Introduction to Integration: Background You’ll Need 1
Find the area of rectangles, triangles, trapezoids, and irregular shapes
Find the Area of a Rectangle
A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, [latex]L[/latex], and the adjacent side as the width, [latex]W[/latex].
Rectangle with all sides labeled
The area of a rectangle is calculated as the product of its length and width. This relationship can be expressed through the formula:
[latex]A=L \times W[/latex]
Consider a rectangular rug that is [latex]2[/latex] feet long by [latex]3[/latex] feet wide.
A rug with length 3 and width 2, divided into unit squares.
The area of this rug would be:
[latex]A = 2 \text{ ft } \times 3 \text{ ft } = 6 \text{ square feet}[/latex]
area of rectangles
Rectangles have four sides and four right [latex]\left(\text{90}^ \circ\right)[/latex] angles.
The lengths of opposite sides are equal.
The area, [latex]A[/latex], of a rectangle is the length times the width. The area will be expressed in square units.
[latex]A=L\cdot W[/latex]
The length of a rectangle is [latex]32[/latex] meters and the width is [latex]20[/latex] meters. Find the area or the rectangle.
Step 1. Read the problem. Draw the figure and label it with the given information.
Rectangle with all sides labeled
Step 2. Identify what you are looking for.
the area of a rectangle
Step 3. Name. Choose a variable to represent it.
Let [latex]A[/latex] = the area
Step 4. Translate. Write the appropriate formula. Substitute.
The area of the rectangle is [latex]640[/latex] square meters.
Find the Area of a Triangle
We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle below, we’ve labeled the length [latex]b[/latex] and the width [latex]h[/latex], so its area is [latex]bh[/latex].
Rectangle with height, base, and area labeled
We can divide this rectangle into two congruent triangles (see the image below). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or [latex]\Large\frac{1}{2}\normalsize bh[/latex]. This example helps us see why the formula for the area of a triangle is [latex]A=\Large\frac{1}{2}\normalsize bh[/latex].
Rectangle split into two triangles with height, base, and area labeled
To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a [latex]\text{90}^ \circ[/latex] angle with the base. The image below shows three triangles with the base and height of each marked.
Examples of how the height of a triangle can be represented relative to its base
area of a triangle
The area of a triangle is one-half the base, [latex]b[/latex], times the height, [latex]h[/latex].
[latex]A={\Large\frac{1}{2}}bh[/latex]
Triangle with key features labeled
Find the area of a triangle whose base is [latex]11[/latex] inches and whose height is [latex]8[/latex] inches.
<tr”>Step 1. Read the problem. Draw the figure and label it with the given information.
Triangle with height and base labeled
Step 7. Answer the question.The area is [latex]44[/latex] square inches.
A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[/latex], and the length of the bigger base [latex]B[/latex]. The height, [latex]h[/latex], of a trapezoid is the distance between the two bases as shown in the image below.
Trapezoid labeled with smaller base, larger base, and height
The formula for the area of a trapezoid is: [latex]{\text{Area}}_{\text{trapezoid}}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex]. Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles.
Trapezoid split into two triangles with smaller base and larger base labeled
The height of the trapezoid is also the height of each of the two triangles.
Trapezoid split into triangles and rectangles to illustrate its area formula
The formula for the area of a trapezoid is
Formula for area of a trapezoid
If we distribute, we get,
Formula for area of blue and red triangles
properties of trapezoids
A trapezoid has four sides.
Two of its sides are parallel and two sides are not.
The area, [latex]A[/latex], of a trapezoid is [latex]\text{A}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex] .
Find the area of a trapezoid whose height is [latex]6[/latex] inches and whose bases are [latex]14[/latex] and [latex]11[/latex] inches.
Step 1. Read the problem. Draw the figure and label it with the given information.
Trapezoid with smaller base, larger base, and height labeled
Step 2. Identify what you are looking for.
the area of the trapezoid
Step 3. Name. Choose a variable to represent it.
Let [latex]A=\text{the area}[/latex]
Step 4.Translate. Write the appropriate formula. Substitute.
If we draw a rectangle around the trapezoid that has the same big base [latex]B[/latex] and a height [latex]h[/latex], its area should be greater than that of the trapezoid. If we draw a rectangle inside the trapezoid that has the same little base [latex]b[/latex] and a height [latex]h[/latex], its area should be smaller than that of the trapezoid.
Using rectangle approximations to verify the trapezoid area formula
The area of the larger rectangle is [latex]84[/latex] square inches and the area of the smaller rectangle is [latex]66[/latex] square inches. So it makes sense that the area of the trapezoid is between [latex]84[/latex] and [latex]66[/latex] square inches Step 7. Answer the question. The area of the trapezoid is [latex]75[/latex] square inches.
Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[/latex] yards and the bases are [latex]8.2[/latex] and [latex]5.6[/latex] yards. How many square yards will be available to plant?
Solution
Step 1. Read the problem. Draw the figure and label it with the given information.
Trapezoid with bases and height labeled
Step 2. Identify what you are looking for.
the area of a trapezoid
Step 3. Name. Choose a variable to represent it.
Let [latex]A[/latex] = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 6. Check: Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of [latex]8.2[/latex] yd and height [latex]3.4[/latex] yd, but more than the area of a rectangle with base [latex]5.6[/latex] yd and height [latex]3.4[/latex] yd.
Comparison of areas of rectangles and a trapezoid with the same height
Step 7. Answer the question.
Vinny has [latex]23.46[/latex] square yards in which he can plant.
Find the Area of Irregular Figures
So far, we have found area for rectangles, triangles, and trapezoids. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.
Find the area of the shaded region.
Shaded region with sides labeled
The given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.
Shaded region split into two rectangles
The blue rectangle has a width of [latex]12[/latex] and a length of [latex]4[/latex]. The red rectangle has a width of [latex]2[/latex], but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is [latex]4[/latex] units long, the length of the red rectangle must be [latex]6[/latex] units.
Shaded region split into two rectangles
Formula for Area
The area of the figure is [latex]60[/latex] square units.
Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area.
Find the area of the shaded region.
Shaded region with sides labeled
We can break this irregular figure into a triangle and rectangle. The area of the figure will be the sum of the areas of the triangle and the rectangle. The rectangle has a length of [latex]8[/latex] units and a width of [latex]4[/latex] units. We need to find the base and height of the triangle.
Since both sides of the rectangle are [latex]4[/latex], the vertical side of the triangle is [latex]3[/latex] , which is [latex]7 - 4[/latex] .
The length of the rectangle is [latex]8[/latex], so the base of the triangle will be [latex]3[/latex] , which is [latex]8 - 5[/latex] .
Shaded region split into a triangle and a rectangle
Now we can add the areas to find the area of the irregular figure.
Formula for Area
The area of the figure is [latex]36.5[/latex] square units.