Graphs of Motion · 7.2.2
Area under a speed-time graph
This standard asks students to understand that the area enclosed between a speed-time graph and the time-axis equals the distance travelled. Students break the region into rectangles, triangles or trapeziums, calculate each area using standard formulae, then sum them to find the total distance covered over a given time interval.
The official learning standard (7.2.2)
“Make a relationship between the area under a speed-time graph and the distance travelled, and hence determine the distance.”
What it means
This standard asks students to understand that the area enclosed between a speed-time graph and the time-axis equals the distance travelled. Students break the region into rectangles, triangles or trapeziums, calculate each area using standard formulae, then sum them to find the total distance covered over a given time interval.
How it is examined
This appears mainly in SPM Paper 2, where a speed-time graph is given or must be sketched, and students calculate the distance travelled during a stated time interval using the area under the graph. It may also appear in Paper 1 as a shorter question involving a simple triangular or rectangular region.
Worked example
A particle moves so that its speed increases uniformly from 0 m/s to 20 m/s in 5 seconds, remains constant at 20 m/s for the next 10 seconds, then decreases uniformly to 0 m/s in the final 5 seconds. Find the total distance travelled by the particle.
- Sketch the speed-time graph: it rises from (0, 0) to (5, 20), stays level to (15, 20), then falls to (20, 0).
- Area of acceleration stage (triangle) = ½ × 5 × 20 = 50 m.
- Area of constant-speed stage (rectangle) = 10 × 20 = 200 m.
- Area of deceleration stage (triangle) = ½ × 5 × 20 = 50 m.
- Total distance = 50 + 200 + 50 = 300 m.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why does the area under a speed-time graph give distance, not something else?
Because area = speed × time for each thin strip of the graph, and speed × time is exactly the formula for distance. Adding up every strip across the whole graph gives the total distance travelled during that time period.
What if the graph goes below the time-axis?
In SPM speed-time graphs, speed is never negative, so the graph never goes below the time-axis. A downward-sloping line means the object is decelerating (slowing down), not moving backwards, and the area under it is still calculated normally as distance.
How do I find the area of a trapezium-shaped region?
Use Area = ½ × (sum of the two parallel sides) × height, where the parallel sides are the two time intervals at the top and bottom of the trapezium and the height is the speed value. This is common when speed is constant for part of the motion.