 = 36\pi$$ The area is $36\pi$, so the whole number multiplying $\pi$ is $\boxed{36}$.

Problem

In the diagram above is a circle with diameter $14$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show a hint

The area formula uses the radius, not the diameter, so first find the radius from $d = 2r$.
With the radius in hand, use $A = \pi r^2$ and read off the whole number sitting in front of $\pi$.

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Halve the diameter to get the radius, $r = \frac{14}{2} = 7$. Then $$A = \pi r^2 = \pi \cdot 7^2 = 49\pi,$$ so the whole number multiplying $\pi$ is $\boxed{49}$. The area formula always runs on the radius, so a diameter has to be halved before it goes in.

Problem

In the diagram above is a circle whose circumference is $26\pi$, with the radius marked by a question mark. Find the radius. Give the number of units.

Show a hint

The circumference of a circle is $C = 2\pi r$. Set that equal to $26\pi$.
From $2\pi r = 26\pi$, cancel the $\pi$ on both sides to get $2r = 26$, then divide by $2$.

Show the full solution

Circumference is $C = 2\pi r$, and here it equals $26\pi$, so $$2\pi r = 26\pi.$$ Cancel the $\pi$ on both sides to get $2r = 26$, then divide by $2$. The radius is $\boxed{13}$.

Problem

In the diagram above is a circle whose area is $64\pi$. Find the radius. Give the number of units.

Show a hint

The area of a circle is $A = \pi r^2$. Set that equal to the area you are given, $64\pi$, and cancel the $\pi$ from both sides.
You are left with $r^2 = 64$. Ask which number times itself gives $64$.

Show the full solution

The area is $A = \pi r^2$, and here it equals $64\pi$, so $$\pi r^2 = 64\pi.$$ Cancel the $\pi$ on both sides to get $r^2 = 64$. The number that squares to $64$ is $8$, so $r = \boxed{8}$.

Problem

In the diagram above is a semicircle with radius $8$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show a hint

A semicircle is half a circle, so its area is half of a full circle's area, $\frac{1}{2}\pi r^2$. Read the radius off the diagram above.
Square the radius $8$ to get $64$, then take half of $64\pi$. The answer is the whole number left in front of $\pi$.

Show the full solution

A semicircle is half a circle, so its area is half of the full circle's $\pi r^2$. With radius $8$, $$A = \frac{1}{2}\pi r^2 = \frac{1}{2}\times \pi \times 64 = 32\pi.$$ We leave the answer in terms of $\pi$, so the area is $32\pi$ and the whole number is $\boxed{32}$.

Problem

In the diagram above is a quarter circle with radius $10$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

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A quarter circle is a fourth of a full circle, so its area is $\frac{1}{4}\pi r^2$. Read the radius off the diagram.
Square the radius first, then take a fourth of it. With $r = 10$ you get $r^2 = 100$.

Show the full solution

A quarter circle is one fourth of a full circle, so its area is $\frac{1}{4}\pi r^2$. With $r = 10$, $$A = \frac{1}{4}\pi r^2 = \frac{1}{4}\times \pi \times 100 = 25\pi.$$ The area is $25\pi$, so the whole number multiplying $\pi$ is $\boxed{25}$.

Problem

In the diagram above is a ring, the shaded space between an outer circle of radius $10$ and an inner circle of radius $6$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

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The ring is what's left when you cut the small circle out of the big one, so its area is $\pi R^2 - \pi r^2$ with $R = 10$ and $r = 6$.
Find each area on its own first. The big circle is $\pi \times 10^2$ and the small one is $\pi \times 6^2$, then subtract.

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A ring is the big circle with the small circle removed, so its area is the difference of the two areas. The big circle has area $\pi \times 10^2 = 100\pi$ and the small circle has area $\pi \times 6^2 = 36\pi$. Subtract to get what's left in the ring. $$100\pi - 36\pi = 64\pi$$ The area is $64\pi$, so the whole number is $\boxed{64}$.

