Angles in polygons

Explore the geometry of these dart and kite shapes!

Make different quadrilaterals on a nine-point pegboard, and work out their angles. What do you notice?

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

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?

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

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

What do you notice when you put three triangles together?

Can you match the Venn diagram with the Carroll diagram that shows the same information?

How can the school caretaker be sure that the tree would miss the school buildings if it fell?