Geometry (B): READ: Area of Triangles and Trapezoids Reading

READ: Area of Triangles and Trapezoids Reading

Triangle

Example 5

How could we find the area of this triangle?

Make it into a parallelogram. This can be done by making a copy of the original triangle and putting the copy together with the original.

The area of the parallelogram is $bh$ , so the area of the triangle is $\frac{bh}{2}$ or $\frac{1}{2}bh.$

Warning: Notice that the height $h$ (also often called the altitude) of the triangle is the perpendicular distance between a vertex and the opposite side of the triangle.

Area of a Triangle

If a triangle has base $b$ units and altitude $h$ units, then the area, $A$ , is $\frac{bh}{2}$ or $\frac{1}{2}$ $bh$ square units.

$A = \frac{bh}{2}$ or $A = \frac{1}{2} bh$

Area of a Trapezoid

Recall that a trapezoid is a quadrilateral with one pair of parallel sides. The lengths of the parallel sides are the bases. The perpendicular distance between the parallel sides is the height, or altitude, of the trapezoid.

To find the area of the trapezoid, turn the problem into one about a parallelogram. Why? Because you already know how to compute the area of a parallelogram.

Make a copy of the trapezoid.
Rotate the copy $180^\circ$ .
Put the two trapezoids together to form a parallelogram.

Two things to notice:

The parallelogram has a base that is equal to $b_1 + b_2$ .
The altitude of the parallelogram is the same as the altitude of the trapezoid.

Now to find the area of the trapezoid:

The area of the parallelogram is $\mathrm{base} \times\;\mathrm{altitude} = (b_1 + b_2) \times h$ .
The parallelogram is made up of two congruent trapezoids, so the area of each trapezoid is one-half the area of the parallelogram.
The area of the trapezoid is one-half of $(b_1 + b_2) \times h$ .

Area of Trapezoid with Bases $b_1$ and $b_2$ and Altitude $h$

Trapezoid with bases $b_1$ and $b_2$ and altitude $h$

$A = \frac{1} {2} (b_1 + b_2) h$ or $A = \frac{(b_1 + b_2)h} {2}$

Notice that the formula for the area of a trapezoid could also be written as the "Average of the bases time the height." This may be a convenient shortcut for memorizing this formula.

Example 1

What is the area of the trapezoid below?

The bases of the trapezoid are $4$ and $6$ . The altitude is $3$ .

$A=\frac{1}{2}({b_1+b_2})h=\frac{1}{2}({4+6})\times 3=15$

Last modified: Tuesday, 29 June 2010, 10:45 AM

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