The Area of a Triangle
The Area of a Triangle
The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional
like a carpet or an area rug. A triangle is a three-sided polygon. We will look at several types
of triangles in this lesson.
To find the area of a triangle, multiply the base by the height, and then divide by 2. The
division by 2 comes from the fact that a paral elogram can be divided into 2 triangles. For
example, in the diagram to the left, the area of each triangle is equal to one-half the area of
the parallelogram.
Since the area of a paral elogram is , the area of a triangle must be one-half the area of a
paral elogram. Thus, the formula for the area of a triangle is:
or
where is the base, is the height and * means multiply.
The base and height of a triangle must be perpendicular to each other. In each of the
examples below, the base is a side of the triangle. However, depending on the triangle, the
height may or may not be a side of the triangle. For example, in the right triangle in Example
2, the height is a side of the triangle since it is perpendicular to the base. In the triangles in
Examples 1 and 3, the lateral sides are not perpendicular to the base, so a dotted line is
drawn to represent the height.
Know More About :- Corresponding Angles Definition


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The number of square units it takes to exactly fil the interior of a triangle.
Most common method
Usually called "half of base times height", the area of a triangle is given by the formula below.
Calculator
where
b is the length of the base
a is the length of the corresponding altitude
You can choose any side to be the base. It need not be the one drawn at the bottom of the
triangle. The altitude must be the one corresponding to the base you choose. The altitude is
the line perpendicular to the selected base from the opposite vertex.
In the figure above, one side has been chosen as the base and its corresponding altitude is
shown.
Any side can be a base, but every base has only one height. The height is the line from the
opposite vertex and perpendicular to the base. In the picture above, the base CB has one and
only one height. The il ustration below shows how any leg of the triangle can be a base and
the height always extends from the vertex of the opposite side and is perpendicular to the
base
Calculating the Area of a Triangle
How to find the area of a triangle:
1. The area of a triangle can be found by multiplying the base times the one-half the height.
2. If a triangle has a base of length 6 inches and a height of 4 inches, its area is 6*2=12
square inches
There are several ways to compute the area of a triangle. For instance, there's the basic
formula that the area of a triangle is half the base times the height. This formula only works, of
course, when you know what the height of the triangle is.
Learn More :- Find Asymptotes


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Another is Heron's formula which gives the area in terms of the three sides of the triangle,
specifically, as the square root of the product s(s - a)(s - b)(s - c) where s is the
semiperimeter of the triangle, that is, s = (a + b + c)/2.
Here, we'll consider a formula for the area of a triangle when you know two sides and the
included angle of the triangle. Suppose we know the values of the two sides a and b of the
triangle, and the included angle C.
As in the proof of the law of sines in the previous section, drop a perpendicular AD from the
vertex A of the triangle to the side BC, and label this height h. Then triangle ACD is a right
triangle, so sin C equals h/b. Therefore, h = b sin C. Since the area of the triangle is half the
base a times the height h, therefore the area also equals half of ab sin C. Although the figure
is an acute triangle, you can see from the discussion in the previous section that h = b sin C
holds when the triangle is right or obtuse as well. Therefore, we get the general formula
Area = ab sin C/2
That is to say, the area of a triangle is half the product of two sides times the sine of the
included angle.


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