Worksheet - Composite Figures

Total Marks: / 40 marks

Booklet A - Multiple Choice Questions (MCQ)

Each question carries 1 mark. Circle the letter of the correct answer.

Q1: A composite figure is made up of 2 identical squares. Each square has a side of 6 cm. What is the perimeter of the composite figure?

1. 24 cm
2. 30 cm
3. 36 cm
4. 48 cm

Q2: A rectangle measuring 12 cm by 8 cm has a square of side 3 cm cut out from one corner. What is the area of the remaining figure?

1. 87 cm²
2. 96 cm²
3. 105 cm²
4. 78 cm²

Q3: The figure below is made up of a rectangle and a triangle. The rectangle measures 10 cm by 6 cm. The triangle has a base of 10 cm and a height of 4 cm. What is the total area of the composite figure?

1. 60 cm²
2. 80 cm²
3. 100 cm²
4. 120 cm²

Q4: A composite figure is formed by joining a square of side 8 cm and a rectangle of length 12 cm. If the perimeter of the composite figure is 56 cm, what is the breadth of the rectangle?

1. 4 cm
2. 6 cm
3. 8 cm
4. 10 cm

Q5: A figure is made up of 3 identical rectangles, each measuring 5 cm by 2 cm, arranged end to end in a straight line. What is the perimeter of the composite figure?

1. 34 cm
2. 30 cm
3. 24 cm
4. 19 cm

Q6: The figure is made up of two identical semicircles and a rectangle. The rectangle measures 14 cm by 10 cm. Each semicircle has a diameter equal to the breadth of the rectangle. What is the area of the composite figure? (Take \(\pi = \frac{22}{7}\))

1. 140 cm²
2. 218.5 cm²
3. 297 cm²
4. 218 cm²

Q7: A rectangular piece of cardboard measuring 20 cm by 15 cm has four identical squares, each of side 2 cm, cut from its four corners. What is the perimeter of the remaining cardboard?

1. 70 cm
2. 54 cm
3. 62 cm
4. 66 cm

Q8: A composite figure is made by removing a triangle from a rectangle. The rectangle measures 16 cm by 9 cm. The triangle removed has a base of 8 cm and a height of 9 cm. What is the area of the remaining figure?

1. 108 cm²
2. 72 cm²
3. 144 cm²
4. 36 cm²

Booklet B - Short Answer Questions (SAQ)

Show your working clearly. Questions carry 2 marks unless stated.

Q9: A composite figure is made up of a square and a rectangle. The square has a side of 7 cm. The rectangle has a length of 10 cm and the same breadth as the side of the square. Find the total area of the composite figure.

Q10: A figure is formed by joining two rectangles. The first rectangle measures 15 cm by 8 cm. The second rectangle measures 9 cm by 8 cm and shares one side with the first rectangle. Calculate the perimeter of the composite figure.

Q11: The figure is made up of a rectangle and a triangle. The rectangle measures 18 cm by 10 cm. The triangle has a base of 18 cm and a height of 6 cm. Find the area of the composite figure.

Q12: A square tile has a side of 12 cm. A smaller square of side 4 cm is removed from one corner. What is the perimeter of the remaining tile?

Q13: A rectangular garden measuring 25 m by 18 m has a rectangular path of width 2 m running around its inside edge. Find the area of the path.

Q14: A composite figure is made by placing a triangle on top of a rectangle. The rectangle has a length of 20 cm and a breadth of 12 cm. The triangle has the same base as the length of the rectangle and a height of 8 cm. Calculate the total area of the figure.

Q15: The figure is made up of 5 identical squares arranged in a cross pattern. Each square has a side of 4 cm. Find the perimeter of the composite figure.

Q16: A rectangular field measures 50 m by 30 m. A triangular flower bed with a base of 15 m and a height of 10 m is planted in one corner of the field. Find the area of the field that is not covered by the flower bed. [3 marks]

Booklet B - Long Answer / Problem Sums (LAQ)

Show all working clearly. Marks are awarded for method and final answer.

