How to find the area of a parallelogram - ISEE Upper Level Mathematics Achievement
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Give the area of the above parallelogram if 
.
Give the area of the above parallelogram if
Multiply height 
by base 
to get the area.
By the 30-60-90 Theorem:
and
The area is therefore
Multiply height
By the 30-60-90 Theorem:
and
The area is therefore
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Give the area of the above parallelogram if 
.
Give the area of the above parallelogram if
Multiply height 
by base 
to get the area.
By the 45-45-90 Theorem,
.
The area is therefore
Multiply height
By the 45-45-90 Theorem,
The area is therefore
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Three of the vertices of a parallelogram on the coordinate plane are 
. What is the area of the parallelogram?
Three of the vertices of a parallelogram on the coordinate plane are
As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:
Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.
The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:
The area of the 12 by 12 square is 
The area of the green triangle is 
.
The area of the blue triangle is 
.
The area of the pink triangle is 
.
The area of the main triangle is therefore
The parallelogram has area twice this, or 
.
As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:
Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.
The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:
The area of the 12 by 12 square is
The area of the green triangle is
The area of the blue triangle is
The area of the pink triangle is
The area of the main triangle is therefore
The parallelogram has area twice this, or
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One of the sides of a square on the coordinate plane has an endpoint at the point with coordinates 
; it has the origin as its other endpoint. What is the area of this square?
One of the sides of a square on the coordinate plane has an endpoint at the point with coordinates
The length of a segment with endpoints 
and 
can be found using the distance formula with 
, 
, 
:
This is the length of one side of the square, so the area is the square of this, or 41.
The length of a segment with endpoints
This is the length of one side of the square, so the area is the square of this, or 41.
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