Perimeter and Area · Lesson 7 of 7
Chapter Summary and Practice
“Connect boundary, enclosed space and rearrangement through a complete chapter review and mixed practice.”
• Choose perimeter or area from what a problem asks you to measure. • Connect all chapter formulas to boundary counting and square counting. • Use tracks, grids, triangles and decompositions to solve mixed problems. • Explain how area and perimeter behave when pieces are rearranged or joined. • Check dimensions, units and totals in floor plans and area puzzles.
Start with what is being measured
The whole chapter connects two questions: how far around, and how much space inside? A boundary problem asks for a length. A covering problem asks for an area. Even when the same rectangle appears in both, the calculation and units are different.
Fencing, lace, frame tape and full laps use perimeter. Carpet, lawn, tiling and room space use area. A layout can require both: first measure the space to cover and then the edge to fence. Read the requested quantity before choosing a formula.
| Quantity | Meaning | Typical units | Examples |
|---|---|---|---|
| Perimeter | Length of one complete boundary | cm, m, ft | Fencing, trim, one lap |
| Area | Amount of enclosed flat region | cm², m², ft² | Floor covering, grass, room space |
Problem
A garden is 18 m long and 12 m wide. Find its fence length and area.
- 1.Fence follows four sides: perimeter = 2 × (18 + 12) = 60 m.
- 2.The interior can be divided into 12 rows of 18 square metres: area = 18 × 12 = 216 m².
- 3.The same dimensions give different answers because the fence measures an edge and the grass covers a region.
Reconnect every formula to its reason
A formula is a shortcut for a relationship you can explain. Perimeter adds outside side lengths, while rectangular area counts equal rows. Equal sides simplify addition; a diagonal cut explains half a rectangle. Remembering these reasons helps you rebuild a formula when you are unsure.
| Shape or situation | Perimeter relationship | Area relationship |
|---|---|---|
| Any polygon | Add all outside sides once | Split into measurable regions when useful |
| Rectangle, length l and width w | P = 2(l + w) | A = l × w |
| Square, side s | P = 4s | A = s × s |
| Regular polygon, n sides of length a | P = n × a | No general area shortcut needed here |
| Triangle, base b and perpendicular height h | Add its three side lengths | A = ½ × b × h |
In this table P means perimeter and A means area. The letters l, w, s, a, b and h name lengths; n counts sides. Use the same length unit for dimensions in a calculation. A triangle’s sloping side is not automatically its height, and base plus height is not its perimeter.
Problem
A rectangle has perimeter 34 cm and width 5 cm. Find its length and area.
- 1.Half the perimeter is 34 ÷ 2 = 17 cm, which equals length + width.
- 2.Length = 17 − 5 = 12 cm.
- 3.Area = 12 × 5 = 60 cm².
- 4.Check perimeter: 2 × (12 + 5) = 34 cm. Finding a missing dimension lets us answer a second question.
Paths, grids and estimates
A perimeter can describe a route as well as a static edge. Remove complete lap lengths before locating a runner along the remaining sides. On a grid, treat straight edges and diagonal edges as different lengths. For area, count full squares and combine partial squares instead of counting outline segments.
Problem
A runner follows a 60 m by 30 m rectangle from bottom-right, travelling left first. Locate the runner after 500 m.
- 1.One lap is 2 × (60 + 30) = 180 m. Two laps use 360 m.
- 2.Remainder = 500 − 360 = 140 m. Travel 60 m along the bottom, then 30 m up the left side.
- 3.That uses 90 m, leaving 50 m along the top. The runner is 10 m before the top-right corner.
- 4.For a common finish on different tracks, trace the required distance backwards from the finish to locate different starts.
A triangle boundary containing six straight units and three diagonal units is 6s + 3d, with d greater than s. Its perimeter is therefore greater than nine straight units. Area uses a different idea: a diagonal through a unit square cuts its area into two equal halves. Do not confuse the diagonal’s length with the half-square’s area.
For a curved or irregular outline, trace it on square paper. Count full squares as 1, more-than-half squares as 1, exact halves as ½ and less-than-half squares as 0. This is an estimate because rounding changes some boundary contributions. Squares are useful area units because they fit without gaps; circles leave gaps, while suitable rectangles and triangles can also tile.
Estimate the perimeter of a cut-paper shape before measuring its sides. Separately estimate its area with square paper. Comparing each estimate with its own measurement helps you see why one outline can have a long boundary but a modest enclosed region.
Add, subtract and rearrange regions
A complicated region becomes manageable when you account for its pieces. Add non-overlapping pieces, subtract an opening from a whole, or use rectangle-and-triangle parts. Moving all pieces into another arrangement preserves total area, but hidden and exposed edges can change perimeter. These ideas connect carpets, tangrams, grid polygons and floor plans.
Problem
A courtyard is 10 m by 8 m and contains a square bed of side 2 m. Pave the remaining region at ₹30 per m² and fence only the outside boundary at ₹20 per metre. Find both costs.
- 1.Courtyard area = 10 × 8 = 80 m². Bed area = 2 × 2 = 4 m².
- 2.Paved area = 80 − 4 = 76 m²; paving cost = 76 × 30 = ₹2,280.
- 3.The outside perimeter is 2 × (10 + 8) = 36 m. The bed does not alter that outside boundary.
- 4.Fence cost = 36 × 20 = ₹720. The two stated jobs cost ₹2,280 + ₹720 = ₹3,000 together.
Problem
A 9 cm by 6 cm rectangle has a corner triangle removed. That triangle has perpendicular legs 3 cm and 4 cm. Find the remaining area.
- 1.Rectangle area = 9 × 6 = 54 cm².
- 2.Removed triangle area = ½ × 3 × 4 = 6 cm².
- 3.Remaining area = 54 − 6 = 48 cm².
