General (1-12)

Computing Area of Two-dimensional Solids
Area is a measurement of the amount of surface of an object. Area is usually expressed in such units as square inches or square centimeters for small surfaces or in square feet or square meters for larger surfaces.

 

Rectangle

A rectangle is a four-sided figure with opposite sides of equal length and parallel. [Figure 1-13] All of the angles are right angles. A right angle is a 90° angle. The rectangle is a very familiar shape in mechanics. The formula for the area of a rectangle is:

 

Area = Length × Width = L × W

 

Example: An aircraft floor panel is in the form of a rectangle having a length of 24 inches and a width of 12 inches. What is the area of the panel expressed in square inches? First, determine the known values and substitute them in the formula.

A = L × W = 24 inches × 12 inches = 288 square inches

 

 

Square
A square is a four-sided figure with all sides of equal length and parallel. [Figure 1-14] All angles are right angles. The formula for the area of a square is:

Area = Length × Width = L × W

 

 

 

 

 Since the length and the width of a square are the same value, the formula for the area of a square can also be written as:

 

Area = Side × Side = S²

 

Example: What is the area of a square access plate whose side measures 25 inches? First, determine the known value and substitute it in the formula.

A = L × W = 25 inches × 25 inches = 625 square inches

 

Triangle
A triangle is a three-sided figure. The sum of the three angles in a triangle is always equal to 180°. Triangles are often classified by their sides. An equilateral triangle has 3 sides of equal length. An isosceles triangle has 2 sides of equal length. A scalene triangle has three sides of differing length. Triangles can also be classified by their angles: An acute triangle has all three
angles less than 90°. A right triangle has one right angle (a 90° angle). An obtuse triangle has one angle greater than 90°. Each of these types of triangles is shown in Figure 1-15.

The formula for the area of a triangle is

Area = 1⁄2 × (Base × Height) = 1⁄2 × (B × H)

 

Example: Find the area of the obtuse triangle shown in Figure 1-16. First, substitute the known values in the area formula.

A = 1⁄2 × (B × H) = 1⁄2 × (2'6" × 3'2")

 

Next, convert all dimensions to inches:

2'6" = (2 × 12") + 6" = (24 + 6) = 30 inches
3'2" = (3 × 12") + 2" = (36 + 2) = 38 inches

 

Now, solve the formula for the unknown value:

 

A = 1⁄2 × (30 inches × 38 inches) = 570 square inches

 

Parallelogram
A parallelogram is a four-sided figure with two pairs of parallel sides. [Figure 1-17] Parallelograms do not necessarily have four right angles. The formula for the area of a parallelogram is:

Area = Length × Height = L × H

 

 

 

 

Trapezoid
A trapezoid is a four-sided figure with one pair of parallel sides. [Figure 1-18] The formula for the area of a trapezoid is:

Area = 1⁄2 (Base1 + Base2) × Height

 

Example: What is the area of a trapezoid in Figure 1-19 whose bases are 14 inches and 10 inches, and whose height (or altitude) is 6 inches? First, substitute the known values in the formula. 

 

 

 

 

A = 1⁄2 (b₁ + b₂) × H
   = 1⁄2 (14 inches + 10 inches) × 6 inches

A = 1⁄2 (24 inches) × 6 inches
   = 12 inches × 6 inches = 72 square inches.

 

Circle
A circle is a closed, curved, plane figure. [Figure 1‑20] Every point on the circle is an equal distance from the center of the circle. The diameter is the distance across the circle (through the center). The radius is the distance from the center to the edge of the circle. The diameter is always twice the length of the radius. The circumference, or distance around, a circle is equal to
the diameter times π.

Circumference = C = d π

 

The formula for the area of a circle is:

Area = π × radius² = π × r²

 

Example: The bore, or “inside diameter,” of a certain aircraft engine cylinder is 5 inches. Find the area of the cross section of the cylinder. 

 

 

 

 

 

 

 

 

First, substitute the known values in the formula:

A = π × r².

 

The diameter is 5 inches, so the radius is 2.5 inches.(diameter = radius × 2)

A = 3.1416 × (2.5 inches)² = 3.1416 × 6.25 square inches = 19.635 square inches

 

Ellipse
An ellipse is a closed, curved, plane figure and is commonly called an oval. [Figure 1-21] In a radial engine, the articulating rods connect to the hub by pins, which travel in the pattern of an ellipse (i.e., an elliptical or obital path).

 

Wing Area
To describe the shape of a wing [Figure 1-23], several terms are required. To calculate wing area, it will be necessary to know the meaning of the terms “span” and “chord.” The wingspan, S, is the length of the wing from wingtip to wingtip. The chord is the average width of the wing from leading edge to trailing edge. If the wing is a tapered wing, the average width, known as the mean chord (C), must be known to find the area.

The formula for calculating wing area is:

Area of a wing = Span × Mean Chord

Example: Find the area of a tapered wing whose span is 50 feet and whose mean chord is 6'8". First, substitute the known values in the formula.

A = S × C
   = 50 feet × 6 feet 8 inches   (Note: 8 inches = 8⁄12 feet = .67 feet)
   = 50 feet × 6.67 feet
   = 333.5 square feet

 

 

 

 

 

Units of Area
A square foot measures 1 foot by 1 foot. It also measures 12 inches by 12 inches. Therefore, one square foot also equals 144 square inches (that is, 12 × 12 = 144). To convert square feet to square inches, multiply by 144. To convert square inches to square feet, divide by 144.

A square yard measures 1 yard by 1 yard. It also measures 3 feet by 3 feet. Therefore, one square yard also equals 9 square feet (that is, 3 × 3 = 9). To convert square yards to square feet, multiply by 9. To convert square feet to square yards, divide by 9. Refer to Figure 1-37, Applied Mathematics Formula Sheet, at the end of the chapter for a comparison of different units of area.
Figure 1-23 summarizes the formulas for computing the area of two-dimensional solids.