Q17: Mrs Tan wants to tile her kitchen floor. The floor is in the shape of a rectangle measuring 6 m by 4 m. There is a rectangular area measuring 2 m by 1.5 m that does not need to be tiled because there is a kitchen cabinet there. Each tile is a square of side 20 cm. How many tiles does Mrs Tan need to buy? [4 marks]

Q18: A composite figure is formed by joining a square and a semicircle. The square has a side of 14 cm. The semicircle is attached to one side of the square, with the diameter of the semicircle equal to the side of the square. Find the perimeter of the composite figure. (Take \(\pi = \frac{22}{7}\)) [4 marks]

Q19: A rectangular piece of paper measures 40 cm by 28 cm. Ali cuts out 6 identical circles from the paper, each with a radius of 4 cm. What is the area of the paper that remains? (Take \(\pi = \frac{22}{7}\)) [5 marks]

Q20: The figure shows a rectangular hall measuring 30 m by 20 m. A stage in the shape of a semicircle is built along one of the shorter sides of the hall. The diameter of the semicircular stage is equal to the width of the hall. The floor of the hall, excluding the stage, is to be carpeted at a cost of $45 per square metre. The perimeter of the stage (excluding the straight edge) is to be decorated with LED lights at a cost of $12 per metre. Find the total cost. (Take \(\pi = 3.14\)) [5 marks]

Answer Key

Multiple Choice Questions

Detailed Solutions

Q1: A composite figure is made up of 2 identical squares. Each square has a side of 6 cm. What is the perimeter of the composite figure?

Ans: C
Explanation: When 2 identical squares are joined, they share one common side.
Total number of sides exposed = 6 sides (not 8, because 2 sides are joined together)
Perimeter = \(6 \times 6 = 36\) cm

Q2: A rectangle measuring 12 cm by 8 cm has a square of side 3 cm cut out from one corner. What is the area of the remaining figure?

Ans: A
Explanation: Area of rectangle = \(12 \times 8 = 96\) cm²
Area of square cut out = \(3 \times 3 = 9\) cm²
Remaining area = \(96 - 9 = 87\) cm²

Q3: The figure below is made up of a rectangle and a triangle. The rectangle measures 10 cm by 6 cm. The triangle has a base of 10 cm and a height of 4 cm. What is the total area of the composite figure?

Ans: B
Explanation: Area of rectangle = \(10 \times 6 = 60\) cm²
Area of triangle = \(\frac{1}{2} \times 10 \times 4 = 20\) cm²
Total area = \(60 + 20 = 80\) cm²

Q4: A composite figure is formed by joining a square of side 8 cm and a rectangle of length 12 cm. If the perimeter of the composite figure is 56 cm, what is the breadth of the rectangle?

Ans: C
Explanation: Assuming the rectangle shares one side with the square (breadth = 8 cm):
Perimeter = \(8 + 8 + 12 + b + 12 + b\) where the shared side is not counted twice
If breadth = 8 cm: Perimeter = \(8 + 12 + 8 + 12 + 8 + 8 = 56\) cm ✓
Breadth of rectangle = 8 cm

Q5: A figure is made up of 3 identical rectangles, each measuring 5 cm by 2 cm, arranged end to end in a straight line. What is the perimeter of the composite figure?

Ans: A
Explanation: When 3 rectangles (5 cm by 2 cm) are arranged end to end in a line:
The composite figure becomes a rectangle of length \(5 + 5 + 5 = 15\) cm and breadth 2 cm
Perimeter = \(2 \times (15 + 2) = 2 \times 17 = 34\) cm

Q6: The figure is made up of two identical semicircles and a rectangle. The rectangle measures 14 cm by 10 cm. Each semicircle has a diameter equal to the breadth of the rectangle. What is the area of the composite figure? (Take \(\pi = \frac{22}{7}\))

Ans: B
Explanation: Area of rectangle = \(14 \times 10 = 140\) cm²
Diameter of each semicircle = 10 cm, so radius = 5 cm
Area of 2 semicircles = Area of 1 full circle = \(\frac{22}{7} \times 5 \times 5 = \frac{22}{7} \times 25 = \frac{550}{7} = 78.57\) cm²
Total area = \(140 + 78.57 \approx 218.5\) cm²

Q7: A rectangular piece of cardboard measuring 20 cm by 15 cm has four identical squares, each of side 2 cm, cut from its four corners. What is the perimeter of the remaining cardboard?

