hey it's Kalei in this lesson we will learn to find the area of polygons
remember that polygons can be regular meaning the side lengths are equal
lateral or they can be irregular where the side lengths can be different sizes
the way we find the area for both of these is different when we have an
irregular polygon like this Pentagon we don't have a go to formula to find the
area so we have to decompose it into different shapes so we do have formulas
decomposing a figure into different shapes means we take our big shape and
break it into smaller shapes that fit inside when you look at this Pentagon do
you see other shapes inside of it we can draw a line right here and turn this
Pentagon into a triangle and a trapezoid now the area of this Pentagon is equal
to the area of the triangle plus the area of the trapezoid and we know the
formula for triangles is base times height divided by 2 and the area for
trapezoids is base one plus base two times the height over two so now I have
my values for my Pentagon and I can see how those values would fit into the
formulas for my triangle and my trapezoid let's do the triangle first
the base of my triangle is gonna be the red line here and it has a value of 5
and the height of this triangle is that right angle line to the opposite vertex
2 so I can find the area for my triangle is gonna be 5 times 2 divided by 2 which
is gonna give me 10 over 2 which is gonna give me 5 so that's the area for
my triangle now let's find the area for my trapezoid so my trapezoid has
a base of 5 which I know appear and another one of three down here so I'm
gonna have base 1 5 plus base 2 3 is multiplied by what's the height of my
trapezoid remember it's a right angle here going to the opposite vertex 4
divided by 2 and this is gonna give me 5 plus 3 gives me 8 8 times 4 is 32 over 2
that's gonna equal to 16 so now if I take the area of my triangle and I add
it to the area of my trapezoid I'll have the area for this entire
Pentagon so the area from my Pentagon is equal to 5 plus 16 and that is equal to
21 great job now what about finding the
area of a regular polygon like this heptagon the first thing we do is
decompose our shape into congruent triangles that means I can't use
diagonals to make triangles since those triangles are different sizes and shapes
instead I start at each vertex and draw a line to the center point of my shape
so now I have 7 triangles and I do know how to find the area of triangles area
equals 1/2 the base times the height but these are not right triangles so what's
the height of these triangles now we need to find the distance from the base
to the opposite vertex with a right angle we call this distance the apothem
the distance from the center of a regular polygon to the side at a right
angle represents the height of the triangles we also call the height of the
triangles the apothem so now that I have the height of my triangles I know the
area of my heptagon is going to be seven times the area of
one triangle but you notice something about the base of the triangle B it is
the same as the side length of our heptagon and the side length times the
number of sides seven is the perimeter of the heptagon so I can rewrite the
formula for my heptagon as area equals one-half times perimeter times height
but since we aren't relying on our triangle anymore we can call the height
the apothem so now that I know this I can rewrite my formula for the area of a
heptagon for any polygon the area of a polygon is one-half times the perimeter
times the height or the apothem so let's find the area of this regular polygon
together here I have my formula area equals one-half times the perimeter
times the apothem now I already know my apothem here it's that distance at a
right angle to the center so this is gonna be seven and what is my perimeter
I have to count how many sides I have one two three four five six two the
number of sides is six times the length of one side is eight so that's my
perimeter and then one half of that all right so my perimeter six times eight
that gives me 48 and I can divide 48 by two that's gonna give me 24 I starts
multiply that by seven which if you don't know off the top of your head you
can use a calculator for 24 times seven is 168 so that is the area of this
regular hexagon great job learning the techniques for finding the area of
regular and irregular polygons practice what you've learned by playing the
online game and quizzes have fun and remember to
always be clever