There are 105 NRICH Mathematical resources connected to Triangles, you may find related items under Angles, polygons, and geometrical proof.
Broad Topics > Angles, polygons, and geometrical proof > Triangles
Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?
Here is a selection of different shapes. Can you work out which ones are triangles, and why?
Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?
Is it possible to find the angles in this rather special isosceles triangle?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?
Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?
This activity focuses on similarities and differences between shapes.
Can you each work out what shape you have part of on your card? What will the rest of it look like?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
What does the overlap of these two shapes look like? Try picturing it in your head and then use some cut-out shapes to test your prediction.
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Can you sort these triangles into three different families and explain how you did it?
This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
Can you work out the fraction of the original triangle that is covered by the inner triangle?
A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?
A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...
How many different triangles can you make on a circular pegboard that has nine pegs?
Can you find all the different triangles on these peg boards, and find their angles?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
How would you move the bands on the pegboard to alter these shapes?
The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?
The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?
Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
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?
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.
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?
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?
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.
Determine the total shaded area of the 'kissing triangles'.
Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.
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
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 .
Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.
Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.
Explore the triangles that can be made with seven sticks of the same length.
Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
