Which volume formulae are given to you on the Formulae List?

The volumes of a sphere , a cone and a pyramid are all given, so none of them needs memorising. The volumes of a cube, cuboid, cylinder and prism are not given, so those do have to be remembered.

Define a composite solid .

A composite solid is a 3D shape made from more than one standard solid. The parts may be stuck together, like a cylinder with a hemisphere on top, or one may be removed from another, like a pyramid with a spherical hollow.

How do you find the volume of a composite solid?

Find the volume of each standard solid separately first, and only then combine them. Whether you add or subtract depends on whether the parts are joined together or one is hollowed out of the other.

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Volume (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontSphere, Cone & Pyramid
True or False?
The $h$ in the cone volume formula is the slant height of the cone.

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Cards in this collection (13)

True or False?
The $h$ in the cone volume formula is the slant height of the cone.
False.
The $h$ is the perpendicular height, measured straight down from the tip to the centre of the base.
The slant height runs along the outside of the cone and is longer, so using it would give too large a volume.
What must you check before substituting into a sphere or cone formula?
Check whether the question has given the diameter rather than the radius, and halve it if so.
A cone given a base diameter of $20$ cm has $r = 10$ cm, and using $20$ instead would multiply the volume by four.
In the pyramid volume formula $V = \frac{1}{3} A h$ what does $A$ stand for?
$A$ is the area of the base, which may be a square, a rectangle or a triangle.
That is why a single formula covers pyramids of every base shape, with $h$ again the perpendicular height.
Why do the cone and pyramid volume formulae look so similar?
A cone is really a pyramid with a circular base, so $\frac{1}{3} π r^{2} h$ is just $\frac{1}{3} A h$ with $A = π r^{2}$ .
Both therefore need the base area and the perpendicular height, and both take a third.
How do you find the volume of a hemisphere?
Find the volume of the whole sphere with the same radius, then halve it.
So a hemisphere has volume $\frac{1}{2} \times \frac{4}{3} π r^{3}$ , which simplifies to $\frac{2}{3} π r^{3}$ .
A sphere has a diameter of $9$ cm, so complete the substitution and the volume to three significant figures.
$V = \frac{4}{3} \times π \times {(______)}^{3} =______{cm}^{3}$
The completed working is:
$V = \frac{4}{3} \times π \times {(4.5)}^{3} = 382 {cm}^{3}$
The $4.5$ is half the diameter, and cubing it is what makes halving correctly so important.
Which volume formulae are given to you on the Formulae List?
The volumes of a sphere, a cone and a pyramid are all given, so none of them needs memorising.
The volumes of a cube, cuboid, cylinder and prism are not given, so those do have to be remembered.
Define a composite solid.
A composite solid is a 3D shape made from more than one standard solid.
The parts may be stuck together, like a cylinder with a hemisphere on top, or one may be removed from another, like a pyramid with a spherical hollow.
Complete the two volume formulae that are not given on the Formulae List.
$cylinder: V =______$
$prism: V =______$
The completed formulae are:
$cylinder: V = π r^{2} h$
$prism: V = A l$
A cylinder is really a prism with a circular cross-section, so its $π r^{2}$ is just the cross-sectional area $A$ .
How do you find the volume of a composite solid?
Find the volume of each standard solid separately first, and only then combine them.
Whether you add or subtract depends on whether the parts are joined together or one is hollowed out of the other.
True or False?
The volume of a frustum is found by subtracting one cone's volume from another's.
True.
A frustum is a cone or pyramid with its top chopped off, so its volume is the large solid minus the small one that was removed.
For a large cone of $4000 π$ and a removed small cone of $500 π$ the frustum is $3500 π {cm}^{3}$ .
For a frustum, how do you find the height of the large cone?
Add the height of the frustum to the height of the small cone that was cut off the top.
A frustum $15$ cm tall with a $15$ cm cone removed came from a cone $30$ cm tall.
A gatepost is a cuboid with a pyramid on top so how do you get the cuboid's height?
Subtract the pyramid's height from the total height, since the total covers both parts together.
A gatepost $2.4$ m tall with a $0.5$ m pyramid on it has a cuboid $1.9$ m tall.

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