### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Cones and Spheres

##### Age 14 to 16 ShortChallenge Level

We need to find the volume of the cone and the volume of metal needed for each sphere.

The volume of a cone is given by $\frac{1}{3}\pi r^2h$, where $r$ is the radius and $h$ is the height. So the volume of the cone, in cubic centimetres, is $\frac{1}{3}\pi \times 6^2\times20=240\pi$.

The volume of a sphere is given by $\frac{4}{3}\pi r^3$. The radius of each of the spheres is 3 cm, so the volume of each of the spheres, in cubic centimetres, is $\frac{4}{3}\pi\times3^3=4\times 9\pi=36\pi$.

$240\div36=6.\dot6=6\frac{2}{3}$, so 6 spheres can be made.
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