Tip:
Highlight text to annotate it
X
Good day students in this clip we are going to be going over two examples, one involves
the calculation of the length of an arc and then the second involves the area of a sector
before we get started with the computations lets go over the formulas real quick so lets
write this down formulas so the first formula is for the arc length arc length so the formula
for the arc length is s , s is the length of the arc equals theta times the radius ok
and then the formula for the area of a sector that is given by A equals theta over two r
squared alright so those are your two formulas alright so geometrically what do these formulas
mean? so lets go ahead and connect these formulas with the this diagram that I have right here
alright so I have a circle and then I have a section cut out if you think of it it could
be like a slice of pizza. or something like that lol!!! now the length of the arc is basically
the distance. It is a one dimensional measure it is the distance from here all the way to
there. So its like the portion of a whole circle so how do you compute that distance?
Well you just need to know what the angle is and what the radius of the circle is and
you can put it into this formula s equals r theta. and that will be the arc length.
Ok so this distance right here this is your arc length arc length. The arc length and
the formula is s equals theta r. alright now what is the sector or the area of the sector?
The area of the sector is a two dimensional measure. it represents the sector that enclosed
by these two radii right here. So the area of this region this portion of the entire
circle is a sector so the region that these two radii enclose is known as your sector
and the area of the sector is given by a equals theta over two times the radius squared ok
now these two formulas you have seen them before. You might not have realized it but
you actually have if you think about the formula for the circumference of a circle c is equal
to 2 pi r. this is the special case of the arc length formula. Do you see what the connection
is here? Well lets say we have a circle with radius ok and then the angle of rotation is
a full circle so lets say we have a circle with radius r and your arc length is the length
of the circumference of the entire circle so if you go full circle like this what will
your theta be? theta is going to be 2 pi or 360 degrees but we are working with radians
here so we
are going to use 2 pi . lets say the radius r is
r
now if i plug
these two values into
the arc length formula then what will
the resulting result be