Jaka EduTech
Jaka EduTech
Parent Sign In

© 2026 Jaka EduTech

Jaka EduTech

📖 Mathematics

Std 6
Curriculum Book Guide
1 2 3 4 5 6 7 8 9 10
Chapters
1 PATTERNS IN MATHEMATICS 2 Lines and Angles 3 Number Play 4 Data Handling and Presentation 5 Idli-Vada Game PRIME TIME 6 PERIMETER AND AREA 7 Fractions 8 Playing with Constructions 9 SYMMETRY 10 Integers (The Other Side of Zero)
6
Chapter 6
Skill: 60%

PERIMETER AND AREA

PERIMETER AND AREA

Chapter at a Glance

This chapter covers perimeter and area for plane figures, including rectangles, squares, triangles, and regular polygons. It introduces how to calculate boundaries and enclosed regions, and how to estimate areas of irregular shapes using grid square counting conventions. Through tangram puzzles, runners' starting-line staggers, floor plans, and area maze logic puzzles, students learn that shapes can have the same area but different perimeters, and vice versa.

Key Definitions & Terminology

  • Perimeter: The distance covered along the outer boundary of a closed plane figure when going around it once.
  • Area: The measure of the region enclosed by a closed plane figure, expressed in square units.
  • Regular Polygon: A closed 2D shape with all sides and all angles equal (e.g. equilateral triangles, squares, regular pentagons, regular hexagons).
  • Tangram: A Chinese puzzle made of a square cut into seven geometric shapes: 2 large triangles (A, B), 1 medium triangle (G), 2 small triangles (C, E), 1 square (F), and 1 parallelogram (D).
  • Straight Unit ($s$): A unit length along the grid line axes.
  • Diagonal Unit ($d$): The length of a diagonal line across a grid unit square, which is longer than a straight unit ($d = \sqrt{2} s \approx 1.41 s$).
  • Area Maze: A logic puzzle where one solves for a missing side length or area in a grid of rectangles without using fractional calculations.

Formulas, Rules & Properties

  • Perimeter Formulas:
  • General Polygon: Sum of all sides.
  • Rectangle: $P = 2 \times (l + b)$, where $l$ is length and $b$ is breadth.
  • Square: $P = 4 \times s$, where $s$ is the side length.
  • Equilateral Triangle: $P = 3 \times s$.
  • Regular $n$-gon: $P = n \times s$.
  • Area Formulas:
  • Rectangle: $A = l \times w$, where $l$ is length and $w$ is width.
  • Square: $A = s \times s = s^2$, where $s$ is the side length.
  • Triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$, derived as exactly half the area of a bounding rectangle.
  • Folding a Square in Half Rule:
  • If a square of perimeter $P$ is cut in half into two identical rectangles, the sum of the perimeters of both rectangles is always $1.5$ times the perimeter of the square:
    $$P_{\text{rect1}} + P_{\text{rect2}} = 1.5 \times P_{\text{square}}$$
  • Tangram Area Ratios (assuming small triangle C has area 1):
  • Shape C (small triangle) $= 1$
  • Shape E (small triangle) $= 1$
  • Shape D (parallelogram) $= 2$
  • Shape F (square) $= 2$
  • Shape G (medium triangle) $= 2$
  • Shape A (large triangle) $= 4$
  • Shape B (large triangle) $= 4$
  • Total Tangram Area $= 16$ units of Shape C's area.
  • Rectangles of a Fixed Area:
  • The rectangle closest to a square has the least perimeter.
  • The most elongated rectangle has the greatest perimeter.

Core Concepts & Topics

  • Diagonal vs. Straight Units on Grids:
  • Boundaries containing diagonal segments cannot be counted as simple grid steps. For instance, a triangle made of 6 grid-edge steps and 3 diagonal steps has a perimeter of $6s + 3d$ units, which is strictly greater than 9 straight units.
  • Grid-based Area Estimation Conventions:
  • Count fully enclosed squares as $1$ sq unit.
  • Ignore squares with less than half of their area enclosed.
  • Count squares with more than half of their area enclosed as $1$ sq unit.
  • Count squares with exactly half of their area enclosed as $0.5$ sq unit.
  • Why Squares are the Unit of Area:
  • Squares tile flat surfaces perfectly without leaving gaps or overlapping. Circles cannot be packed tightly without leaving empty spaces.
  • Charan vs. Sharan Floor Plans Comparison:
  • Charan's plot ($35\text{ ft} \times 30\text{ ft}$): Area $= 1050$ sq ft; Perimeter $= 130$ ft.
  • Sharan's plot ($42\text{ ft} \times 25\text{ ft}$): Area $= 1050$ sq ft; Perimeter $= 134$ ft.
  • Inference: Both plots have the same area, but Sharan's house has a larger perimeter because its shape is more elongated (further from a square).

