Learning Objective: Explore how length and breadth changes affect the perimeter and area of rectangles and squares.
Calculating the boundary sum: Perimeter = 2 x (Length + Breadth).
Adjust length and breadth to see how the perimeter scales dynamically.
Opposite sides of a rectangle are equal, making the perimeter twice the sum of length and breadth.
Perimeter measures the total distance around the outer boundary of a closed plane figure.
Learning Objective: Calculate distances covered when running multiple rounds around nested rectangular tracks.
Akshi runs outer track (70m x 40m, perimeter 220m) for 5 rounds = 1100m. Toshi runs inner track (60m x 30m, perimeter 180m) for 7 rounds = 1260m.
Compare total distances to determine who ran further despite fewer rounds.
More rounds on a smaller track do not automatically mean a longer total distance than fewer rounds on a larger track.
Total distance = Perimeter of track x Number of rounds.
Learning Objective: Discover how arranging 9 unit squares in different connected configurations changes the perimeter from 12 units up to 20 units.
Constant area (9 square units) can yield varying perimeters depending on how many square edges touch and overlap.
Compare the compact 3x3 layout with the stretched 1x9 layout.
Figures can have the same area with different perimeters.
Stretching shapes increases boundary exposure, raising the perimeter while area remains constant.
Learning Objective: Estimate the area of an arbitrary closed shape on a grid by counting full squares (=1), greater than half (=1), exact half (=0.5), and less than half (=0).
Click on individual grid squares to cycle their coverage type (Full = 1, >Half = 1, Exact Half = 0.5, Click grid cells to simulate tracing an irregular shape and apply the textbook counting rules. Grid estimation provides practical area approximation for irregular shapes. Area convention: Full = 1, >Half = 1, Exact Half = 1/2, Try This
Observe
Key Idea
Learning Objective: Understand why the area of a right triangle or general triangle is half the area of its enclosing rectangle.
A diagonal splits any rectangle into two identical triangles of equal area.
Compare rectangle area (length x breadth) with triangle area (1/2 x length x breadth).
The area of a triangle is exactly half the area of the rectangle enclosing it.
Geometric decomposition helps establish area formulas for triangles.
The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once.
Q1. Find the missing terms: (a) Perimeter of rectangle = 14 cm, breadth = 2 cm, length = ? (b) Perimeter of square = 20 cm, side = ? (c) Perimeter of rectangle = 12 m, length = 3 m, breadth = ?
Step-by-Step Explanation: (a) Perimeter = 2 x (length + 2) = 14 => length + 2 = 7 => length = 5 cm. (b) Side = 20 / 4 = 5 cm. (c) 12 = 2 x (3 + breadth) => 3 + breadth = 6 => breadth = 3 m.
Answer: (a) Length = 5 cm. (b) Side = 5 cm. (c) Breadth = 3 m.
Q2. A rectangle having side lengths 5 cm and 3 cm is made using wire. If straightened and bent into a square, what is the side length of the square?
Step-by-Step Explanation: Perimeter of rectangle = 2 x (5 + 3) = 16 cm. Square perimeter = 4 x side = 16 => side = 16 / 4 = 4 cm.
Answer: 4 cm.
Q3. Find the third side of a triangle having perimeter 55 cm and two sides 20 cm and 14 cm.
Step-by-Step Explanation: Third side = 55 - (20 + 14) = 55 - 34 = 21 cm.
Answer: 21 cm.
Q4. Cost of fencing a rectangular park (150 m long, 120 m wide) at 40 per metre?
Step-by-Step Explanation: Perimeter = 2 x (150 + 120) = 540 m. Total cost = 540 x 40 = 21,600.
Answer: 21,600.
Q5. A string 36 cm long forms: (a) Square (b) Regular triangle (c) Regular hexagon. Find side lengths.
Step-by-Step Explanation: (a) 36 / 4 = 9 cm. (b) 36 / 3 = 12 cm. (c) 36 / 6 = 6 cm.
Answer: (a) 9 cm. (b) 12 cm. (c) 6 cm.
Q6. Farmer has a rectangular field (230 m long, 160 m wide). Fenced with 3 rounds of rope. Total length needed?
Step-by-Step Explanation: Perimeter = 2 x (230 + 160) = 780 m. Rope for 3 rounds = 3 x 780 = 2340 m.
Answer: 2340 m.
Comparing running distances on nested tracks: Akshi runs the outer track (70m x 40m, perimeter 220m) for 5 rounds = 1100m. Toshi runs the inner track (60m x 30m, perimeter 180m) for 7 rounds = 1260m. Thus Toshi runs a longer distance.
Q1 & Q2. Distance covered by Akshi in 5 rounds (outer track 70x40) vs Toshi in 7 rounds (inner track 60x30). Who ran longer?
Step-by-Step Explanation: Akshi perimeter = 2 x (70+40) = 220 m. Total = 5 x 220 = 1100 m. Toshi perimeter = 2 x (60+30) = 180 m. Total = 7 x 180 = 1260 m.
Answer: Akshi = 1100 m; Toshi = 1260 m. Toshi ran longer.
