An empty cylindrical container has radius \(8\) cm and height \(71\) cm. \(6\) identical balls are stacked directly on top of one another inside it. Each ball is a sphere of radius \(8\) cm.

Will all \(6\) balls fit inside the container? Show working to support your answer.

A cup is in the shape of a cone. It is full of cola. The diagram shows the shape of the cup, which is used with its point downwards.

The full depth of the cup is \(15\) cm. Ben drinks some of the cola, until the depth of cola left in the cup is \(10\) cm.

Ben says, "The fraction of the cola still in the cup, by volume, is \(\frac{ 4 }{ 9 }\)."

Is the fraction that Ben gives correct? Tick a box.

Yes / No

Give a reason for your answer.

The diagram shows a solid sphere of radius \(10\) cm.

Surface area of a sphere \(= 4\pi r^2\)

Work out the surface area of the sphere. Give your answer in terms of \(\pi\).

Here is a solid sphere. It has a radius of \(12\) cm, so its diameter is \(24\) cm.

The sphere is cut exactly in half, making a solid hemisphere.

Volume of a sphere \(= \frac{4}{3}\pi r^3\)

Amara works out the volume of the hemisphere, in terms of \(\pi\). Here is her working:

Volume of hemisphere \(= \frac{4}{3} \times \pi \times 12^{2} = 192\pi\) cm³

Amara has made two mistakes in this working.

Write down the two mistakes she has made.

The diagram shows a solid sphere. The diameter of the sphere is \(18\) cm.

Volume of a sphere \(= \frac{4}{3}\pi r^3\)

Work out the volume of the sphere. Give your answer correct to 1 decimal place.

A solid wooden ball is a sphere of radius \(6\) cm.

The ball is cut exactly in half through its centre, making two identical solid hemispheres.

Volume of a sphere \(= \frac{4}{3}\pi r^3\)

Surface area of a sphere \(= 4\pi r^2\)

Here is a solid cone. The radius of its base is \(8\) cm and its slant height is \(17\) cm.

Volume of a cone \(= \frac{1}{3}\pi r^2 h\)

The diagram shows a cube with edges of length \(x\) cm, and a sphere of radius \(9\) cm.

Surface area of a sphere \(= 4\pi r^2\)

The surface area of the cube is equal to the surface area of the sphere.

Show that \(x = \sqrt{ 54\pi }\).

The diagram shows a sphere-shaped exercise ball. The ball has a radius of \(23\) cm.

The ball already contains \(20.9\) litres of air. An electric pump then adds more air to the ball at a constant rate of \(2\) litres per minute.

Volume of a sphere \(= \frac{4}{3}\pi r^3\)

\(1\) litre \(= 1000\ \text{cm}^3\)

Does it take less than \(11\) minutes to fill the ball completely? You must show your working.

A bowl is a hemisphere with radius \(9\) cm. The diagram shows the full sphere it is exactly half of, cut through its centre.

Volume of a sphere \(= \frac{4}{3}\pi r^3\)

Water is poured into the bowl at a rate of \(39.2\) cm³ per second for \(23\) seconds.

Does the water fill less than \(55\%\) of the bowl? You must show your working.