Worked Examples

  • RECTANGLE WIRE TO SQUARE:
  • Question: A wire is bent to form a rectangle of dimensions $5\text{ cm} \times 3\text{ cm}$. If the wire is straightened and then bent to form a square, what is the side length of the square?
    Solution:
    • $P_{\text{rect}} = 2 \times (5 + 3) = 16\text{ cm}$.
    • Since the wire length remains the same, $P_{\text{square}} = 16\text{ cm}$.
    • Side of square $= 16 / 4 = \mathbf{4\text{ cm}}$.
  • FENCING ROTATIONS:
  • Question: A farmer has a rectangular field $230\text{ m}$ long and $160\text{ m}$ wide. He fences it with 3 rounds of rope. Find the total length of the rope.
    Solution:
    • $P = 2 \times (230 + 160) = 780\text{ m}$.
    • Length for 3 rounds $= 780 \times 3 = \mathbf{2340\text{ m}}$.
  • CARPET LAYING:
  • Question: A rectangular floor is $5\text{ m}$ long and $4\text{ m}$ wide. A square carpet of side $3\text{ m}$ is laid on the floor. Find the uncarpeted area.
    Solution:
    • Floor Area $= 5 \times 4 = 20\text{ sq m}$.
    • Carpet Area $= 3 \times 3 = 9\text{ sq m}$.
    • Uncarpeted Area $= 20 - 9 = \mathbf{11\text{ sq m}}$.
  • COCONUT GROVE SPACING:
  • Question: A rectangular grove is $100\text{ m} \times 50\text{ m}$. If each tree requires $25\text{ sq m}$, what is the maximum number of trees that can be planted?
    Solution:
    • Total Area $= 100 \times 50 = 5000\text{ sq m}$.
    • Maximum Trees $= 5000 / 25 = \mathbf{2 00\text{ trees}}$.
  • SUM OF AREAS RECTANGLE:
  • Question: Find the dimensions of a rectangle whose area is the sum of the areas of two rectangles: $5\text{ m} \times 10\text{ m}$ and $2\text{ m} \times 7\text{ m}$.
    Solution:
    • Total Area $= (5 \times 10) + (2 \times 7) = 50 + 14 = 64\text{ sq m}$.
    • Possible dimensions of the rectangle: $8\text{ m} \times 8\text{ m}$, $16\text{ m} \times 4\text{ m}$, or $32\text{ m} \times 2\text{ m}$.
  • 9 UNIT SQUARES CONFIGURATIONS:
  • Question: Using 9 unit squares, what are the smallest and largest possible perimeters if they form a single connected shape?
    Solution:
    • Smallest perimeter: $12\text{ units}$ (achieved by a $3 \times 3$ square layout).
    • Largest perimeter: $20\text{ units}$ (achieved by a $9 \times 1$ linear layout).
  • AREA MAZE SOLUTION:
  • Question: A grid has four quadrants. Top-left area $= 13\text{ sq cm}$, top-right area $= 15\text{ sq cm}$, bottom-left area $= 26\text{ sq cm}$. Find the bottom-right area.
    Solution:
    • Let the top row height be $h_1$ and bottom row height be $h_2$.
    • Let the left column width be $w_1$ and right column width be $w_2$.
    • $w_1 \times h_1 = 13$ and $w_1 \times h_2 = 26 \rightarrow h_2 / h_1 = 2$.
    • The bottom-right area is $w_2 \times h_2 = w_2 \times (2h_1) = 2 \times (w_2 \times h_1) = 2 \times 15 = \mathbf{30\text{ sq cm}}$.

Practical Activities & Experiments

  • Tangram Assembly: Cutting out the 7 tangram pieces and fitting them together to show that a square, a rectangle, and other shapes made from them all occupy the exact same area.
  • Newspaper Shapes Fencing: Cutting out irregular newspaper shapes, estimating their boundary lengths, and measuring them with string to verify perimeters.
  • Squared Paper Leaf Tracing: Tracing real leaves onto grid paper and applying the grid square conventions to estimate their surface areas.
  • Paper Fold Verification: Folding a paper square in half and cutting along the crease to prove that the sum of the perimeters of the two resulting rectangles is $1.5$ times the square's perimeter.
Previous Chapter 6 of 10 Next