![Tilted Squares](/sites/default/files/styles/medium/public/thumbnails/content-id-2293-icon.png?itok=q-hpU9S_)
problem
Tilted Squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
![Garden Shed](/sites/default/files/styles/medium/public/thumbnails/content-id-11190-icon.png?itok=1_kaF7QW)
![Where is the dot?](/sites/default/files/styles/medium/public/thumbnails/content-id-5615-icon.jpg?itok=tWV7bskh)
problem
Where is the dot?
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?
![Three cubes](/sites/default/files/styles/medium/public/thumbnails/content-01-04-15plus4-icon.gif?itok=bcqrGNhy)
![Semi-detached](/sites/default/files/styles/medium/public/thumbnails/content-03-09-six12-icon.gif?itok=yxHs7zkf)
problem
Semi-detached
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.
![Inscribed in a Circle](/sites/default/files/styles/medium/public/thumbnails/content-04-02-six1-icon.png?itok=SKI9K0iZ)
problem
Inscribed in a Circle
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?
![The Spider and the Fly](/sites/default/files/styles/medium/public/thumbnails/content-id-2365-icon.png?itok=Yh6Zz6SK)
problem
The Spider and the Fly
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?
![Where to Land](/sites/default/files/styles/medium/public/thumbnails/content-id-2658-icon.png?itok=F_qramHi)
problem
Where to Land
Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?
![Ladder and Cube](/sites/default/files/styles/medium/public/thumbnails/content-99-11-15plus2-icon.jpg?itok=HbFFdTEK)
problem
Ladder and Cube
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](/sites/default/files/styles/medium/public/thumbnails/content-00-05-15plus1-icon.gif?itok=Ss3g1b4V)
problem
Bendy Quad
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.
![Compare Areas](/sites/default/files/styles/medium/public/thumbnails/content-98-01-six3-icon.png?itok=CMzZ1xZ2)
problem
Compare Areas
Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
![Hexy-Metry](/sites/default/files/styles/medium/public/thumbnails/content-98-04-15plus2-icon.jpg?itok=5bs1o7nw)
problem
Hexy-Metry
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
![Far Horizon](/sites/default/files/styles/medium/public/thumbnails/content-id-2357-icon.png?itok=j6L59i9x)
problem
Far Horizon
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](/sites/default/files/styles/medium/public/thumbnails/content-98-04-15plus3-icon.jpg?itok=ZyshD-ER)
problem
Three by One
There are many different methods to solve this geometrical problem - how many can you find?
![Cubestick](/sites/default/files/styles/medium/public/thumbnails/content-id-2690-icon.png?itok=NideabI_)
problem
Cubestick
Stick some cubes together to make a cuboid. Find two of the angles
by as many different methods as you can devise.
Image
![Pythagoras' Theorem & Trigonometry STEM footer Pythagoras' Theorem & Trigonometry STEM footer](/sites/default/files/styles/large/public/thumbnails/content-id-13477-STEMlogo.png?itok=O48bf6Vs)
You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.