Formula for Area of a Triangle - Why?

let's talk about the formula for finding

the area of a triangle area equals

one-half base times height we could just

plug the numbers into the formula and

get an answer but what does it mean

where does that 1/2 come from

we'll get to that but first let's get

rid of that half wait a minute that

looks like the formula for finding the

area of a rectangle area equals length

times width and that tells us how many

square units are inside the rectangle

rectangles are a cinch because of all

those right angles interestingly area

for parallelograms that don't have right

angles like these are solved in a

similar fashion we just focus on base

and height instead of length and width

but watch this area equals base times

height if you cut off this piece and

move it over here you can see we have a

rectangle with the length of 6 and a

height of 4 the areas of both the

rectangle and this parallelogram are the

same okay moving on why am I talking

about different types of parallelograms

this is supposed to be about triangles

all right here we go

area of a triangle equals one-half base

times height take this right triangle

here with a base of 10 inches and a

height of 8 inches we want to know the

area or how many square inches the space

inside can hold so I'm going to take

another triangle same size and shape or

congruent to this one just in a

different color to help you see now if I

line these up with their hypotenuse is

completely touching you see I now have a

rectangle with a length of 10 inches and

a width of 8 inches so you see it takes

two of the triangles here to make this

rectangle therefore the area of this

right triangle is going to be 1/2 the

area of this rectangle there it is the

1/2 from the formula of area of a

triangle area equals 1/2 base times

height as further proof if half across

this diagonal is difficult for you to

see look at it like this if I decompose

this triangle like this you'll see that

the triangle can fit into half of the

rectangle in this way to 1/2 of the base

times height not all triangles have

right angles though don't worry we'll

talk about those too let's copy and

rotate this

triangle just like we did before but

this time instead of getting a rectangle

we get a parallelogram you know that the

area of a parallelogram is base times

height and half of a parallelogram is a

triangle so no matter what angles are in

the triangle the area must be one-half

base times height