Problem

In the diagram above are two pizzas, a small one of radius $6$ and a large one of radius $12$. How many times as much pizza (area) is the large one? Give the number.

Show a hint

Area grows with the radius squared, since $A = \pi r^2$. Start by comparing the radii. How many times bigger is the large radius than the small one?
The radius goes from $6$ to $12$, so it doubles. When the radius is multiplied by a number, the area is multiplied by that number squared.

Show the full solution

Area is $A = \pi r^2$, so it depends on the radius squared. The radius doubles from $6$ to $12$, which multiplies the area by $$2^2 = 4.$$ The large pizza is $\boxed{4}$ times as much. Checking directly, the small pizza is $36\pi$ and the large one is $144\pi$, which is $4$ times as big.

Problem

In the diagram above, the small circular garden has radius $10$ and needs $3000$ seeds to fill it. The larger garden has three times that radius. How many seeds does it need to fill? Give the number of seeds.

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Area grows with the square of the radius. If the radius is multiplied by $3$, the area is multiplied by $3^2$.
Work out $3^2$, then multiply the number of seeds for the small garden by that factor.

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Seeds needed match the area, and the area of a circle depends on the radius squared. Tripling the radius multiplies the area by $$3^2 = 9.$$ So the larger garden needs $9$ times as many seeds as the small one, $$9 \times 3000 = 27000.$$ The larger garden needs $\boxed{27000}$ seeds.

Problem

In the diagram above is a dartboard of three circles with radii $4$, $6$, and $9$. Find the area of the outer ring, the shaded space between the radius-$6$ circle and the radius-$9$ circle. That area is a whole number times $\pi$. Give the whole number multiplying $\pi$.

Show a hint

A ring is a big circle with a smaller one punched out of the middle. Its area is the outer circle's area minus the inner circle's area, $\pi R^2 - \pi r^2$. Here the innermost radius-$4$ circle is a distraction, and the outer ring only uses $R = 9$ and $r = 6$.
Work out each area on its own first. The big circle has area $\pi \times 9^2$, and the circle you subtract has area $\pi \times 6^2$. Then subtract the whole numbers in front of $\pi$.

Show the full solution

The outer ring is the radius-$9$ circle with the radius-$6$ circle taken out, so subtract the two areas. The big circle has area $\pi \times 9^2 = 81\pi$ and the one removed has area $\pi \times 6^2 = 36\pi$, so $$81\pi - 36\pi = 45\pi.$$ The whole number multiplying $\pi$ is $\boxed{45}$. The radius-$4$ circle sits inside the part already removed, so it never enters the calculation.

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Practice

In the diagram above is a circle with radius $11$. Its diameter is a length you can read off. Find the diameter. Give the number of units.

Show the solution

The diameter runs all the way across a circle, and it is exactly twice the radius, so $d = 2r$. Here the radius is $11$, so $$d = 2 \times 11 = 22.$$ The diameter is $\boxed{22}$.

Practice

In the diagram above is a circle with diameter $8$. Its circumference is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

The circumference is the diameter times $\pi$. Read the diameter off the diagram above, $d = 8$, and use $$C = \pi d = 8\pi.$$ So the whole number multiplying $\pi$ is $\boxed{8}$.

Practice

In the diagram above is a circle with radius $9$. Its circumference is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

The circumference of a circle is $C = 2\pi r$. Here the radius is $9$, so $$C = 2\pi \times 9 = 18\pi.$$ Leaving the answer in terms of $\pi$, the whole number multiplying $\pi$ is $\boxed{18}$.

Practice

In the diagram above is a circle with radius $5$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

The area of a circle is $A = \pi r^2$. Read the radius $5$ off the diagram above and square it. $$A = \pi \times 5^2 = 25\pi$$ The area is $25\pi$, so the whole number multiplying $\pi$ is $\boxed{25}$.

