This is part of our Secondary Curriculum collection of favourite rich tasks arranged by topic.
Scroll down to see the complete collection, or explore our subcollections on Perimeter and Area in two dimensions, and Surface Area and Volume in three dimensions.
problem
Favourite
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?
problem
Favourite
Garden shed
Age
11 to 14
Challenge level
Can you minimise the amount of wood needed to build the roof of my garden shed?
problem
Favourite
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?
problem
Favourite
Three cubes
Age
14 to 16
Challenge level
Can you work out the dimensions of the three cubes?
problem
Favourite
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.
problem
Favourite
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?
problem
Favourite
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?
problem
Favourite
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?
problem
Favourite
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.
problem
Favourite
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?
problem
Favourite
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?
problem
Favourite
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?
problem
Favourite
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?
problem
Favourite
Three by one
Age
16 to 18
Challenge level
There are many different methods to solve this geometrical problem - how many can you find?
problem
Favourite
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.
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You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.