On a speed-time graph, the area between the line and the time axis is the distance travelled. Speed × time gives distance, and area is height times width.
This skill extends finding acceleration from a gradient and completes the graph toolkit in motion and graphs.
Why does area give distance?
Take an object at a steady 5 m/s for 10 s. It travels 5 × 10 = 50 m. On the graph, this is a rectangle 10 units wide and 5 units tall, with area 50. The multiplication that gives distance is the same multiplication that gives the area.
When the line slopes, the speed changes, but the same idea holds. You split the shape into parts whose areas you can find.
How do I find the distance step by step?
- Split the region under the line into rectangles, triangles and trapezia.
- Read the dimensions of each shape from the axes: width in seconds, height in m/s.
- Find each area: rectangle = base × height, triangle = ½ × base × height, trapezium = ½ × (a + b) × width.
- Add the areas. The total is the distance in metres.
- Write the unit, which is metres because m/s × s = m.
Worked example
Use the invented scooter graph from the last lesson. Speed rises from 0 to 12 m/s in 6 s, stays at 12 m/s until 14 s, then falls to 0 m/s at 18 s.
0 to 6 s (triangle): area = ½ × 6 × 12 = 36 m.
6 to 14 s (rectangle): width = 14 − 6 = 8 s. Area = 8 × 12 = 96 m.
14 to 18 s (triangle): width = 4 s. Area = ½ × 4 × 12 = 24 m.
Total distance = 36 + 96 + 24 = 156 m.
Average speed for the whole 18 s = 156 ÷ 18 = 8.67 m/s (to 3 significant figures).
Check for sense: the average is below the maximum speed of 12 m/s, as it should be, because part of the journey is slower.
The mistake to watch for
A student wants the distance from the same graph and multiplies the maximum speed by the total time.
Mistaken answer: 12 × 18 = 216 m.
This treats the whole journey as a rectangle at maximum speed, but the scooter was not at 12 m/s all the time.
A second, related slip is to use a triangle without the ½, which doubles that piece. Correction: split into the three shapes above, apply the correct formula to each, and add. The total is 156 m, well below 216 m.
Check yourself
Try these, then open each answer.
1. An object’s speed rises steadily from 0 to 8 m/s in 5 s, then stays at 8 m/s for 10 s. Find the total distance.
Show answer
Triangle: ½ × 5 × 8 = 20 m. Rectangle: 10 × 8 = 80 m. Total = 20 + 80 = 100 m.
2. Speed rises steadily from 4 m/s to 10 m/s over 6 s. Find the distance using a trapezium, and check it with the average speed.
Show answer
Trapezium: ½ × (4 + 10) × 6 = ½ × 14 × 6 = 42 m. Check: average speed = (4 + 10) ÷ 2 = 7 m/s, and 7 × 6 = 42 m. Both agree.
3. On a curved speed-time graph, one small square is 2 s wide and 1 m/s tall. The region under the curve covers about 37 small squares. Estimate the distance.
Show answer
Each square represents 2 × 1 = 2 m. Distance ≈ 37 × 2 = 74 m. This is an estimate, because partial squares were counted by eye.
Where this leads next
The last lesson in this module adds direction: distinguishing displacement from total distance. The graph-model and residual explorer and the rate and energy graph interpreter let you check whether you chose a gradient or an area correctly.
Students who find graph questions slow often gain from online one-to-one Physics tuition, where a teacher can show how to choose the quickest shape split.