Calculate perimeters and areas of plots of land
Perimeter
The perimeter of a plot of land is the total length of the boundary.
We can work out the perimeter by adding up all the lengths of the pieces that make up the boundary. This assumes that someone, possibly a surveyor, has done the measurements and has made them available.
However, sometimes certain measurements have been carried out but not all of the boundary lengths are known until one or more calculations using trigonometry or Pythagoras's theorem have been done.
Sometimes it may be necessary to use an accurate scale diagram of the plot of land to recover missing measurements. This can be done by measuring distances on the diagram and scaling them up by the correct proportion.
Diagrams of plots of land can include measurements other than the required boundary measurements. If so, one must be careful to include only the relevant measurements in the perimeter calculation.
One should also take care not confuse measurements of perimeter with measurements of area. Perimeter can be thought of as the length of a fence that would fully separate the plot from other pieces of land.
Example 1
In this diagram of a plot of land, the measurements of all of the boundary segments have been given. To calculate the perimeter, all that is needed is to add the five boundary measurements.
$\text{perimeter}=22.4+11.6+34.6+2.9+16.0=87.5$perimeter=22.4+11.6+34.6+2.9+16.0=87.5 m
Radial surveys
One method that surveyors can use to obtain the essential measurements of a piece of land is called a radial survey. A reference point is chosen and measurements are made from it to the corners of the plot, together with the angles between the directions of the corners.
The lengths of the boundary segments that have not been directly measured are then calculated using the cosine rule.
Example 2
In this diagram, the same plot of land as in Example 1 has been measured by the radial survey method using point A as the reference point.
We need to calculate the lengths BC, CD and DE. We use the cosine rule.
| $\text{BC}^2$BC2 | $=$= | $22.4^2+25.2^2-2\times22.4\times25.2\times\cos27.5^\circ$22.42+25.22−2×22.4×25.2×cos27.5° |
| $=$= | $135.4$135.4 | |
| $\text{BC}$BC | $=$= | $11.6$11.6 |
| $\text{CD}^2$CD2 | $=$= | $25.2^2+17.9^2-2\times25.2\times17.9\times\cos105.2^\circ$25.22+17.92−2×25.2×17.9×cos105.2° |
| $=$= | $1191.99$1191.99 | |
| $\text{CD}$CD | $=$= | $34.5$34.5 |
| $\text{DE}^2$DE2 | $=$= | $17.9^2+16.0^2-2\times17.9\times16.0\times\cos7.4^\circ$17.92+16.02−2×17.9×16.0×cos7.4° |
| $=$= | $8.38$8.38 | |
| $\text{DE}$DE | $=$= | $2.9$2.9 |
We now add the lengths as before and get a total of $87.4$87.4 m. The small difference between this and the previous result can be attributed to imprecision in the angle measurements.
Scale drawings
Think of a line segment representing a piece of the boundary of some land. Suppose the line, on paper, is $3.5$3.5 cm long.
There is a scale included with the drawing that tells us that $1$1 cm on the drawing represents $2.5$2.5 m on the ground. This means that the $3.5$3.5 cm line would be equivalent to $3.5\times2.5=8.75$3.5×2.5=8.75 m on the ground.
To make use of this technique, you may need to physically measure the distances in a scale drawing with a ruler before multiplying each measurement by the scale factor.
Area
The area of a plot of land is the amount of flat space inside the boundary of the plot.
When the plot of land for which we want to measure the area is a rectangle, the area is simply $\text{length}\times\text{width}$length×width.
Sometimes a plot of land can be seen as a collection of rectangles. In this case, we find the area of each rectangle separately and then add them.
In other situations, the plot of land is irregular in shape but as long as its boundary is made up of straight edges it can be split into a number of triangles. We use a formula to calculate the area of each triangle and then add the separate areas.
For a triangle that has a right-angle, the area is just half a rectangle: $\frac{1}{2}\text{base}\times\text{height}$12base×height.
A more generally useful formula for the area of a triangle that does not necessarily have a right-angle, requires the measurement of two sides and the angle between them. If the two sides have lengths $a$a and $b$b and the angle between them is $C$C, we have $\text{area}=\frac{1}{2}ab\sin C$area=12absinC.
Example 3
Look again at the diagram for Example 2. The shape ABCDE has been conveniently divided into three triangles by the radial survey process and we have all the information needed to calculate the area.
We find the areas of the three triangles ABC, ACD and ADE.
ABC
$\text{area}_{ABC}=\frac{1}{2}\times22.4\times25.2\times\sin27.5^\circ=130.3$areaABC=12×22.4×25.2×sin27.5°=130.3 m$^2$2.
ACD
$\text{area}_{ACD}=\frac{1}{2}\times25.2\times17.9\times\sin105.2^\circ=217.6$areaACD=12×25.2×17.9×sin105.2°=217.6 m$^2$2.
ADE
$\text{area}_{ADE}=\frac{1}{2}\times17.9\times16.0\times\sin7.4^\circ=18.4$areaADE=12×17.9×16.0×sin7.4°=18.4 m$^2$2.
Finally, we add the separate areas:
$\text{total area}_{ABCDE}=130.3+217.6+18.4=366.3$total areaABCDE=130.3+217.6+18.4=366.3 m$^2$2.
Worked Examples
Question 1
Calculate the perimeter in metres of the plot of land pictured here on this site plan. All measurements are given in metres.
Question 2
Answer the following questions using the aerial photo of a block of land. All measurements are in metres.
Calculate the length $x$x to the nearest metre.
Using your answer to part (a), calculate the total perimeter of the land.
Question 3
Consider the following radial survey of a block of land. In the survey, $B$B is directly east of $A$A.
From the radial survey provided, which diagram would describe the block of land?
A
B
C
DCalculate the length of $AB$AB to the nearest metre.
Calculate the length of $BC$BC to the nearest metre.
Hence calculate the perimeter of the block of land.