Pythagoras' theorem

There are 108 NRICH Mathematical resources connected to Pythagoras' theorem
Pythagoras Proofs
problem
Favourite

Pythagoras proofs

Age
11 to 16
Challenge level
filled star filled star empty star

Can you make sense of these three proofs of Pythagoras' Theorem?

Garden Shed
problem
Favourite

Garden shed

Age
11 to 14
Challenge level
filled star filled star empty star
Can you minimise the amount of wood needed to build the roof of my garden shed?
Kite in a Square
problem
Favourite

Kite in a square

Age
14 to 18
Challenge level
filled star filled star empty star
Can you make sense of the three methods to work out what fraction of the total area is shaded?
Generating Triples
problem
Favourite

Generating triples

Age
14 to 16
Challenge level
filled star empty star empty star
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Belt
problem
Favourite

Belt

Age
16 to 18
Challenge level
filled star empty star empty star
A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
Inscribed in a Circle
problem
Favourite

Inscribed in a circle

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Where is the dot?
problem
Favourite

Where is the dot?

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Baby Circle
problem
Favourite

Baby circle

Age
16 to 18
Challenge level
filled star empty star empty star
A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
LOGOSquares
problem
Favourite

LOGOsquares

Age
16 to 18
Challenge level
filled star empty star empty star

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Nicely Similar
problem
Favourite

Nicely similar

Age
14 to 16
Challenge level
filled star filled star empty star

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?