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The main difference is that using graph paper will give less precise measurements. Also, due to human errors, using graph paper is not reliable; results vary between different attempts to measure the same leaf. This is because grid squares that are partly included in the area must be approximated so introduce inaccuracies.

Leaf Area Meters are scientific instrument that are specially designed to measure the area of leaves. E.g. AM300 Leaf Area Meter

My AM300 can store the image, and will analyse it to measure all sorts of useful parameters, like perimeter. I need a lightweight instrument which I can carry for hours, and it only weighs 1.8kg . Other models which are called 'portable' weigh 6.8kg!

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Q: What is differences in leaf areas using a graph paper and leaf area meter?
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