How to Find the Area Of A Shape With Simpson’s 1/3 Rule

Simpson’s Rule for Area of a Shape: Video Lesson

Simpson’s 1/3 Rule For Area Formula

Simpson’s rule for finding the area of a shape is: Area=(w/3)×[yfirst + 4(yodd) + 2(yeven) + ylast], where w is the equal width between each measurement, yfirst and ylast are the first and last measurements, yodd is the sum of the odd measurements and yeven is the sum of the even measurements.

The Simpson Rule Formula is used for calculating the approximate area of shapes with curved boundaries.

A r e a. equals w over 3 times open bracket y sub f of i r s t plus 4 times open paren y sub o d d close paren plus 2 times open paren y sub e v e n close paren plus y sub l a. s t close bracket

Simpson’s 1/3 Rule for Calculating Area
  • w = the equal width of each subinterval (width between each measurement)
  • yfirst = the first measurement
  • yfirst = the last measurement
  • yodd = the sum of the odd measurements
  • ylast = the sum of the even measurements
simpsons rule for calculating area formula

Simpson’s rules are several different equations named after the mathematician Thomas Simpson which can be used for finding areas.

Whilst many other forms of Simpson’s rule involve integration, Simpson’s 1/3 Rule is a formula that does not involve integration and can be used to approximate the area of an irregular shape by taking equally-spaced measurements of the height of the shape.

Simpson’s 1/3 rule (also known as Composite Simpson’s rule) involves a quadratic interpolation of a parabola between the points given and is more accurate than the trapezoidal rule for area. Alternate formulae can be used for Simpson’s rule involving definite integrals.

The area of the shape is broken into ‘n‘ equally-sized strips of width ‘w’. These are sometimes known as the subintervals.

To improve the accuracy of the Simpson’s Rule approximation, a greater number of measurements should be taken. Essentially, a larger n value (or smaller strip width) leads to a more accurate result.

Limitations with Simpson’s rule include:

  • The need to have equally-spaced measurements, which may not always be possible.
  • Several measurements may be needed to sufficient accuracy, which may not always be possible.
  • Poor accuracy when considering shapes with sharp corners or rapid changes in measurements.

How to Calculate the Area of a Shape Using Simpson’s Rule

To calculate the area of a shape using Simpson’s rule use the formula Area=(w/3)[yfirst + 4(yodd) + 2(yeven) + ylast], where w is the width between each measurement, yfirst and ylast are the first and last meaurements, yodd is the sum of the odd measurements and yeven is the sum of the even measurements.

For example, calculate the area of the irregular shape shown below with measurements taken every 6 metres:

Here, w = 6 m.

  • yfirst = 5 m
  • y1 = 25 m
  • y2 = 26 m
  • y3 = 18 m
  • y4 = 20 m
  • y5 = 16 m
  • y6 = 12 m
  • ylast = 2 m

Therefore: y sub o d d equals y sub 1 plus y sub 3 plus y sub 5 and so, y sub o d d equals 25 plus 18 plus 16 equals 59.

y sub e v e n equals y sub 2 plus y sub 4 plus y sub 6 and so, y sub e v e n equals 26 plus 20 plus 12 equals 58.

how to use simpsons rule for area

The Simpson’s rule of A r e a. equals w over 3 times open bracket y sub f of i r s t plus 4 times open paren y sub o d d close paren plus 2 times open paren y sub e v e n close paren plus y sub l a. s t close bracket becomes:

A r e a. equals six thirds times open bracket 5 plus 4 times 59 plus 2 times 58 plus 2 close bracket

Evaluating this, we obtain: A r e a. equals six thirds times open bracket 5 plus 236 plus 116 plus 2 close bracket which simplifies to A r e a. equals 2 times 359 and so, A r e a. equals 718 m squared.

Therefore the approximate area of the irregular shape is 718 m2.

Simpson’s Rule Examples: Questions and Answers

Here are some further examples of using Simpson’s one-third rule to calculate the area of irregular shapes.

For example, the length of a curved island is measured in subintervals every 20 m as shown in the diagram. Approximate the area.

In this example, the first (yfirst) and last (ylast) measurements are 0 m as represented by the dots on each end of the diagram below.

Since the measurements are taken every 20 metres, the value of w = 20.

  • yfirst = 0
  • y1 = 30
  • y2 = 40
  • y3 = 50
  • y4 = 60
  • y5 = 22
  • ylast = 0

Therefore y sub o d d equals y sub 1 plus y sub 3 plus y sub 5 and so, y sub o d d equals 30 plus 50 plus 22 equals 102.

y sub e v e n equals y sub 2 plus y sub 4 and so, y sub e v e n equals 40 plus 60 equals 100.

simpsons rule for area question and answer

The Simpson’s rule of A r e a. equals w over 3 times open bracket y sub f of i r s t plus 4 times open paren y sub o d d close paren plus 2 times open paren y sub e v e n close paren plus y sub l a. s t close bracket becomes:

A r e a. equals 20 over 3 times open bracket 0 plus 4 times 102 plus 2 times 100 plus 0 close bracket which simplifies to A r e a. equals 20 over 3 times open bracket 408 plus 200 close bracket.

