Perimeter and Area · Lesson 2 of 7
Perimeter in Paths and Changing Shapes
“Follow runners around tracks and see how diagonal edges and shared edges affect boundary length.”
• Compare distances on tracks with different lap lengths. • Locate positions after complete laps and a remaining distance. • Distinguish straight grid units from diagonal grid units. • Generalise perimeter for regular polygons. • Explain how joining shapes removes shared edges from the boundary.
Lap length and total distance
A runner on a larger track travels farther in each lap, but another runner may complete more laps. Neither track size nor lap count alone decides the total distance. We must calculate the perimeter of each track and then count how many times it is travelled.
Problem
Akshi completes 5 laps of a 70 m by 40 m track. Toshi completes 7 laps of a 60 m by 30 m track. Compare their distances.
- 1.Akshi’s lap is 2 × (70 + 40) = 220 m. Her total is 5 × 220 = 1,100 m.
- 2.Toshi’s lap is 2 × (60 + 30) = 180 m. Her total is 7 × 180 = 1,260 m.
- 3.Toshi runs 1,260 − 1,100 = 160 m farther, even though her track is smaller.
Where is a runner after part of a lap?
Returning to the start uses one whole perimeter. After removing all complete laps from a distance, only the remaining distance determines the runner’s position. Follow that remainder along the sides in the stated direction; do not treat it as a straight line across the track.
Problem
From the outer bottom-right start, Akshi runs left, up, right and down. Where is she after 1,000 m?
- 1.Four complete laps use 4 × 220 = 880 m. Five would use 1,100 m, which is too far.
- 2.The remaining distance is 1,000 − 880 = 120 m.
- 3.Travel 70 m along the bottom and 40 m up the left side. This uses 110 m, leaving 10 m.
- 4.She is on the top side, 10 m to the right of its top-left corner.
The same method works for every distance. Starting points and directions matter: a position described as “10 m along the top” is meaningful only when we know which corner the runner reached first. The table gives positions for both runners using the directions in the track diagram.
| Runner and distance | Complete laps | Remainder | Position after the remainder |
|---|---|---|---|
| Akshi: 250 m | 1 | 30 m | Bottom side, 30 m left of her start |
| Akshi: 500 m | 2 | 60 m | Bottom side, 60 m left of her start |
| Akshi: 1,000 m | 4 | 120 m | Top side, 10 m right of top-left corner |
| Toshi: 250 m | 1 | 70 m | Left side, 10 m above bottom-left corner |
| Toshi: 500 m | 2 | 140 m | Top side, 50 m right of top-left corner |
| Toshi: 1,000 m | 5 | 100 m | Top side, 10 m right of top-left corner |
Problem
Two square tracks have sides 100 m and 150 m. Runners travel 350 m, finishing at the middle of the bottom side, moving right along that bottom side. Find their starts.
- 1.The inner perimeter is 400 m. Begin at the inner bottom-right corner: up 100 m, left 100 m, down 100 m, then right 50 m. The total is 350 m.
- 2.The outer runner finishes by going 75 m right along the bottom after travelling 150 m down the left side. These last parts total 225 m.
- 3.Before them, travel 350 − 225 = 125 m left along the top. Therefore start 125 m right of the top-left corner, or 25 m left of the top-right corner.
- 4.Trace both routes to check that their total lengths match despite their different starts.
A diagonal grid edge is longer
On a square grid, a horizontal or vertical edge of one square is one straight unit. A diagonal crosses that square from one corner to the opposite corner. It is longer than one straight edge. Counting the number of segments is therefore not enough when their lengths differ.
Here s means the length of one horizontal or vertical grid edge; d means the length of a diagonal across one grid square. These symbols name two different lengths, with d greater than s.
A right triangle spanning three grid edges horizontally and three vertically has two straight sides totalling 6s. Its sloping side contains three one-cell diagonal segments, totalling 3d. Its perimeter is 6s + 3d. Because d is longer than s, this is greater than nine straight units; no diagonal-length formula is needed to see why.
Problem
Find the perimeter of outline A in the diagram in straight and diagonal units.
- 1.Follow the boundary: 3 straight units along the top, then 1 diagonal unit down-left.
- 2.Continue 1 straight unit left, 1 straight unit down, and 1 diagonal unit down-left.
- 3.Return up the long left side: 3 straight units.
