Volume & Surface Area Flashcards

All 22 cards in this deck

What is the formula for the volume of a cuboid?

$V = \text{length} \times \text{width} \times \text{height}$

e.g. $8 \times 5 \times 4 = 160$ cm$^3$

What are the steps to find a missing dimension of a cuboid from its volume?

e.g. volume $200$ cm$^3$, length $10$ cm, width $4$ cm — find the height

1. Multiply the two known dimensions: $10 \times 4 = 40$
2. Divide the volume by this: $200 \div 40 = 5$
3. Height $= 5$ cm

What is the formula for the volume of any right prism?

e.g. a triangular prism of cross-sectional area $6$ cm$^2$ and length $10$ cm

$V = \text{area of cross section} \times \text{length}$

$6 \times 10 = 60$ cm$^3$

What is the formula for the volume of a cylinder of radius $r$ and height $h$?

$V = \pi r^2 h$

e.g. $r = 3$, $h = 5$ gives $\pi \times 3^2 \times 5 = 45\pi$ cm$^3$

What is the formula for the volume of a sphere of radius $r$?

$V = \frac{4}{3}\pi r^3$

What is the formula for the volume of a pyramid or cone?

$V = \frac{1}{3} \times \text{base area} \times \text{perpendicular height}$

For a cone this gives $V = \frac{1}{3}\pi r^2 h$

How many cm$^3$ and how many ml are there in $1$ litre?

$1$ litre $= 1000$ cm$^3 = 1000$ ml (so $1$ ml $= 1$ cm$^3$)

True or false? $1$ m$^3 = 100$ cm$^3$.

False. $1$ m$^3 = 100 \times 100 \times 100 = 1\,000\,000$ cm$^3$.

What are the steps to find the volume of a composite solid?

e.g. a cuboid $5$ cm $\times$ $4$ cm $\times$ $2$ cm with a cube of side $2$ cm on top

1. Split the solid into simple solids: cuboid + cube
2. Find each volume: $5 \times 4 \times 2 = 40$ and $2^3 = 8$
3. Add them: $40 + 8 = 48$ cm$^3$

What are the steps to find how many cups can be completely filled from a partly full container?

e.g. a container of volume $3420$ cm$^3$ that is $\frac{2}{3}$ full, cups of $275$ ml

1. Volume of water: $3420 \times \frac{2}{3} = 2280$ cm$^3$ ($= 2280$ ml)
2. Divide by the cup capacity: $2280 \div 275 = 8.29...$
3. Round down: $8$ cups

When a volume calculation for how many whole boxes fit into a container gives a decimal answer, what do you do with it?

Round down to the whole number below — a part box does not count.

True or false? Surface area is measured in cubic units such as cm$^3$.

False. Surface area is the total area of all the faces, so it is measured in square units such as cm$^2$.

What is the total surface area of a cube with edge length $x$?

$6x^2$

e.g. edge $5$ cm: $6 \times 5^2 = 150$ cm$^2$

What are the steps to find the total surface area of a cuboid?

e.g. a cuboid $6$ cm by $5$ cm by $3$ cm

1. Area of three different faces: $6 \times 5 = 30$, $6 \times 3 = 18$, $5 \times 3 = 15$
2. Add them: $30 + 18 + 15 = 63$
3. Double (faces come in pairs): $63 \times 2 = 126$ cm$^2$

What is the formula for the total surface area of a solid cylinder of radius $r$ and height $h$?

$2\pi r h + 2\pi r^2$

Curved surface plus the two circular ends.

What is the formula for the surface area of a sphere of radius $r$?

$4\pi r^2$

What are the steps to find the volume of a cube from its total surface area?

e.g. total surface area $150$ cm$^2$

1. Area of one face: $150 \div 6 = 25$
2. Side length: $\sqrt{25} = 5$
3. Volume: $5 \times 5 \times 5 = 125$ cm$^3$

True or false? Two cuboids with the same volume can have different surface areas.

True.
e.g. a $1$ cm by $1$ cm by $8$ cm cuboid and a $2$ cm cube both have volume $8$ cm$^3$ but their surface areas are $34$ cm$^2$ and $24$ cm$^2$.

What are the steps to find the volume of a cone when the formula is given in the question?
e.g. $V = \frac{1}{3}\pi r^2 h$ with $r = 3$ cm, $h = 10$ cm

1. Square the radius: $3^2 = 9$
2. Multiply by $\pi$ and by the height: $9 \times \pi \times 10 = 90\pi$
3. Multiply by $\frac{1}{3}$: $30\pi = 94.2$ cm$^3$ (3 s.f.)

A question gives the volume of a pyramid as $V = \frac{1}{3} \times \text{base area} \times \text{height}$. Which height must you substitute?
e.g. a square-based pyramid

The perpendicular (vertical) height from the base up to the apex.
Not the slant height along a sloping edge or face.

What are the steps to decide whether a claim about a volume is correct, showing your working?
e.g. Vera says 6 cubes of side $2$ cm have a total volume of $48$ cm$^3$

1. Work out the volume of one solid: $2 \times 2 \times 2 = 8$ cm$^3$
2. Scale up or combine as the claim describes: $8 \times 6 = 48$ cm$^3$
3. Compare with the claim and write a conclusion: $48 = 48$, so Vera is correct

True or false? Doubling the edge length of a cube doubles its volume.

False.
All three dimensions double, so the volume becomes $2 \times 2 \times 2 = 8$ times larger.