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There are **127** NRICH Mathematical resources connected to **2D shapes and their properties**, you may find related items under Angles, polygons, and geometrical proof.

Problem
Primary curriculum
Secondary curriculum
### Seeing Parallelograms

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rhombus It

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Parallelogram It

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Name That Triangle!

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle or No Triangle?

Here is a selection of different shapes. Can you work out which ones are triangles, and why?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seeing Squares

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Always, Sometimes or Never? KS1

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's Happening?

Shapes are added to other shapes. Can you see what is happening? What is the rule?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Olympic Rings

This problem is intended to get children to look really hard at something they will see many times in the next few months.

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Poly Plug Rectangles

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Opposite Vertices

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Track Design

Where should runners start the 200m race so that they have all run the same distance by the finish?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shaping It

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sorting Logic Blocks

This activity focuses on similarities and differences between shapes.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shapely Lines

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Shape?

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Jig Shapes

Can you each work out what shape you have part of on your card? What will the rest of it look like?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Curvy Areas

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Overlapping Again

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 2 Rings

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Torn Shapes

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Board Block Challenge

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Efficient Cutting

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square It

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Salinon

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rolling Around

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Age 11 to 14

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
### Where Are They?

Use the isometric grid paper to find the different polygons.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Approximating Pi

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Blue and White

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Age 11 to 14

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

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadarc

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Age 14 to 16

Challenge Level

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

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
### Chain of Changes

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Age 5 to 11

Challenge Level

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
### 2D-3D

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lawnmower

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Area I'n It

Triangle ABC has altitudes h1, h2 and h3. The radius of the inscribed circle is r, while the radii of the escribed circles are r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1 + 1/r2 + 1/r3 .

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Age 16 to 18

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

Problem
Primary curriculum
Secondary curriculum
### Two by One

An activity making various patterns with 2 x 1 rectangular tiles.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rectangles with Dominoes

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

Age 5 to 7

Challenge Level