# How to Calculate the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## Introduction to Area Approximation

Are you having trouble solving areas of complicated and irregular-shaped curve figures? If yes, this is the perfect article for you. Many methods and formulas are used to approximate the area of irregularly shaped curves, as shown in the figure below. Among these are Simpson’s Rule, the Trapezoidal Rule, and Durand’s Rule.

## Three Rules

The **Trapezoidal Rule** is an integration rule where you divide the total area of the irregular shaped figure into little trapezoids before evaluating the area under a specific curve.

**Durand’s Rule** is a slightly more complicated but more precise integration rule than the trapezoidal rule. This method of area approximation uses the Newton-Cotes formula, which is an extremely useful and straightforward integration technique.

Lastly, **Simpson’s Rule** gives the most accurate approximation as compared to the other two mentioned formulas. It is also important to note that the greater the value of n in Simpson’s Rule, the greater accuracy of area approximation.

## What Is Simpson’s 1/3 Rule?

Simpson’s Rule is named after the English mathematician Thomas Simpson who was from Leicestershire, England. But for some reason, the formulas used in this method of area approximation were similar to Johannes Kepler’s formulas used over 100 years prior. That is the reason why many mathematicians call this method Kepler’s Rule.

Simpson’s Rule is considered a very diverse numerical integration technique. It is entirely based on the type of interpolation you will use. Simpson’s 1/3 Rule or Composite Simpson’s Rule is based upon a quadratic interpolation, while Simpson’s 3/8 Rule is based upon a cubic interpolation. Among all methods of area approximation, Simpson’s 1/3 Rule gives the most accurate area because parabolas are used to approximate each part of the curve, and not rectangles or trapezoids.

Simpson's 1/3 Rule states that if y_{0}, y_{1}, y_{2},...,y_{3} (n is even) are the lengths of a series of parallel chords of uniform interval d, the area of the figure enclosed above is given approximately by the formula below. Note that if the figure ends with points, then take y_{0} = y_{n} = 0.

**A = (1/3)(d) [(y**_{0}** + y**_{n}**) + 4(y**_{1}** + y**_{3}** + ... + y**_{n-1}**) + 2(y**_{2}** + y**_{4}** + ... + y**_{n-2}**)]**

_{0}

_{n}

_{1}

_{3}

_{n-1}

_{2}

_{4}

_{n-2}

## Problem 1

### Solution

a. Given the value of n = 10 of the irregularly shaped figure, identify the height values from y_{0} to y_{10}. Create a table and list all the height values from left to right for a more organized solution.

Variable (y) | Height Value |
---|---|

y0 | 10 |

y1 | 11 |

y2 | 12 |

y3 | 11 |

y4 | 6 |

y5 | 7 |

y6 | 4 |

y7 | 8 |

y8 | 4 |

y9 | 3 |

y10 | 0 |

b. The given value of the uniform interval is d = 0.75. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y_{0} + y_{n}) + 4(y_{1} + y_{3} + ... + y_{n-1}) + 2(y_{2} + y_{4} + ... + y_{n-2})]

A = (1/3)(3) [(10 + 0) + 4(11 + 11 + 7 + 8 + 3) + 2(12 + 6 + 4 + 4)]

A = 222 square units

c. Find the area of the right triangle formed from the irregular shape. Given a height of 10 units and an angle of 30°, find the length of adjacent sides and compute for the area of the right triangle using Scissors formula or Heron's formula.

Length = 10 / tan (30°)

Length = 17.32 units

Hypotenuse = 10 / sin (30°)

Hypotenuse = 20 units

Semi-Perimeter (s) = (10 + 20 + 17.32) / 2

Semi-Perimeter (s) = 23. 66 units

Area (A) = √s (s - a) (s - b) (s - c)

Area (A) = √23.66 (23.66 - 10) (23.66 - 20) (23.66 - 17.32)

Area (A) = 86.6 square units

d. Subtract the area of the right triangle from the area of the whole irregular figure.

Shaded Area (S) = Total Area - Triangular Area

Shaded Area (S) = 222 - 86.6

Shaded Area (S) = 135.4 square units

**Final Answer: **The approximate area of the irregular figure above is 135.4 square units.

## Problem 2

### Solution

a. Given the value of n = 6 of the irregularly shaped figure, identify the height values from y_{0} to y_{6}. Create a table and list all the height values from left to right for a more organized solution.

