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There are **96** NRICH Mathematical resources connected to **Pythagoras' theorem**, you may find related items under Pythagoras and trigonometry.

Problem
Primary curriculum
Secondary curriculum
### Garden Shed

Can you minimise the amount of wood needed to build the roof of my garden shed?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Kite in a Square

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Generating Triples

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras Proofs

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

Age 11 to 16

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Problem
Primary curriculum
Secondary curriculum
### Partly Circles

What is the same and what is different about these circle questions? What connections can you make?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Nicely Similar

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 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?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Cubestick

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Where to Land

Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### 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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tilted Squares

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### 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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Belt

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras for a Tetrahedron

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hex

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Zig Zag

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Napkin

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Compare Areas

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Orthogonal Circle

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

Age 16 to 18

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Problem
Primary curriculum
Secondary curriculum
### 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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Logosquares

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Baby Circle

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Chordingly

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Floored

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round and Round

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Trice

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Isosceles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circle Packing

Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rectangular Pyramids

Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ball Packing

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

Age 14 to 16

Challenge Level

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Primary curriculum
Secondary curriculum
### Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

Age 11 to 14

Challenge Level

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Primary curriculum
Secondary curriculum
### Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Equilateral Areas

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Retracircles

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### 30-60-90 Polypuzzle

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square World

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

Age 16 to 18

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Problem
Primary curriculum
Secondary curriculum
### Get Cross

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Strange Rectangle

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Reach for Polydron

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Age 16 to 18

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Problem
Primary curriculum
Secondary curriculum
### Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Medallions

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Age 14 to 16

Challenge Level