Area and volume

The last line of the question usually names a unit, and that tells you what to find.

• “What is the area, in square inches, of the triangle?” Square units, so it's an area.
• “What is the volume, in cubic centimeters, of the cone?” Cubic units, so it's a volume.
• “What is the length, in inches, of an edge of the base?” A plain unit, so it's a length, often worked back from an area or a volume.
• “The volume of the sphere is $n\pi$ cubic feet. What is the value of $n$?” It wants the number in front of $\pi$.

Two more cues tell you the kind of work. “Similar” or “each side is 3 times as long” means a scaled-up figure. “Total” or “not taken up” means two pieces to add or subtract.

Desmos does the arithmetic, but it can't pick the formula or tell you which side is the height.

The reference sheetThe page of formulas you can open at any point in the Math section.Read the lesson gives the area of a rectangle, a triangle and a circle, and the volume of a box, a cylinder, a sphere, a cone and a rectangular pyramid. Copy the formula onto scratch paper before you put any numbers in.

A square is a rectangle with equal sides, so its area is $s^2$. A cube is a box with three equal edges, so its volume is $s^3$. A cube with edge 5 has volume $5\times5\times5=125$. The wrong choices 25 and 30 come from $5^2$ and $6\times5$.

The sheet's box formula works only when the base is a rectangle. A prism is any solid with the same shape all the way through, like a tent. Multiply the area of one end by the length.

Say the end is a triangle with base 6 and height 4, and the solid is 10 long.

1. Find the area of the triangle at the end.

   $$B=\tfrac{1}{2}(6)(4)=12$$

2. Multiply by the length, which is $h$ here.

   $$V=12\cdot10=120$$

Multiplying 6, 4 and 10 gives 240, which treats the triangle as a rectangle.

PerimeterThe distance all the way around a flat shape., the distance around, isn't on the sheet either. It's $2\ell+2w$ for a rectangle and $4s$ for a square.

Surface area is the total area of every outside face. It isn't on the sheet, so you build it face by face.

The box's volume, $2\times4\times5=40$, is in cubic units, so it can't be a surface area. A cube has six equal square faces, so its surface area is $6s^2$. A box with no lid has only five faces.

The four side faces of a box unroll into one rectangle as wide as the base's perimeter. Standing on its 2 by 4 face, this box's sides make $12\times5=60$.

A closed cylinder has two circles for lids and a curved side. Unroll the side and it's a rectangle. Its width is the distance around the circle, $2\pi r$, and its height is the cylinder's height.

A closed cylinder has a base area of $9\pi$ square inches and a height of 10 inches. What's its surface area?

1. Get the radius from the base area: $r^2=9$.

   $$\pi r^2=9\pi,\ \text{so}\ r=3$$

2. Two lids, each $9\pi$.

   $$2(9\pi)=18\pi$$

3. The side: the distance around times the height.

   $$2\pi(3)(10)=60\pi$$

4. Add the lids and the side.

   $$18\pi+60\pi=78\pi$$

Some questions give the height as a letter. With height $h$, the same steps give $18\pi+6\pi h$.

Write down each length the formula needs before you multiply. Some take a step to find.

A rectangle is 6 wide, and its length is 3 more than twice its width.

In the same way, “12 longer than the width” means $\ell=w+12$. Multiplying the width by 12 is the slip to watch for.

The height in any area or volume formula meets the base at a right angle. This triangle has base 10, a height of 6, and a slanted side of 8.

A cone works the same way. Its height runs straight down the middle, and the slanted edge you can see is longer.

If the height isn't given, you may have to find it first. In a right triangle, get the missing side from $c^2=a^2+b^2$, which the right triangles lesson covers. On a graph, a side that runs straight across or straight up and down is as long as the difference between its ends' coordinates.

Converting an area or a volume takes the conversion squared or cubed. A foot is 12 inches, so a square foot is $12^2=144$ square inches and a cubic foot is $12^3=1{,}728$ cubic inches.

Watch for two unit words in one question, like a tank measured in feet with the water's depth in inches, or a price per square yard for a floor measured in feet.

The diameterThe distance straight across a circle through the center, which is twice the radius. goes all the way across through the center, so it's twice the radius. Halve a diameter before it goes into any formula.

Which is the volume of a cone with base diameter 10 and height 12?

