How to find the area of a triangle


The most common way to find the area of a triangle is to take half of the base times the height. Numerous other formulas exist, however, for finding the area of a triangle, depending on what information you know. Using information about the sides and angles of a triangle, it is possible to calculate the area without knowing the height. 
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 parallelogram 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 parallelogram is A = B * H, the area of a triangle must be one-half the area of a parallelogram. Thus, the formula for the area of a triangle is:
triangle or triangle
where b is the base, h 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.
Example 1: Find the area of an acute triangle with a base of 15 inches and a height of 4 inches.
Solution:

triangle
a_0.gif = one_half.gif· (15 in) · (4 in)
a_0.gif = one_half.gif· (60 in2)
a_0.gif = 30 in2

Example 2: Find the area of a right triangle with a base of 6 centimeters and a height of 9 centimeters.
Solution:

triangle
a_0.gif = one_half.gif· (6 cm) · (9 cm)
a_0.gif = one_half.gif· (54 cm2)
a_0.gif = 27 cm2

Example 3: Find the area of an obtuse triangle with a base of 5 inches and a height of 8 inches.
Solution:
triangle
a_0.gif = one_half.gif· (5 in) · (8 in)
a_0.gif = one_half.gif· (40 in2)
a_0.gif = 20 in2

Example 4: The area of a triangle shaped mat is 18 square feet and the base is 3 feet. Find the height. (Note: The triangle in the illustration to the right is NOT drawn to scale.)
Solution:
In this example, we are given the area of a triangle and one dimension, and we are asked to work backwards to find the other dimension.
triangle

18 ft2 = one_half.gif\B7 (3 ft) · h
Multiplying both sides of the equation by 2, we get:
36 ft2 = (3 ft) · h
Dividing both sides of the equation by 3 ft, we get:
12 ft = h
Commuting this equation, we get:
h = 12 ft

Summary: Given the base and the height of a triangle, we can find the area. Given the area and either the base or the height of a triangle, we can find the other dimension. The formula for area of a triangle is:

triangle or triangle where b is the base and h is the height.

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