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If a ball is rolled into the corner of a room how far is its centre from the corner?

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A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Conical Bottle

Age 14 to 16
Challenge Level

A right circular cone is filled with liquid to a depth of half its vertical height.

The cone is inverted. How high up the vertical height of the cone will the liquid rise?