• A. $25\pi$
• B. $100\pi$
• C. $300\pi$
• D. $400\pi$

1. Halve the diameter.

   $$r=10\div2=5$$

2. Copy the cone formula from the sheet and put the numbers in.

   $$V=\tfrac{1}{3}\pi(5)^2(12)$$

3. Square the radius.

   $$=\tfrac{1}{3}\pi(25)(12)$$

4. $25\times12=300$, and a third of 300 is 100.

   $$=100\pi$$

B is correct. A is the area of the base alone, and C leaves off the one-third.

When the choices have $\pi$ in them, keep it. Some questions ask for just the number in front of $\pi$, its coefficientThe number multiplying a letter, like the 5 in 5x.Read the lesson: “The volume is $n\pi$. What is the value of $n$?” Keep fractions as fractions. Here's a sphere 3 inches across.

1. Halve the 3.

   $$r=\tfrac{3}{2}$$

2. Put it into the sphere formula from the sheet.

   $$V=\tfrac{4}{3}\pi\left(\tfrac{3}{2}\right)^3$$

3. Cube the top and the bottom: $3^3=27$ and $2^3=8$.

   $$=\tfrac{4}{3}\cdot\tfrac{27}{8}\pi$$

4. $\tfrac{4\cdot27}{3\cdot8}=\tfrac{108}{24}$, which simplifies to $\tfrac{9}{2}$.

   $$=\tfrac{9}{2}\pi$$

5. Enter the number in front of $\pi$.

   $$n=4.5$$

About 14.14 is the whole volume with $\pi$ multiplied in, so it doesn't answer this question.

Leave pi out when you want the coefficient. If the question is a grid-inA question where you type your own answer instead of picking one of four.Read the lesson, the lesson on entering answers covers what fits in the box.

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A lighthouse lamp that Salma monitors runs 11 hours nightly. Its housing is shaped like the right circular cone shown.

What is its volume, in cubic inches, in terms of $\pi$?

1. A

   $136\pi$

2. B

   $320\pi$

3. C

   $960\pi$

4. D

   $1{,}280\pi$

Show the answer

• A

  $136\pi$

  This computes the lateral surface area: the slant height is $\sqrt{8^2+15^2}=17$, so $\pi r\ell=\pi(8)(17)=136\pi$. The question asks for volume.

• B

  $320\pi$

  Correct answer

  The radiusThe distance from the center of a circle out to its edge.Read the lesson is half the 16-inch diameterThe distance straight across a circle through the center, which is twice the radius.Read the lesson, so $r=8$. Using $V=\frac{1}{3}\pi r^2h$ gives $V=\frac{1}{3}\pi(8^2)(15)=320\pi$ cubic inches.

• C

  $960\pi$

  This drops the $\frac{1}{3}$ from the cone volume formula. Computing $\pi r^2h=\pi(8^2)(15)$ gives $960\pi$, three times the cone's volume.

• D

  $1{,}280\pi$

  This substitutes the 16-inch diameterThe distance straight across a circle through the center, which is twice the radius.Read the lesson for the radiusThe distance from the center of a circle out to its edge.Read the lesson in the volume formula. Using $r=16$ gives $\frac{1}{3}\pi(16^2)(15)=1{,}280\pi$, four times the volume.

Solution

The cone's diameterThe distance straight across a circle through the center, which is twice the radius.Read the lesson is 16 inches, so its radiusThe distance from the center of a circle out to its edge.Read the lesson is $16\div2=8$ inches. Its height is 15 inches.

Use the cone volume formula: $$V=\frac{1}{3}\pi r^2h$$. Substitute $r=8$ and $h=15$: $$V=\frac{1}{3}\pi(8^2)(15)=\frac{1}{3}\pi(64)(15)=320\pi.$$. The housing's volume is $320\pi$ cubic inches.

Report a problem

When the question gives a volume and asks for a length, put in what you know and undo the formula one step at a time. If the unknown is squared, the last step is a square root.

A cone holds $96\pi$ cubic centimeters and stands 8 centimeters tall. What's its radius?

1. Put the volume and the height into the cone formula.

   $$96\pi=\tfrac{1}{3}\pi r^2(8)$$

2. Divide both sides by $\pi$.

   $$96=\tfrac{1}{3}r^2(8)$$

3. Multiply both sides by 3.

   $$288=8r^2$$

4. Divide both sides by 8.

   $$36=r^2$$

5. Take the square root. A length is positive, so keep 6.

   $$r=6$$

A pyramid with a square base works the same way.