- 4.This uses exact simple regions, rather than estimating boundary squares.
In the tangram, C and E each have one small-triangle unit of area; D, F and G have two; A and B have four. All seven total 16. A rectangle or another shape made from all the pieces has that same area if there are no overlaps or gaps. Two triangles on the same base and at the same perpendicular height also have equal areas, even when their side lengths and shapes differ.
Floor plans use aligned edges to reveal missing lengths. A known rectangle area divided by one side gives the other side. After calculating room areas, add every region and compare with the whole plot. Charan’s hall includes a 5 ft by 5 ft section above its main 20 ft by 12 ft part, making 265 ft². Accounting for that section makes all plot regions total 1,050 ft², matching Sharan’s different-shaped plot.
Equal area and equal perimeter are separate ideas
Different arrangements can keep one measurement while changing the other. Whole-number factor pairs let us compare rectangles with fixed area. Shared-edge counting lets us compare figures made of the same number of unit squares. These are two ways to explain why a larger area does not always need a longer boundary.
Problem
Give one example of equal areas with different perimeters and one of equal perimeters with different areas.
- 1.A 3 by 8 rectangle and a 4 by 6 rectangle both have area 24 square units. Their perimeters are 22 and 20 units.
- 2.A 1 by 5 rectangle and a 2 by 4 rectangle both have perimeter 12 units. Their areas are 5 and 8 square units.
- 3.The first comparison keeps the product fixed; the second keeps the sum of adjacent sides fixed.
- 4.For positive whole-number rectangles of fixed area, listing factor pairs helps find the smallest and largest perimeters within those choices.
Nine connected unit squares without holes always have area 9 square units. A 3 by 3 arrangement has minimum perimeter 12; a 1 by 9 row has maximum perimeter 20. Other arrangements can have perimeter 18, and more than one shape can share such a perimeter. Adding one more square along one, two or three full edges changes perimeter by +2, 0 or −2 while area always increases by one square unit.
Problem
Two 6 cm by 2 cm rectangles share a 5 cm edge without overlapping. Find their joined perimeter and area.
- 1.Each separate perimeter is 16 cm, so their total is 32 cm.
- 2.The shared edge is counted twice in that total, so joined perimeter = 32 − 2 × 5 = 22 cm.
- 3.Each area is 12 cm²; joined area = 12 + 12 = 24 cm².
- 4.Shared edges disappear from perimeter, while non-overlapping piece areas add.
Equal area does not imply equal perimeter, and equal perimeter does not imply equal area. Keep length units and square units distinct. For cut pieces, the sum of separate boundaries includes both copies of a cut edge; for joined pieces, internal contact edges are not outside boundary.
Quiz
A job charges per square metre of floor. Which measurement determines its cost?
A square has side 6 cm. Which pair gives perimeter and area in that order?
A 180 m lap is repeated for a total distance of 1,000 m. What are the full laps and remainder?
Two triangles share base 8 cm and perpendicular height 4 cm. What follows?
Which fixed-area-24 rectangle has the largest perimeter among positive whole-number dimensions?
An added unit square shares two full edges. What changes?
Two separate piece perimeters total 40 cm. Their shared contact is 4 cm. What is the joined perimeter?
Why should all room areas add to the whole plot area in these mathematical plans?
Practice Problems
- A 28 m by 16 m field needs three complete rounds of rope. Find the total rope length.
- A regular hexagon is made from a 54 cm string. Find its side length and explain your division.
- A rectangle has area 96 cm² and width 8 cm. Find its length and perimeter.
- A 12 m by 9 m floor has a 4 m square carpet. Find the uncovered area and its covering cost at ₹25 per m².
- A runner starts at the bottom-right of a 50 m by 30 m track and travels left first. Find the position after 390 m.
- Draw a gridded polygon containing both straight and diagonal edges. Express its perimeter using s and d and find its area by decomposition.
- Estimate a circle of diameter 3 grid units by counting squares. State which parts of the count are approximate.
- List all whole-number rectangles of area 32 cm² and order them by perimeter.
- Construct two nine-square connected figures with no holes and different perimeters. Can you find two different figures with perimeter 20? Explain.
- Arrange two 6 cm by 2 cm pieces with contact length 3 cm. Find joined perimeter and area.
- Find the area of a triangle with base 14 cm and perpendicular height 6 cm. Draw another triangle with the same base and height but a different top-vertex position.
- Find an inner rectangle of area 48 cm² fitting strictly inside a 12 cm by 8 cm rectangle. Explain the margins.
- A 12 cm square is folded and cut into two equal rectangles. Compare combined area and combined separate perimeters with the original.
- Make a rectangular two-room plan of total area 120 m². Give enough measurements to infer a missing length, then verify all areas.
- For a page of height 30 cm and width 21 cm, draw a border 1 cm from top and bottom and 1.5 cm from both sides. Find its perimeter.
- Charan and Sharan have plot dimensions 35 ft by 30 ft and 42 ft by 25 ft. Compare areas and outside perimeters, and explain why indoor totals must specify whether garden and parking are included.
Ask what quantity was requested, check that dimensions use the same unit, recompute by a second method when possible, and check the final unit. A sum of room areas should match the plot only when every plotted region is included once.
Key Takeaways
• Perimeter measures a boundary in length units; area measures an enclosed region in square units. • Rebuild rectangle and square formulas from side counting and rows of squares. • Triangle area uses half the base times its perpendicular height. • Tracks, diagonal grid edges and shared edges require careful boundary counting. • Add or subtract simple regions, infer missing dimensions, and check floor-plan totals. • Rearranging pieces can preserve area while changing perimeter; neither measurement determines the other. • Use grid estimates for curved regions and explain approximations.
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Area Puzzles and Floor Plans
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