Slope is the y quantity divided by the x quantity, and area is the y quantity multiplied by the x quantity. Read the axes and their units, and the choice follows.
This page is for students who mix up slope and area on graph questions. It is one of the help pages for SPM Physics.
What does this sound like?
A student might say, “I found the gradient, but the question wanted the distance.” Or, “I know there are two things you can do with a graph, and I pick the wrong one.”
The cause is usually reading the graph by its shape, not by its axes.
What usually causes it?
Three academic causes show up.
- The rules are memorised for one graph type, for example velocity-time, and then applied to other graphs without checking the axes.
- Units are not checked, so the wrong calculation gives an answer in the wrong unit and nobody notices.
- The gradient of a straight line is known, but the area of a composite shape is avoided.
A worked example
A velocity-time graph for a cyclist has three parts. The velocity rises from 0 to 10 m s⁻¹ in 4 s, stays at 10 m s⁻¹ for 6 s, then falls to 0 in 2 s.
Slope: during the first 4 s, the slope is 10 ÷ 4 = 2.5 m s⁻². The unit m s⁻¹ divided by s is acceleration.
Area: the distance is the total area, which is ½ × 4 × 10 = 20 m, then 6 × 10 = 60 m, then ½ × 2 × 10 = 10 m. The total is 20 + 60 + 10 = 90 m, because m s⁻¹ times s is metres.
The same graph answers two questions, and the units tell you which calculation each one needs.
What can you do on your own?
Make the axes decide. Write the units of the slope and the area before calculating anything.
| Graph | Slope gives | Area gives |
|---|---|---|
| Displacement-time | Velocity | Nothing useful |
| Velocity-time | Acceleration | Displacement |
| Force-extension | Spring constant | Elastic energy stored |
| Force-distance | Not needed | Work done |
For a straight line, review calculating a straight line gradient. To see how the slope and the area change as a graph changes, try the motion graph explorer. For graphs from experiments, see explaining a supplied practical result.
When might one-to-one tuition help?
Graph skills improve with deliberate practice, and most of that can be done alone with the table above. If you still switch between slope and area on new graph types, you may need to see a worked example with someone who asks why.
A one-to-one teacher can give you an unfamiliar graph and ask you to predict the unit of the answer before calculating anything. That makes the reasoning visible so that it can be corrected. See online one-to-one Physics tuition or the one-hour trial class (from RM50).
What to read next
Practise the written side of graph questions in explaining a supplied practical result.