# Shapes inside circles

Here’s a circle:

Sponsored Links

Now there are some other *terms* you need to know when
you’re dealing with circles.

### Chords and segments

A *chord* is a straight line that joins any two points
on the boundary of a circle. When you draw a chord, you divide the circle into
two areas. These areas are called *segments*. The smaller area is called
a *minor segment*. The larger area is called a *major* *segment*.

### Arcs

We already know that sectors are like wedges of a pie or
cake. Well, an *arc* is the part of the circle’s circumference taken up
by the sector:

We know that the circumference of a circle is just two pi multiplied by the radius of the circle:

_{}

The length of an arc is always going to be some *fraction*
of the circumference. Say we had an arc with an angle of 180°. This would
mean the arc would stretch halfway around the circle – so the arc’s length
would be *half* the circumference. The arc’s angle, 180°, is also half
the total angle you need to rotate through to get a circle – 360°. This is no
co-incidence – the fraction that the arc’s angle is of 360° is the same as the
fraction that the arc’s length is of the circumference of the circle:

_{}

### Triangles inside circles

Sometimes you’ll get questions which have triangles,
sometimes more than one, drawn inside a circle. You’ll usually get supplied
with *some* information about the angles and side lengths. Also you might
get information telling you that a corner of a triangle is located at the
centre of the circle for instance.

Usually these questions will ask you to find another angle
in the diagram, or the length of a side. Alternatively they might ask you to
name whether the triangle is an *acute* triangle or an *obtuse*
triangle, or perhaps to count how many *isosceles* triangles there are in
the diagram. Here’s an example diagram:

There are a few things that will help you solve a question
like this. First up, remember that the sum of the three interior angles of a
triangle is 180°. Also, *any* straight line from the centre of the circle
to the circle’s circumference is a radius of the circle. Make sure what you
think is the centre of the circle is clearly said in the question. Some
questions might try and trick you by drawing lines from what seems to be the
centre of the circle, but not *specifically* say anywhere that it is the
centre. What you’re looking for is a statement like in the diagram like, “A is
the circle centre.”

In this diagram, lines AD, AC and AB are all the same
identical length, since they are all circle radii (‘radii’ is the plural form
of ‘radius’). This helps you if you are asked how many isosceles triangles
there are in the circle – an isosceles triangle has two sides exactly the same
length. Triangle ACD has two sides the same length – AD and AC. Triangle ABC
has two sides the same length as well – AC and AB. So there are *two*
isosceles triangles in the circle.

### Triangles with a circle diameter as one side

A triangle drawn inside a circle with one side as the
diameter of the circle has a special *property*. The angle in the
triangle opposite the diameter side is always 90°:

There’s a pretty easy way to prove this is always true. All you need to do is split a triangle in two by drawing another radius from the origin to the corner opposite the diameter side. Then you just label all the angles, and write some equations about what the sum of angles inside the triangles must be:

First take a triangle inside the circle… |
…and add a line from the circle centre to the corner opposite the diameter side |

Now, we know that the interior angles inside any triangle add up to 180°. So we can straight away write two equations, one for each triangle:

_{}

We are trying to prove that the angle of the original triangle opposite the diameter side is 90°. In our new diagram, this angle is made up of ‘b’ and ‘c’. So we’re trying to prove that b + c = 90°.

At the moment, we have two equations, but we have *six*
unknowns – a, b, c, d, e, and f. We need more information – as in more
equations.

There is a straight angle at the centre of the circle – ‘e’ and ‘f’ together make up a straight angle, or 180°. So that’s another equation we can write:

_{}

Now, what types of triangles are these? Well, both have two sides the same length, so they are isosceles triangles. Another property of isosceles triangles is that they have two angles exactly the same size. The identical angles are opposite the corner where the two identical length sides meet. So in the left triangle, angles ‘a’ and ‘b’ are the same. In the right triangle, angles ‘c’ and ‘d’ are the same. So that’s two more equations:

_{}

We now have a total of *five* equations, but we’ve
still got *six* unknowns. So we don’t have enough information to work out
the value of all the unknowns. However, because we’ve only got *one less*
equation than unknown (5 versus 6), we should be able to work out some simple
relationships, like ‘a + c = …’ or ‘f – b = …’ What we’re interested
in is what ‘b + c’ equals.

If we start with these two equations:

_{}

We can add them together to get:

_{}

Now, we know that a = b and c = d. All we’re interested in is ‘b’ and ‘c’, we want to get rid of all the other variables. So what we can try and do is instead of writing ‘a’s in the equation, we can write ‘b’s, and instead of ‘d’s in the equation we can write ‘c’s:

_{}

We’re interested in ‘b’ + ‘c’, so we can rearrange the equation so that only terms with ‘b’ or ‘c’ in them are on the left hand side.:

_{}

Okay, so now all we’ve got to do is get rid of the ‘f’ and
the ‘e’. There’s another equation we haven’t used yet – the _{} one. If we
rewrite our current equation a little:

_{}

Now instead of ‘e + f’, we can write 180°:

_{}

The last step was just to divide both sides by 2. Now
we’ve got an equation saying that b and c together make up 90°. This is our
proof! Notice that this proof applies for *any* triangle inside a circle
where *one side is the diameter*, and the *corner opposite the diameter
touches the circumference* of the circle.