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.

Here is a solid sphere of radius \(5\) cm.

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

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

The diagram shows a solid sphere. It has a radius of \(15\) cm, so its diameter is \(30\) cm.

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

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

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

Volume of hemisphere \(= \frac{2}{3} \times \pi \times 30^{2} = 600\pi\) cm³

Marcus has made two mistakes in this working.

Write down the two mistakes he has made.

The diagram shows a solid sphere. The diameter of the sphere is \(20\) 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 rubber ball is a sphere of radius \(12\) 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 \(= 4\pi r^2\)

The diagram shows a cube with edges of length \(x\) cm, and a sphere of radius \(3\) 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 \(x = \sqrt{ 6\pi }\).

The diagram shows a sphere-shaped exercise ball. The ball has a radius of \(22\) cm.

The ball already contains \(15.6\) litres of air. An electric pump then adds more air to the ball at a constant rate of \(1.5\) litres per minute.

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

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

Does it take less than \(15\) minutes to fill the ball completely? You must show your working.

A bowl is a hemisphere with radius \(10\) 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 \(46.4\) cm³ per second for \(18\) seconds.

Does the water fill more than \(45\%\) of the bowl? You must show your working.

Two solid cones are shown before they are joined. Both cones have base radius \(r\) cm. One has slant height \(3r\) cm and the other has slant height \(4r\) cm.

The two cones are then joined base to base to make one solid.

Curved surface area of a cone \(= \pi r l\), where \(l\) is the slant height.

The total surface area of the solid is \(323.68\pi\) cm².

Calculate the radius \(r\).

A container in the shape of a sphere has radius \(8\) cm. A second container is a cone. The diagram shows the shape of the cone, which is used with its point downwards. The cone has base radius \(8\) cm and perpendicular height \(29\) cm. Both containers are to be filled with water. The diagrams are not drawn to scale.

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

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

Which container holds more? You must show your working.