A r e a. equals 20 over 3 times 608 can be evaluated by multiplying 20 by 608 and dividing by 3 to obtain A r e a. almost equals 4053 m squared.

Here is another example of using Simpson’s 1/3 rule to find the area of a shape.

The distance across a swimming pool is measured every 150 cm as shown in the diagram. Approximate the area.

Since the distances are measured every 150 cm, the value of w = 150.

The first measurement is zero as represented by the dot at the start of the diagram below. Therefore yfirst = 0.

  • yfirst = 0
  • y1 = 180
  • y2 = 240
  • y3 = 320
  • ylast = 350

Therefore y sub o d d equals y sub 1 plus y sub 3 and so, y sub o d d equals 180 plus 320 equals 500.

y sub e v e n equals y sub 2 equals 240.

simpsons rule for area example

The Simpson’s rule of A r e a. equals w over 3 times open bracket y sub f of i r s t plus 4 times open paren y sub o d d close paren plus 2 times open paren y sub e v e n close paren plus y sub l a. s t close bracket becomes:

150 over 3 times open bracket 0 plus 4 times 500 plus 2 times 240 plus 350 close bracket.

This simplifies to 50 times open bracket 2000 plus 480 plus 350 close bracket which becomes 50 times 2830 and so, A r e a. equals 141500 c m squared.

This is equivalent to an area of 14.15 m2.

Simpson’s Rule From a Table

Simpsons rule, Area=(w/3)[yfirst + 4(yodd) + 2(yeven) + ylast] can be used to calculate the area of a shape when the values are given in a table. yfirst is the first value in the table, ylast is the last value in the table. yodd is the sum of the values in the odd rows of the table and yeven is the sum of the even rows of the table.

For example, use Simpson’s rule to calculate the area of the shape with measurements taken every 9 cm as given in the following table:

yfirst6 cm
y13 cm
y27 cm
y39 cm
y45 cm
y54 cm
y68 cm
ylast0 cm

y sub o d d equals y sub 1 plus y sub 3 plus y sub 5 and so, y sub o d d equals 3 plus 9 plus 4 equals 16.

y sub e v e n equals y sub 2 plus y sub 4 plus y sub 6 and so, y sub e v e n equals 7 plus 5 plus 8 equals 20.

simpsons rule for area from a table

The Simpson’s rule area formula of A r e a. equals w over 3 times open bracket y sub f of i r s t plus 4 times open paren y sub o d d close paren plus 2 times open paren y sub e v e n close paren plus y sub l a. s t close bracket becomes

A r e a. equals nine thirds times open bracket 6 plus 4 times 16 plus 2 times 20 plus 0 close bracket.

This simplifies as A r e a. equals 3 times open bracket 6 plus 64 plus 40 plus 0 close bracket or A r e a. equals 3 times 110.

Therefore the area of the shape is 330 cm2.

Using Simpson’s Rule in Surveying

Simpson’s rule is a numerical method for approximating irregularly shaped areas. In surveying, Simpson’s rule can be used to approximate the area of plots of land by dividing the land into an even number of equally-sized intervals and measuring the height at each interval.

Here are the steps for using Simpson’s rule in surveying:

  1. Divide the land into ‘n‘ equally-spaced intervals along the x-axis and record the width between them as ‘w‘
  2. Measure the height (elevation) of the land at each of the intervals, f(xi).
  3. Use Simpson’s rule: Area ≈ (w/3) [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + … + 2f(xn-2) + 4f(xn-1) + f(xn)]

The number of ordinates is the number of measurements made. In this example, we have 6 ordinates.

We multiply the odd ordinates by 4 and the even ordinates by 2. The first and last ordinates are not multiplied.

For example, the following elevations were measured along the curve every 30 m (so ‘w’ = 30)

  • f(x0) = 15
  • f(x1) = 18
  • f(x2) = 24
  • f(x3) = 31 – this is f(xn-2)
  • f(x4) = 32 – this is f(xn-1)
  • f(x5) = 35 – this is f(xn)
simpsons rule in surveying

Therefore, Area ≈ (w/3) [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + … + 2f(xn-2) + 4f(xn-1) + f(xn)] becomes:

Area ≈ (30/3) [15 + 4×18 + 2×24 + 4×31 + 2×32 +35] ≈ 3580 m2

Therefore the cross area of the surveyed land is approximately 3580 m2.