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This essential trains for: AMC-8, AMC-10, AMC-12, AIME, Math Kangaroo 7-8, Math Kangaroo 9-10.
First, a word of warning relating to areas: A vast number of errors on SAT exams stem from the fact that students often compute the area for a problem that asks for the perimeter! This common error is exploited by the examiner who invariably provides the correct area as an answer choice. The wise student makes it a habit to double-check that he/she is indeed being asked to calculate an area before applying the area formulas discussed in this section to any problem.
Let's start with a review of perimeter, to emphasize the difference between perimeter and area.
The perimeter of a figure is the sum of the lengths of its sides (the lengths must be expressed in the same unit).
The semi-perimeter is simply half the perimeter.
The area of a triangle can be computed in several ways.
Method using a side and the corresponding altitude: The area of a triangle can be computed using the length of a side and the length of the altitude that is perpendicular onto that side.
The following formula is justified by recognizing that a triangle is the half of some parallelogram:
Method using two sides and the angle between them:
Consider the triangle with side lengths a, b, c and angle x formed by the sides a and c. If we could compute the height H corresponding to side c from this information, we could use the previous formula to find the area.
We can achieve this by using the definition of the sine ratio:
Solving for H:
we can substitute this expression for H in:
Method using the coordinates of the three vertices (sometimes known as the "encasement method"):
If the coordinates of the vertices are given:
then the triangle ABC can be encased in a rectangle of area:
The area of the triangle ABC can be obtained by subtracting from the area of the rectangle, the areas of the three shaded right triangles. The areas of the shaded triangles are easy to find, since the triangles are right angled:
The area of the triangle ABC is the difference:
Note that we've only worked through one example rather than deriving a formula that works in all cases. This is because, while a formula exists, it does not have a simple algebraic form. Instead, the formula has many cases depending on how the triangle is shaped. Rather than memorizing and applying a complicated formula, it's usually easier to work through the method based on the specific coordinates given.
Method using Heron's formula:
If the lengths of the sides a, b, and c of a triangle are given, then Heron's formula can be used to derive the area.
First, find the semi-perimeter p:
Then find the area:
Method using the cross-product of vectors in 2D:
This method is equivalent to the encasement method.
If the coordinates of the three vertices are given, then two vectors that share an initial point can easily be defined:
The area of the triangle formed by the two vectors is equal to half the magnitude of the cross-product:
This follows from the definition of the cross-product, which calls for a magnitude of:
Clearly, from one of the area computations derived above, the magnitude of the cross product is related to the area.
The cross product can also be derived using the components of the two vectors:
which can be used to compute the area:
Using the cross-product of vectors in 3D
This method is useful when we know the coordinates of the three vertices of a triangle in 3D.
Choose two vectors that share an initial point and correspond to two of the sides of the triangle, and compute the lengths of their x, y, z components. The area of the triangle formed thus is: