Scale drawings, such as maps, are useful for finding measurements without physically measuring the real life object. We can find the perimeter or area of land directly by reading the map and applying the scale factor, and more - by using our knowledge of trigonometry and geometry we can obtain other measurements that aren’t otherwise given on the map.
Exploration
Finding the perimeter of a certain section of a scale drawing is performed in the same way as calculating the perimeter of a shape - sum up all the side lengths of the shape. The only difference is that the sum of the side lengths will be the real life perimeter of the shape, and not the perimeter of the shape on the scale drawing. Consider the following aerial view of a paddock that is bordered by a river:
 |
|
Aerial view of paddock |
If we wanted to fence the paddock, how much fencing would we need? We find the perimeter of the paddock by adding up each side length:
| Perimeter | $=$= | $530+320+440+210$530+320+440+210 | (Summing all edges) |
| $=$= | $1500$1500 m | (Simplifying the expression) |
In other words, we would need $1500$1500 metres of fencing.
Finding the area of the land from a scale drawing uses the same method as for any other shape. The only difference in this context is that we often encounter odd shapes that have no easy area formula. The paddock above is a good example - its top side follows the curved edge of a river bank, so we need to approximate the lengths involved by drawing straight lines. We effectively make trapeziums, and the more trapeziums we divide the area into, the more accurate our approximation will be.
In this case, we divide the paddock down the middle and make two trapeziums. Finding the area of a shape this way is called the trapezoidal rule.
| Trapezoidal rule | $=$= | $\frac{h\left(a+b\right)}{2}$h(a+b)2 | (Formula for the area of a trapezium) |
| Area 1 | $\approx$≈ | $\frac{220\left(210+258\right)}{2}$220(210+258)2 | (Substituting values for Trapezium 1 into formula) |
| $=$= | $51480$51480 m2 | (Simplifying the expression) | |
| Area 2 | $\approx$≈ | $\frac{220\left(258+320\right)}{2}$220(258+320)2 | (Substituting values for Trapezium 2 into formula) |
| $=$= | $63580$63580 m2 | (Simplifying the expression) | |
| Total Area | $=$= | Area 1 $+$+ Area 2 | (Formula for the total area) |
| $\approx$≈ | $51480+63580$51480+63580 | (Summing the areas of the trapeziums) | |
| $=$= | $115060$115060 m2 | (Simplifying the expression) |
We can go even further and use this rule to find volumes. A common application of this is finding rainfall, which is often reported as a height in millimetres. If we know the area, written as $A$A, and the given rainfall, written as $h$h, then we calculate the volume of rainfall over a given area using $V=Ah$V=Ah. In other words, the volume equals the area of the land multiplied by the rainfall. This calculation tells us how much water would be sitting above the land if the rain did not run off or pass through the ground.
When finding the volume of rainfall, we want to make sure the units for the height of the rainfall matches the units of the area. For instance, if the average rainfall of this month last year was $80$80 mm, we can calculate the volume of water on the paddock by using the formula:
| Volume | $=$= | $Ah$Ah | (Formula for the volume of rainfall) |
| $h$h | $=$= | $\frac{80}{1000}$801000 | (Converting the rainfall to metres) |
| $=$= | $0.08$0.08 m | (Simplifying the expression) | |
| Volume | $=$= | $115060\times0.8$115060×0.8 | (Substituting the values into formula) |
| $=$= | $9204.8$9204.8 m3 | (Simplifying the expression) |
That is a lot of water! Of course the water mostly is soaked up by the ground, but knowing this amount precisely is very important for agriculture, as well as disaster response in the case of a flood or drought.
Practice questions
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
The image below shows the outline of a construction site for the foundations of a large office building.
Find the area of the construction site in square metres.
Prolonged rainstorms bring $0.21$0.21 m of rain to the region and the foundations of the building are flooded.
What is the volume of water in the construction site in cubic metres?
Give your answer as a decimal.
question 3
A surveyor provided the following diagram with measurements for a property she was mapping out.
Find the approximate total area of the property, rounded to the nearest square metre, by using three applications of the trapezoidal rule.
The surveyor is reading a meteorological report that lists the average monthly rainfall in the region. According to the report, August sees $13.8$13.8 cm of rainfall on average.
What volume of rainfall can the surveyor expect to fall over the property next August? Give your answer to the nearest cubic metre.