On a speed-time graph, the area under the line is the distance travelled and the gradient is the acceleration. Both come straight from the units: speed × time is distance, and change in speed ÷ time is acceleration.
This skill appears in journey questions where a vehicle speeds up, cruises and slows down. It builds on reading a distance-time graph.
How do you find a distance from the graph?
- Check the units. Speed and time must match, for example m/s with s. If they do not, convert first using the speed unit lesson.
- Split the area into simple shapes. Triangles, rectangles and trapeziums cover most graphs.
- Work out each area. Triangle: ½ × base × height. Rectangle: base × height. Trapezium: ½ × (sum of parallel sides) × width.
- Add the parts. The total is the distance.
- For acceleration, use the gradient. Change in speed ÷ change in time.
Worked example
A car’s speed is 0 at time 0. It increases steadily to 12 m/s at 6 s, stays at 12 m/s until 16 s, then decreases steadily to 0 at 20 s.
Part 1, 0 to 6 s (triangle): ½ × 6 × 12 = 36 m.
Part 2, 6 s to 16 s (rectangle): 10 s at 12 m/s = 10 × 12 = 120 m.
Part 3, 16 s to 20 s (triangle): ½ × 4 × 12 = 24 m.
Total distance: 36 + 120 + 24 = 180 m.
Average speed: 180 ÷ 20 = 9 m/s.
Acceleration in the first part: 12 ÷ 6 = 2 m/s². Deceleration in the last part: 12 ÷ 4 = 3 m/s².
Check by a different route: the whole shape is a trapezium with parallel sides 20 s and 10 s and height 12. Area = ½ × (20 + 10) × 12 = 180 m. The two methods agree.
The mistake to watch for
A common slip is to multiply the greatest speed by the total time.
Mistaken answer: 12 × 20 = 240 m
The student treated the whole journey as if the car travelled at 12 m/s all the time. It spent part of the time going slower.
The correction is to find the area of the actual shape, which is smaller than the rectangle around it: 180 m. A quick look at the shape tells you the answer must be less than 240.
Check yourself
Try these, then open each answer.
1. Speed rises steadily from 0 to 8 m/s in 4 s, then stays at 8 m/s for 5 s. Find the total distance.
Show answer
Triangle: ½ × 4 × 8 = 16 m. Rectangle: 5 × 8 = 40 m. Total = 16 + 40 = 56 m.
2. A train travels at a constant 60 km/h for 45 minutes. Find the distance in km.
Show answer
Convert the time: 45 min = 0.75 h. Distance = 60 × 0.75 = 45 km.
3. A motorbike’s speed rises steadily from 10 m/s to 30 m/s in 8 s. Find the distance and the acceleration.
Show answer
Distance = ½ × (10 + 30) × 8 = 160 m. Acceleration = (30 − 10) ÷ 8 = 20 ÷ 8 = 2.5 m/s².
Where this leads next
Next, distinguishing an average rate from an instantaneous reading uses both graph types together. Try the graph-model explorer and the quadratic structure explorer for extra graph reading, and return to the module page.
Some students get the areas right and still forget to say what the number means. Our teachers look for that in online one-to-one Mathematics tuition.