Variable (y) | Height Value |
---|---|

y0 | 5 |

y1 | 3 |

y2 | 4 |

y3 | 6 |

y4 | 4.5 |

y5 | 1.5 |

y6 | 0 |

b. The given value of the uniform interval is d = 1.00. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y_{0} + y_{n}) + 4(y_{1} + y_{3} + ... + y_{n-1}) + 2(y_{2} + y_{4} + ... + y_{n-2})]

A = (1/3)(1.00) [(5 + 0) + 4(3 + 6 + 1.5) + 2(4 + 4.5)]

A = 21.33 square units

**Final Answer: **The approximate area of the irregular figure above is 21.33 square units.

## Problem 3

### Solution

a. Given the value of n = 6 of the irregularly shaped figure, identify the height values from y_{0} to y_{6}. Create a table and list all the height values from left to right for a more organized solution.

Variable (y) | Upper Value | Lower Value | Height Value (Sum) |
---|---|---|---|

y0 | 0 | 0 | 0 |

y1 | 3 | 2 | 5 |

y2 | 1.5 | 1.75 | 3.25 |

y3 | 1.75 | 4 | 5.75 |

y4 | 3 | 2.75 | 5.75 |

y5 | 2.75 | 3 | 5.75 |

y6 | 0 | 0 | 0 |

b. The given value of the uniform interval is d = 1.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y_{0} + y_{n}) + 4(y_{1} + y_{3} + ... + y_{n-1}) + 2(y_{2} + y_{4} + ... + y_{n-2})]

A = (1/3)(1.50) [(0 + 0) + 4(5 + 5.75 + 5.75) + 2(3.25 + 5.75)]

A = 42 square units

**Final Answer: **The approximate area of the irregular shape above is 42 square units.

## Problem 4

### Solution

a. Given the value of n = 8 of the irregularly shaped figure, identify the height values from y_{0} to y_{8}. Create a table and list all the height values from left to right for a more organized solution.

Variable (y) | Height Value |
---|---|

y0 | 10 |

y1 | 9 |

y2 | 8 |

y3 | 7 |

y4 | 6 |

y5 | 5 |

y6 | 4 |

y7 | 3 |

y8 | 0 |

b. The given value of the uniform interval is d = 1.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

_{0} + y_{n}) + 4(y_{1} + y_{3} + ... + y_{n-1}) + 2(y_{2} + y_{4} + ... + y_{n-2})]

A = (1/3)(1.50) [(10 + 0) + 4(9 + 7 + 5 +3) + 2(8 + 6 +4)]

A = 71 square units

**Final Answer: **The approximate area of the irregular shape above is 71 square units.

## Problem 5

### Solution

a. Given the equation of the irregular curve, identify the height values from y_{0} to y_{8} by substituting each value of x to solve for the corresponding value of y. Create a table and list all the height values from left to right for a more organized solution. Use an interval of 0.5.

Variable (y) | X-Value | Height Value |
---|---|---|

y0 | 1.0 | 1.732050808 |

y1 | 1.5 | 1.870828693 |

y2 | 2.0 | 2.0000000 |

y3 | 2.5 | 2.121320344 |

y4 | 3.0 | 2.236067977 |

y5 | 3.5 | 2.34520788 |

y6 | 4.0 | 2.449489743 |

b. Use the uniform interval d = 0.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

_{0} + y_{n}) + 4(y_{1} + y_{3} + ... + y_{n-1}) + 2(y_{2} + y_{4} + ... + y_{n-2})]

A = (1/3)(0.50) [(1.73 + 2.45) + 4(1.87 + 2.12 + 2.35) + 2(2 + 2.23)]

A = 6.33 square units

**Final Answer: **The approximate area of the irregular shape above is 6.33 square units.

## Problem 6

### Solution

a. Given the value of n = 8 of the irregularly shaped figure, identify the height values from y_{0} to y_{8}. Create a table and list all the height values from left to right for a more organized solution.

Variable (y) | Height Value |
---|---|

y0 | 50 |

y1 | 40 |

y2 | 30 |

y3 | 27 |

y4 | 28 |

y5 | 38 |

y6 | 40 |

y7 | 45 |

y8 | 48 |

b. The given value of the uniform interval is d = 5.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

_{0} + y_{n}) + 4(y_{1} + y_{3} + ... + y_{n-1}) + 2(y_{2} + y_{4} + ... + y_{n-2})]

A = (1/3)(5.50) [(50 + 48) + 4(40 + 27 +38 +45) + 2(30 +28 + 40)]

A = 1639 square units

**Final Answer: **The approximate area of the irregular shape above is 1639 square units.

## Other Topics About Area and Volume

- How to Solve for the Surface Area and Volume of Prisms and Pyramids

This guide teaches you how to solve the surface area and volume of different polyhedrons such as prisms, pyramids. There are examples to show you how to solve these problems step-by-step. - Finding the Surface Area and Volume of Truncated Cylinders and Prisms

Learn how to compute for the surface area and volume of truncated solids. This article covers concepts, formulas, problems, and solutions about truncated cylinders and prisms.

**© 2020 Ray**