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].

A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.

 

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 rectangle made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.

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.

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].

A rectangle with the side labeled h and the bottom labeled b. The center says A equals bh.

 

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].

A rectangle with a diagonal line drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says "Area of each triangle A = one-half bh".

 

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.

Three triangles. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.

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]

 

A triangle, with vertices labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.

 

Find the area of a triangle whose base is [latex]11[/latex] inches and whose height is [latex]8[/latex] inches.


Find the Area of a Trapezoid

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.

A trapezoid, with the top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.

 

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.

A trapezoid, with the top labeled with a small b and the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.

 

The height of the trapezoid is also the height of each of the two triangles.

A trapezoid, with the top labeled with a small b and the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.

 

The formula for the area of a trapezoid is

The formula for the area of a trapezoid, which is one half h times the quantity of lowercase b plus capital B

 

If we distribute, we get,

The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.

 

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.

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?


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.

An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.

Find the area of the shaded region.

A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7.