Volume & Surface Area Flashcards

All 19 cards in this deck

What is the formula for the volume of a prism?

e.g. a prism with cross-sectional area $12\text{ cm}^2$ and length $5$ cm

Volume = area of cross-section $\times$ length

$12 \times 5 = 60\text{ cm}^3$

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

$V = \pi r^2 h$

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 cone of base radius $r$ and height $h$?

$V = \frac{1}{3}\pi r^2 h$, where $h$ is the perpendicular (vertical) height, not the slant height.

What is the formula for the volume of a pyramid?

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

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

$1000\text{ cm}^3$, since $1\text{ cm}^3 = 1\text{ ml}$.

What are the steps to find a missing height when the volume of a prism is given?

e.g. volume $60\text{ cm}^3$, length $5$ cm, triangular cross-section of base $4$ cm

1. Volume $\div$ length = cross-section area: $60 \div 5 = 12\text{ cm}^2$
2. Put this into the area formula: $\frac{1}{2}\times 4 \times h = 12$
3. Solve for $h$: $h = 6$ cm

What are the steps to find how long a container takes to fill at a given rate?

e.g. a tank of volume $6000\text{ cm}^3$ filled at $250\text{ cm}^3$ per second

1. Work out the volume of the container: $6000\text{ cm}^3$
2. Check the volume and the rate use the same units.
3. Time = volume $\div$ rate: $6000 \div 250 = 24$ seconds

What is the formula linking mass, density and volume of a solid?

mass = density $\times$ volume, so density = mass $\div$ volume.

True or false? When a solid is melted down and recast into a different shape, its volume stays the same.

True. So you can equate the two volumes to find a missing length.

How do you find the total surface area of a prism?

e.g. a cuboid $7$ cm by $6$ cm by $5$ cm

Add the areas of all its faces.

$2(7\times6 + 7\times5 + 6\times5) = 214\text{ cm}^2$

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

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

The curved surface area alone is $2\pi r h$.

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

$4\pi r^2$

What is the formula for the total surface area of a solid cone?

$\pi r l + \pi r^2$, where $l$ is the slant height.

True or false? The total surface area of a solid hemisphere of radius $r$ is $2\pi r^2$.

False. $2\pi r^2$ is only the curved part; the total is $2\pi r^2 + \pi r^2 = 3\pi r^2$.

What are the steps to find the surface area of a composite solid?

e.g. a cylinder of radius $r$, height $h$, with a hemisphere of radius $r$ on top

1. List the exposed surfaces of each part: curved cylinder, base circle, curved hemisphere.
2. Leave out any joined or hidden faces: the top circle of the cylinder.
3. Add the areas: $2\pi r h + \pi r^2 + 2\pi r^2$

True or false? $1\text{ m}^3 = 100\text{ cm}^3$.

False. $1\text{ m}^3 = 1\,000\,000\text{ cm}^3$, since you multiply by $100$ for each of the three dimensions.

What are the steps to find the volume of a frustum of a cone?
e.g. a cone of radius $6$ cm and height $10$ cm with the top cone of radius $3$ cm and height $5$ cm removed

1. Volume of the full cone: $\frac13\pi\times6^2\times10=120\pi$
2. Volume of the small cone removed: $\frac13\pi\times3^2\times5=15\pi$
3. Subtract: $120\pi-15\pi=105\pi\text{ cm}^3$

True or false? The volume of a composite solid is always found by adding the volumes of the standard solids it can be split into.

False. If part of a solid has been removed (e.g. a drilled hole or the top of a cone), that volume is subtracted instead.