Practice

In the diagram above is a circle with diameter $20$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

The diameter is $20$, and the radius is half of that, so $r = 10$. Area is $\pi r^2$, so $$A = \pi \times 10^2 = 100\pi.$$ We leave the answer in terms of $\pi$ and give just the whole number in front, so the answer is $\boxed{100}$.

Practice

In the diagram above is a circle whose circumference is $30\pi$. Find its radius. Give the number of units.

Show the solution

Circumference is $2\pi r$, and here it equals $30\pi$, so $$2\pi r = 30\pi.$$ Divide both sides by $2\pi$ to get $r = 15$. The radius is $\boxed{15}$.

Practice

In the diagram above is a circle whose area is $49\pi$. Find its radius. Give the number of units.

Show the solution

The area of a circle is $\pi r^2$. Here that equals $49\pi$, so $$\pi r^2 = 49\pi.$$ Divide both sides by $\pi$ to get $r^2 = 49$. The number that squares to $49$ is $7$, so the radius is $\boxed{7}$.

Practice

In the diagram above is a semicircle with radius $10$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

A semicircle is half a full circle, so its area is half of $\pi r^2$. With $r = 10$ the radius squared is $r^2 = 100$, so $$A = \frac{1}{2}\pi r^2 = \frac{1}{2}\times 100\pi = 50\pi.$$ The area is $50\pi$, so the whole number multiplying $\pi$ is $\boxed{50}$.

Practice

In the diagram above is a quarter circle with radius $8$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

A quarter circle covers a fourth of the whole circle, so its area is $\frac{1}{4}\pi r^2$. Reading the radius off the diagram above, $r = 8$, so $r^2 = 64$ and $$A = \frac{1}{4}\pi r^2 = \frac{1}{4}\times 64\pi = 16\pi.$$ We leave the answer in terms of $\pi$, so the whole number in front of $\pi$ is $\boxed{16}$.

Practice

In the diagram above is a ring between an outer circle of radius $13$ and an inner circle of radius $5$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

A ring is the big disk with the small disk punched out, so its area is the outer area minus the inner area. Using $A = \pi r^2$ for each, $$\pi \times 13^2 - \pi \times 5^2 = 169\pi - 25\pi = 144\pi.$$ The area is $144\pi$, so the whole number multiplying $\pi$ is $\boxed{144}$.

Practice

In the diagram above are two circles, one with radius $5$ and one with radius $10$. The radius of the larger circle is twice the radius of the smaller. How many times as big as the smaller circle's area is the larger circle's area? Give the number.

Show the solution

Area is $\pi r^2$, so doubling the radius multiplies the area by $$2^2 = \boxed{4}.$$ Checking it, the small circle has area $\pi \cdot 5^2 = 25\pi$ and the large one has $\pi \cdot 10^2 = 100\pi$, which is $4$ times as much. Doubling the radius does not double the area.

Practice

In the diagram above, a small circular flower bed of radius $7$ needs $150$ bags of soil to fill it. The larger bed beside it has twice the radius. How many bags of soil does the larger bed need? Give the number of bags.

Show the solution

The area of a circle is $\pi r^2$, so it grows with the square of the radius. Doubling the radius multiplies the area by $2^2 = 4$. The larger bed covers $4$ times the area, so it needs $4$ times the soil. $$4 \times 150 = 600$$ The larger bed needs $\boxed{600}$ bags.

Practice

In the diagram above is a dartboard of three circles with radii $5$, $8$, and $12$. Find the area of the outer ring, the region between the circle of radius $8$ and the circle of radius $12$. Its area is a whole number times $\pi$. Find that whole number. Give the whole number multiplying $\pi$.

Show the solution

A ring is one circle with a smaller one taken out of it, so its area is the big area minus the small area, $\pi R^2 - \pi r^2$. The outer edge is at radius $12$ and the inner edge is at radius $8$, so $$\pi \times 12^2 - \pi \times 8^2 = 144\pi - 64\pi = 80\pi.$$ The area is $80\pi$, so the whole number is $\boxed{80}$.

Circles

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