Some questions put two shapes together. Find each piece and write both numbers down. Then add for “total” or “combined,” and subtract for “remaining” or “not taken up.”

Tank A is a cylinder with radius 3 feet and height 10 feet. Tank B holds $50\pi$ cubic feet. How much do they hold together?

1. Tank A's volume.

   $$\pi(3)^2(10)=90\pi$$

2. Add tank B's volume as it's given.

   $$90\pi+50\pi=140\pi$$

You don't need tank B's radius.

When one solid sits inside another, get the inner one's size from the outer one.

A box is 20 by 20 by 30 centimeters, and a cylinder 30 centimeters tall just fits inside. To the nearest cubic centimeter, how much of the box isn't taken up by the cylinder?

1. The box.

   $$20\cdot20\cdot30=12{,}000$$

2. The cylinder, with radius 10.

   $$\pi(10)^2(30)=3{,}000\pi$$

3. Box minus cylinder. Keep $\pi$ for now.

   $$12{,}000-3{,}000\pi$$

4. Round once, at the end. The full value is about 2,575.22.

   $$\approx2{,}575$$

In Desmos, type 20*20*30-pi*10^2*30 on one line. The result starts 2575.22, and nothing gets rounded along the way.

Picture a cube 1 inch on each edge next to one 2 inches on each edge. Each edge doubled. Each face went from 1 square inch to 4, since it's 2 by 2. The volume went from 1 cubic inch to 8, since 8 of the small cubes fit inside the big one.

Two figures are similar when one is a scaled-up copy of the other, with every length multiplied by the same number, the scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.Read the lesson $k$. Perimeters multiply by $k$ too, areas and surface areas by $k^2$, and volumes by $k^3$. The lines, angles, and triangles lesson works the area part with triangles.

Two prisms are similar, and each edge of the larger one is 3 times the matching edge of the smaller one. The smaller one has a volume of 20 cubic inches. What's the volume of the larger one?

1. The scale factor.

   $$k=3$$

2. Volume grows by $k^3$.

   $$3^3=27$$

3. Multiply the smaller volume by 27.

   $$20\cdot27=540$$

So volumes of 5 and 40 give $40\div5=8$, and the cube root of 8 is 2.

A perimeter is a length, so two perimeters give $k$ directly. Two similar flower beds have perimeters of 18 meters and 45 meters, and the small bed's shortest side is 4 meters.

1. Divide the perimeters.

   $$k=45\div18=2.5$$

2. The big bed's shortest side is 10 meters.

   $$4\times2.5=10$$

3. If the small bed's area is 12, the big bed's area uses $k^2=6.25$.

   $$12\times2.5^2=75$$

Multiplying the side by 6.25 gives 25, which uses an area's power on a length.

A count can scale too. Anything spread over a surface grows with the area. A mosaic that takes 150 tiles needs $150\times2^2=600$ tiles of the same size when every length is doubled.

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A coffee roaster uses similar square trays. The graph relates batch time to the factor $k$ multiplying each side length of a tray whose area is 105 square centimeters.

What is the area, in square centimeters, of the tray used for a 24-minute batch?

1. A

   $105$

2. B

   $420$

3. C

   $1{,}680$

4. D

   $6{,}720$

Show the answer

• A

  $105$

  This reports the original tray’s area, $105$ square centimeters, without enlarging it for the $24$-minute batch.

• B

  $420$

  This comes from $105\times 4=420$, multiplying the area by the length scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.Read the lesson only once. Both side lengths are multiplied by $4$, so the area scales by $4^2$.

• C

  $1{,}680$

  Correct answer

  The graph shows that a $24$-minute batch uses a length scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.Read the lesson of $4$. The tray’s area is therefore $105\times 4^2=1{,}680$ square centimeters.

• D

  $6{,}720$

  This comes from $105\times 4^3=6{,}720$, applying volume scaling to area. A square’s area scales by the square of the length factor.

Solution

Read the point $(4,24)$ on the graph: a batch time of $24$ minutes corresponds to $k=4$.

Each side of the square tray is multiplied by $4$. Since area is length times width, the area is multiplied by $4\times 4=16$.

Multiply the original area by $16$: $105\times 16=1{,}680$ square centimeters.

Report a problem

Before each one, say what the question wants, and check that your answer's unit matches it.