The67Sprint — LearningMap: Perimeter and Area

📏 Perimeter — The Boundary

Perimeter is the total distance covered going once around the outer boundary of a closed figure
Rectangle: 2 × (Length + Breadth)
Square: 4 × side
Equilateral Triangle: 3 × side
Any Regular Polygon: Number of sides × Side length
Two figures can share the same perimeter but have different areas, or share the same area but have different perimeters

🟩 Area — The Surface Region

Area is the amount of flat surface enclosed inside a closed 2D figure
Rectangle: Length × Breadth
Square: Side × Side
A triangle cut from a rectangle's diagonal has exactly half the rectangle's area

🔲 Grid-Paper Estimation

Full and more-than-half enclosed squares count as 1
Exactly half-enclosed squares count as ½
Less-than-half enclosed squares count as 0 — never ½
Squares tile without gaps, which is why they're used to measure area

1 · ½ · 0

Memory Cheat Code

Same P, Different A: A fixed perimeter can enclose many different areas — closer to a square gives more area, longer and thinner gives less.

Quick Recall — Sprint Facts

Rectangle P = 2(L+B) · A = L×B

Square P = 4S · A = S×S

Fixed String 36 cm string → 9 cm square side, 12 cm triangle side, 6 cm hexagon side

Unit Conversion 1 m² = 10,000 cm²

🔑 Key Terms to Know

Perimeter

The distance around a closed figure

Area

The square units contained within a closed 2D figure

Unit Square

A square of side 1 unit, with area 1 square unit

Regular Polygon

A closed figure with all sides equal and all angles equal

🏆 Olympiad & Challenger Edge

Algebraic Sides

If a rectangle's sides are given as expressions like 6x+9 and x+8, use the perimeter or area equation to solve for x, then find the actual dimensions

Pythagorean Diagonals

Given one side and the diagonal, find the missing side using Length = √(Diagonal² − Breadth²), then compute the area

Hexagon & Octagon Areas

A regular hexagon's area splits into 6 equilateral triangles: (3√3/2) × side². A regular octagon uses 2 × side² × (1+√2)

Tangram Ratios

In the 7-piece tangram, the big square is 16× the smallest triangle (Shape C); Shape D is exactly twice the area of Shape C

Unit Mismatch

Always convert length and breadth to the same unit before calculating — never multiply metres by centimetres directly

Grid-Counting Error

Counting less-than-half squares as ½ instead of 0 inflates the area — this is the most common grid mistake

Joined Shapes Mistake

When two shapes are joined, the shared inner boundary disappears — don't add both sides to the perimeter

Skipping the Formula

Always write the formula first, then substitute values — jumping straight to numbers loses working marks

Boxing Ring Ropes

A square ring with two ropes per side needs 2 × perimeter of rope in total

House Floor Plans

Architects use room dimensions like a 15 ft × 15 ft bedroom to calculate total house area and individual room perimeters

Tiling & Carpeting

Finding how many 20 cm × 20 cm tiles cover a 3 m × 5 m floor uses area, not perimeter

Farming Grove Spacing

A farmer plants trees needing 25 sq m each — total area divided by space-per-tree gives the maximum count

Wrong Unit Labels

Perimeter answers need linear units (cm, m) — area answers need square units (cm², m²). Examiners penalise swaps

The Tricky Square

Attaching a new square to a shape can increase, decrease, or leave the perimeter unchanged, depending on where it's placed

Constant P ≠ Constant A

Bending the same wire into a different shape keeps the perimeter fixed, but the enclosed area still changes

Cut Paper Myth

Cutting a piece out of paper always reduces area — but the perimeter can actually increase, not decrease

Logic Chain — How to Choose Perimeter or Area

Step 1

Identifying the shape and checking that length and breadth share the same unit connects to knowing which formula applies

Step 2

Recognising whether the question asks for a border, fence, or boundary leads to choosing Perimeter = 2(L+B)

Step 3

Recognising whether the question asks for surface, tiling, or planting space leads to choosing Area = L×B

Step 4

Computing the answer connects to appending the correct unit — linear for perimeter, square for area

🦉

Mentor Byte

"Perimeter is the baseboard running along the walls; Area is the floor tiles you walk on. For tricky compound rooms, use Chop and Drop — slice the shape into rectangles with a dotted line, find each area, then add them together. That one image outlasts a dozen formulas."

For Parents

Give your child a length of string or ribbon and let them bend it into a square, then a triangle, then a hexagon on a table. The perimeter never changes, but ask them to guess which shape encloses the most space — then check by counting grid squares underneath.