Part 1: Composite Shapes and Circles

Instructions: Show all working. Give units with every answer. Use π = 3.14 unless an exact form is requested. Diagrams are not drawn to scale.

Success criteria: I can decompose irregular shapes, select suitable formulas, calculate unknown dimensions, and explain whether an answer is reasonable.

Formula reminder: Rectangle area = length × width; rectangle perimeter = 2 × (length + width).

Triangle area = 1/2 × base × perpendicular height; trapezium area = 1/2 × (sum of parallel sides) × height.

Circle circumference = 2πr; circle area = πr²; semicircle area = 1/2πr².

1. Find the area and perimeter of the rectangle.

11 cm

┌──────────────┐

6 cm│ │

└──────────────┘

2. The L-shape is made from two rectangles. Find its total area and perimeter.

10 m

┌────────────────┐

│ │ 4 m

9 m │ └──────┐

│ │ 5 m

└───────────┐ │

└───────────┘

6 m

3. Calculate the area and perimeter of the trapezium.

14 cm

┌──────────┐

5 cm│ │

└──────────────┘

22 cm

4. A rectangle has length 12 cm and width 7 cm. A semicircle with diameter 7 cm is attached to one 7 cm side. Find the outside perimeter and total area.

12 cm

┌────────────────┐

7 cm│ │

└───────╮ ╭─────┘

╰──╯

semicircle, diameter 7 cm

5. A circular pond is cut from a 16 m by 10 m rectangular garden. The pond has radius 3 m. Find the shaded garden area remaining.

16 m

┌────────────────────┐

10 m │ ◯ r = 3 m │

└────────────────────┘

6. Each small square in the irregular polygon diagram measures 1 cm by 1 cm. Find the area and perimeter of the polygon.

┌───────┐

│ └───┐

┌───┘ │

│ │

└───────────────┘

Grid coordinates: (0,0), (0,2), (3,2), (3,4), (6,4), (6,0)

Part 2: Algebra, Scale and Reasoning

7. A rectangle has area 96 cm² and width 8 cm. Find its length and perimeter.

8. A rectangle has length (x + 4) cm and width x cm. Its perimeter is 32 cm. Find x, then calculate the rectangle’s area.

9. A circular running track has radius 14 m. Find its circumference to the nearest metre and its area to the nearest square metre. Use π = 22/7 and give the area in exact form before rounding.

10. A park map uses a scale of 1 cm : 25 m. A rectangular garden measures 7.2 cm by 4.8 cm on the map. Find its actual dimensions, area and perimeter.

11. Sam says, “If the radius of a circle doubles, its area doubles.” Is Sam correct? Explain using the area formula and a numerical example.

12. Extension challenge. Design a composite shape with an area between 150 cm² and 170 cm² and a perimeter between 50 cm and 60 cm. Include at least one curved edge. Label all dimensions, calculate the exact area and perimeter using π = 3.14, and justify that your design meets both conditions.

Differentiation supports

Support: Draw auxiliary lines to split each composite shape into rectangles, triangles or circles. Highlight known lengths and use a formula list. Check that all lengths use the same unit before calculating.

Advanced learners: Solve questions 8, 11 and 12 algebraically, compare two possible decompositions, and investigate how changing one dimension affects area and perimeter.

Answer Key

1. Area = 11 × 6 = 66 cm². Perimeter = 2(11 + 6) = 34 cm.

2. Split into 10 m × 4 m and 6 m × 5 m rectangles. Area = 40 + 30 = 70 m². Missing vertical side = 9 m and missing horizontal side = 10 − 6 = 4 m. Perimeter = 10 + 4 + 6 + 5 + 4 + 9 = 38 m.

3. Area = 1/2(14 + 22) × 5 = 90 cm². The sloping side difference is 8 cm; using the diagram’s right-triangle split gives a sloping length of approximately 9.43 cm. Perimeter = 14 + 22 + 5 + 9.43 = 50.43 cm approximately.

4. Semicircle radius = 3.5 cm. Rectangle area = 84 cm². Semicircle area = 1/2 × 3.14 × 3.5² = 19.235 cm². Total area = 103.235 cm². Outside perimeter = 12 + 12 + 7 + 3.14 × 3.5 = 41.99 cm.

5. Rectangle area = 16 × 10 = 160 m². Pond area = 3.14 × 3² = 28.26 m². Shaded area = 160 − 28.26 = 131.74 m².

6. Using the grid, area = (3 × 2) + (3 × 4) = 18 cm². Perimeter = 3 + 2 + 3 + 4 + 6 + 2 = 20 cm.

7. Length = 96 ÷ 8 = 12 cm. Perimeter = 2(12 + 8) = 40 cm.

8. 2(x + 4) + 2x = 32; 4x + 8 = 32; x = 6. Dimensions are 10 cm by 6 cm. Area = 60 cm².

9. Circumference = 2 × 22/7 × 14 = 88 m, so 88 m. Area = 22/7 × 14² = 616 m², so 616 m².

10. Actual length = 7.2 × 25 = 180 m. Actual width = 4.8 × 25 = 120 m. Area = 180 × 120 = 21,600 m². Perimeter = 2(180 + 120) = 600 m.

11. Sam is incorrect. Area depends on r². If r = 2 cm, area = 12.56 cm²; if r = 4 cm, area = 50.24 cm², which is four times as large, not twice as large.

12. Answers will vary. A valid design must have an area from 150 cm² to 170 cm² and a perimeter from 50 cm to 60 cm, with clearly labelled dimensions and correct calculations.

Final reflection

Which strategy helped you most: decomposing a shape, using a formula, writing an equation, or checking whether your answer was reasonable? Explain briefly.