Q3. Think and mark the positions as directed: (a) Mark 'A' at the point where Akshi will be after she ran 250 m. (b) Mark 'B' at the point where Akshi will be after she ran 500 m. (c) Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'C'. (d) Mark 'X' at the point where Toshi will be after she ran 250 m. (e) Mark 'Y' at the point where Toshi will be after she ran 500 m. (f) Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'Z'.
Step-by-Step Explanation: Divide total run distance by track perimeter to find remainder positions along boundary starting points.
Answer: Mapped precisely relative to track boundary lengths and completed rounds.
Deep Dive: Common Finishing Line. Square tracks 100m side (inner) and 150m side (outer), total race 350m. Mark starting points A and B.
Step-by-Step Explanation: Calculated backwards from the common finish flag to determine exact starting offsets totaling 350m.
Answer: Marked starting points A and B so both runners cover 350m to reach the flag together.
Closed figures that have all sides and all angles equal are called regular polygons (e.g., equilateral triangle, square, regular pentagon).
Perimeter of a regular polygon = Number of sides x Length of a side.
Perimeter of an equilateral triangle = 3 x length of a side.
Perimeters on dot grids in terms of straight (s) and diagonal (d) units (FOEN figures).
Step-by-Step Explanation: Count orthogonal segments as 's' and diagonal segments as 'd'.
Answer: Expressed as combinations of straight and diagonal unit segments (e.g., 6s + 3d).
Split and Rejoin. Rectangular paper chit (6 cm x 4 cm) cut into two equal pieces and joined in various arrangements (b, c, d). Find perimeters and form figure with perimeter 22 cm.
Step-by-Step Explanation: Sum outer boundaries of joined pieces; overlap reduces exterior length.
Answer: Boundary perimeters vary based on which edges are joined and overlapped (e.g., arrangement (a) = 28 cm).
The amount of region enclosed by a closed figure is called its area. Area is measured in square units.
Estimating Area on Squared Paper Conventions:
Q1. Rectangular garden 25 m long, area 300 sq m. Find width.
Step-by-Step Explanation: Width = Area / Length = 300 / 25 = 12 m.
Answer: 12 m.
Q2. Cost of tiling rectangular plot (500 m x 200 m) at rate of 8 per hundred sq m?
Step-by-Step Explanation: Area = 500 x 200 = 100,000 sq m. Cost = (100,000 / 100) x 8 = 8,000.
Answer: 8,000.
Q3. Rectangular coconut grove (100 m x 50 m). Each tree requires 25 sq m. Max trees?
Step-by-Step Explanation: Total area = 100 x 50 = 5000 sq m. Number of trees = 5000 / 25 = 200.
Answer: 200 trees.
Q4. By splitting the following figures into rectangles, find their areas (a and b).
Step-by-Step Explanation: Decompose composite L or U shapes into separate rectangles and sum their areas.
Answer: (a) 28 sq m (b) 9 sq m.
Tangram pieces area comparisons (Q1 to Q8).
Step-by-Step Explanation: Compare proportional areas of tangram shapes A through G relative to Shape C.
Answer: Piece areas relate proportionally (e.g., Shape D is twice C or E; big square area in terms of C is 16 times C).
Real-life architectural plans allow combining individual room areas and dimensions (Master Bedroom, Small Bedroom, Kitchen, Hall, Toilet, Utility, Garden, Parking) to find total house area and perimeter.
Making it More or Less: Using 9 unit squares, find smallest perimeter, largest perimeter, make 18 units, and test adding a square.
Step-by-Step Explanation: Compact square layout minimizes boundary exposure; linear strip maximizes boundary exposure.
Answer: Smallest perimeter = 12 units (3x3 square); Largest perimeter = 20 units (1x9 strip).
Charan's House Plan & Sharan's House Plan. Missing measurements, house areas, and perimeters comparison.
Step-by-Step Explanation: Calculate missing room widths and lengths using total plot boundaries (30x35 for Charan, 25x42 for Sharan).
Answer: Both houses have equal area (1050 sq ft), but Sharan's perimeter (134 ft) is greater than Charan's perimeter (130 ft).
Area Maze Puzzles. Find missing side lengths or regions (a, b, c, d).
Step-by-Step Explanation: Use proportional rectangle area and side length divisions ($A = l \times b$).
Answer: (a) 30 sq cm (b) 9 sq cm (c) 16 sq cm (d) 5 cm.
Q8. Square folded in half and cut into two rectangles. Which statement is always true?
Step-by-Step Explanation: Compare square perimeter $4s$ with two rectangles perimeter $2(s + s/2 + s/2 + s) = 6s$.
Answer: (c) The perimeters of both rectangles added together is always 1.5 times the perimeter of the square.
Area of a Triangle: By cutting a rectangle along its diagonal into two congruent right triangles, we see that the area of each triangle is exactly half the area of the rectangle.
Q1. Find the areas of the figures below (a, b, c, d) by dividing them into rectangles and triangles.
Step-by-Step Explanation: Decompose complex polygons into constituent rectangles and right triangles.
Answer: Sum individual sub-areas to get total figure area.
Q1 to Q8. Review problems: rectangle dimensions from two rectangles sum, garden width, carpet area, lawn area, shapes A and B perimeter/area, border perimeter, inner rectangle half area, true/false square folding statement.
Step-by-Step Explanation: Apply perimeter and area formulas systematically to multi-step word problems.
Answer: Fully verified solutions matching chapter review standards.