Tilted Squares
Age 11 to 14
Challenge Level
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Garden Shed
Age 11 to 14
Challenge Level
Can you minimise the amount of wood needed to build the roof of my garden shed?
Where Is the Dot?
Age 14 to 16
Challenge Level
A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?
Compare Areas
Age 14 to 16
Challenge Level
Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
The Spider and the Fly
Age 14 to 16
Challenge Level
A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?
Semi-detached
Age 14 to 16
Challenge Level
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
Where to Land
Age 14 to 16
Challenge Level
Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?
Inscribed in a Circle
Age 14 to 16
Challenge Level
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
Hexy-metry
Age 14 to 16
Challenge Level
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Ladder and Cube
Age 14 to 16
Challenge Level
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?
Bendy Quad
Age 14 to 16
Challenge Level
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.
Far Horizon
Age 14 to 16
Challenge Level
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Three by One
Age 16 to 18
Challenge Level
There are many different methods to solve this geometrical problem - how many can you find?
Cubestick
Age 16 to 18
Challenge Level
Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
Pythagoras' Theorem and Trigonometry - Short Problems
Age 11 to 16
A collection of short problems on Pythagoras's Theorem and Trigonometry.
You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.
NRICH is part of the family of activities in the Millennium Mathematics Project.