Geometry
Understanding quadrilaterals
Work out the angles in a polygon and use the properties of parallelograms, rhombuses, rectangles and squares.
A quadrilateral is any closed shape with four straight sides. Some of them have extra properties and their own names — parallelogram, rhombus, rectangle, square, kite, trapezium — and the useful thing is that the names stack. Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a quadrilateral. So anything true of the wider family is automatically true of the narrower one.
Two totals worth memorising
The interior angles of a polygon with n sides add up to (n - 2) × 180 degrees, because that is how many triangles it splits into from one corner. The exterior angles of any convex polygon — one with no dent inwards — add up to 360 degrees, no matter how many sides it has — one full turn as you walk right round it.
Have a play
The four corners of this parallelogram hold four angles. Find out how big each one is — nothing here is right or wrong.
Tap any part of the picture.
Worked example
Three angles of a quadrilateral are 110, 85 and 70 degrees. Find the fourth.
Add the three given angles: 110 + 85 + 70 = 265.
Everything in this kind of question comes from one total, so the first move is always to find how much of it is already used.
Try it together
Now let us find the angles of a regular octagon together.
A regular polygon has all its sides and all its angles equal, so once you know the total you only have to share it out.
1.An octagon has 8 sides. How many triangles does it split into from one corner?
Have a go
Have a go on your own: what do the interior angles of a regular nonagon, which has 9 sides, add up to, in degrees? Give just the number.
Hint: Two fewer triangles than it has sides, and each triangle is worth 180 degrees.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.