12.3: Areas of Triangles

what's the area of a triangle

1/2 (b * h) square units

explain how units are to be used when calculating the area formula for triangles

base b and height h must be described w/ the same unit for example both the b and h must be in ft or in cm if one length is in ft and the other is in in, convert both lengths to a common unit the area of the triangle is then in square units of whatever common unit you used for the base and height. so if the base and height are both in cm, then the area resulting from the formula is sq cm

what's the base of a triangle

base of a triangle can be any of its three sides in a formula the word base, or a letter, B, represents base, really means length of the base

Given a triangle, what's the most primitive method for determining its area...

count how many 1 unit by 1 unit sq it takes to cover the shape without gaps or overlaps..

ex: figure 12.19 method B: what's the advanced way to determine the area of these two triangles

either by moving a big chunk or by embedding the triangle in a rectangle. these methods lead to the triangle area formulas (1/2 b * h)

what about the triangle in figure 12.22?

for this triangle, it's not clear how to turn it into half of a b-by-h rectangle, however the formula for the area of a triangle is still valid. enclose the triangle in a rectangle, as shown in figure 12.23a. the rectangle consists of two copies of the original triangle (white) and two copies of the another triangle, lightly shaded. the rectangle has area (b + a) * h, which is equal to b x h + a x h by the distributive property. if we put the two shaded triangles together, as in figure 12.23(b), they form a rectangle of area a * h. if we take the area away from the area of the large rec, the remaining area is the area of the two copies of the original triangle combined (by the moving and additivity principle_ therefore, the area of the two copies of the original triangle is (b * h + a * h) - a * h = b * h so, original triangle has half this area, namely 1/2 (b * h)

suppose we have a triangle, and suppose we have chosen a base b and height h for the triangle. why is the area of the triangle equal to one half the base times the height

in some cases, such as those shown in figure 12.21, two copies of the triangle can be subdivided if necessary and recombined, without overlapping, to form a b-by-h rectangle. in such cases because of the moving and additivy principles about area, 2 * area of triangle = b * h but since the rectangle has area b x h 2* area of triangle = b * h therefore, area of triangle = 1/2 (b * h) `

ex: figure 12.19 method A: what's a primitive way to determine the area of this triangle.

move small pieces and count the total number of squares

example 12.20 shows two copies of a triangle ABC, what are two of the three ways to choose the base (b) and height (h)....

notice that the base and the height have different lengths for the different choices in the second choice the height is the dashed line segment CE. even though the height h doesn't meet b itselv, it meets an extension of b

ex: if a triangle has a base that is 5 in long, and if the corresponding height of the triangle is 3 inches long, then the area of the triangle is

plug in appropriate units using the area formula for triangles

what's the advanced way for determining the area of a give triangle

rely on relating the triangle to a rectangle and applying the rec area formulas

what does the primitive method for determining areas of triangles require

requires squares to be cut apart and pieces to be moved and recombined w. other pieces to make as many full squares as possible

once the base has been chosen, the height is ..

the line segment that is perpendicular to the base and connects the base, or an extension of the base, to the vertex of the triangle that is not on the base in a formula, the word height, or a letter, such as h, which represents the height, actually means the length of the height

no matter which side is chosen to be the height of a triangle, what formula applies

there is a single formula that produces the triangle's area