- 4.The total is (3 + 1 + 1 + 3)s + (1 + 1)d = 8s + 2d.
| Outline | Straight units | Diagonal units | Perimeter |
|---|---|---|---|
| A | 8 | 2 | 8s + 2d |
| B | 4 | 6 | 4s + 6d |
| C | 12 | 6 | 12s + 6d |
| D | 18 | 6 | 18s + 6d |
A one-square diagonal and a one-square edge are not equal lengths. Keep s and d separate; adding their counts as though every segment were the same unit underestimates the perimeter.
Regular polygons and boundary measurement
Equal side lengths make a perimeter calculation simpler. A regular polygon has both equal sides and equal angles, so its perimeter repeats the same side length. The number of repetitions is the number of sides. This generalises the square and equilateral triangle shortcuts.
A polygon whose sides are all equal in length and whose angles are all equal.
An equilateral triangle has three equal sides, so its perimeter is 3 × side length. A regular pentagon has five equal sides, so its perimeter is 5 × side length. A regular hexagon has six. Let n name the number of sides and a name the length of each side.
Problem
A regular hexagon has each side 7 cm. Find its perimeter.
- 1.A hexagon has six sides. Regularity tells us all six measure 7 cm.
- 2.Perimeter = 6 × 7 = 42 cm.
- 3.Check by imagining six copies of the 7 cm length placed end to end.
Cut a few irregular shapes from scrap paper. For each shape, record an estimated perimeter before measuring. Measure every straight edge with a ruler; use a flexible tape along curved parts. Add the measured lengths, compare with your estimate and explain which boundary parts you underestimated. Look for regular polygon objects around you and check whether their sides really match.
Cutting and rejoining changes the boundary
Cutting a rectangle creates new exposed edges. Joining pieces hides some edges where the pieces touch. The material can stay exactly the same while its outside boundary changes. Trace only the outline of the joined figure, including inward corners.
Cut a 6 cm by 4 cm rectangle into two 6 cm by 2 cm rectangles. Each small piece has perimeter 2 × (6 + 2) = 16 cm, so the separate boundaries total 32 cm. If they share a contact edge of length c cm, that length appears once on each separate piece. Both copies disappear from the outside boundary when joined.
Problem
Join the two 6 cm by 2 cm pieces end to end, then compare with an L arrangement, a T arrangement and a staggered arrangement sharing 3 cm.
- 1.End to end, the shared edge is 2 cm. Perimeter = 32 − 2 × 2 = 28 cm; this also matches a 12 cm by 2 cm rectangle.
- 2.An L arrangement or a T arrangement with contact length 2 cm also has perimeter 28 cm. Their appearance differs, but the same length of boundary is hidden.
- 3.If the shared contact length is 3 cm, perimeter = 32 − 2 × 3 = 26 cm.
- 4.The contact length, rather than the shape name, determines the change.
Problem
Arrange the same two pieces to make perimeter 22 cm.
- 1.The separate perimeters total 32 cm. We need to remove 32 − 22 = 10 cm from that total.
- 2.Each centimetre of contact removes two centimetres of separate boundary, so contact must be 10 ÷ 2 = 5 cm.
- 3.Place one horizontal piece above the other and slide it 1 cm sideways. Their 6 cm edges now share 5 cm, giving perimeter 22 cm.
Quiz
How far are seven laps of a 60 m by 30 m track?
A lap is 220 m. After 500 m, what distance remains after complete laps?
Which expression measures a triangle with six straight and three diagonal grid units?
A regular pentagon has side 8 cm. What is its perimeter?
Two pieces have separate perimeters totalling 32 cm and share a 3 cm edge. What is the joined perimeter?
Why are different starting points needed for the 350 m race?
Practice Problems
- Find Akshi’s and Toshi’s positions after 750 m, using the diagram’s starts and directions.
- On a square track of side 80 m, start at the bottom-right and travel left first. Locate the position after 500 m.
- Trace outlines B, C and D and verify their straight and diagonal counts.
- Draw a regular octagon with side 3 cm written on every side. Find its perimeter.
- Build L and T outlines from two 6 cm by 2 cm paper pieces. Measure their contact edges and verify perimeter 28 cm.
- Estimate three cut-paper boundaries, measure them and record the differences.
Key Takeaways
• Distance travelled equals lap length times lap count, plus any partial lap. • Remove complete laps and follow the remaining distance along the actual route. • Straight grid units and diagonal grid units have different lengths. • A regular polygon has perimeter equal to side count times side length. • Joining pieces removes the shared edge twice from their separate perimeter total.