Triangles

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?

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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'.