Ans: A
Explanation: Original perimeter = \(2 \times (20 + 15) = 70\) cm
When a square is cut from a corner, 2 sides of length 2 cm are removed, but 2 sides of length 2 cm are added back.
The perimeter remains the same = 70 cm

Q8: A composite figure is made by removing a triangle from a rectangle. The rectangle measures 16 cm by 9 cm. The triangle removed has a base of 8 cm and a height of 9 cm. What is the area of the remaining figure?

Ans: A
Explanation: Area of rectangle = \(16 \times 9 = 144\) cm²
Area of triangle = \(\frac{1}{2} \times 8 \times 9 = 36\) cm²
Remaining area = \(144 - 36 = 108\) cm²

Short Answer Questions

Q9: A composite figure is made up of a square and a rectangle. The square has a side of 7 cm. The rectangle has a length of 10 cm and the same breadth as the side of the square. Find the total area of the composite figure.

Ans:
Area of square = \(7 \times 7 = 49\) cm²
Breadth of rectangle = 7 cm
Area of rectangle = \(10 \times 7 = 70\) cm²
Total area = \(49 + 70 = 119\) cm²

Q10: A figure is formed by joining two rectangles. The first rectangle measures 15 cm by 8 cm. The second rectangle measures 9 cm by 8 cm and shares one side with the first rectangle. Calculate the perimeter of the composite figure.

Ans:
If the rectangles share the 8 cm side, the composite figure forms an L-shape or a longer rectangle.
Assuming they form a longer rectangle: Total length = \(15 + 9 = 24\) cm, breadth = 8 cm
Perimeter = \(2 \times (24 + 8) = 2 \times 32 = 64\) cm

Q11: The figure is made up of a rectangle and a triangle. The rectangle measures 18 cm by 10 cm. The triangle has a base of 18 cm and a height of 6 cm. Find the area of the composite figure.

Ans:
Area of rectangle = \(18 \times 10 = 180\) cm²
Area of triangle = \(\frac{1}{2} \times 18 \times 6 = 54\) cm²
Total area = \(180 + 54 = 234\) cm²

Q12: A square tile has a side of 12 cm. A smaller square of side 4 cm is removed from one corner. What is the perimeter of the remaining tile?

Ans:
Original perimeter = \(4 \times 12 = 48\) cm
When a corner square is removed, 2 sides of 4 cm are removed, but 2 sides of 4 cm are added.
Perimeter remains = 48 cm

Q13: A rectangular garden measuring 25 m by 18 m has a rectangular path of width 2 m running around its inside edge. Find the area of the path.

Ans:
Area of outer rectangle = \(25 \times 18 = 450\) m²
Dimensions of inner rectangle = \((25 - 2 - 2) \times (18 - 2 - 2) = 21 \times 14\) m
Area of inner rectangle = \(21 \times 14 = 294\) m²
Area of path = \(450 - 294 = 156\) m²

Q14: A composite figure is made by placing a triangle on top of a rectangle. The rectangle has a length of 20 cm and a breadth of 12 cm. The triangle has the same base as the length of the rectangle and a height of 8 cm. Calculate the total area of the figure.

Ans:
Area of rectangle = \(20 \times 12 = 240\) cm²
Area of triangle = \(\frac{1}{2} \times 20 \times 8 = 80\) cm²
Total area = \(240 + 80 = 320\) cm²

Q15: The figure is made up of 5 identical squares arranged in a cross pattern. Each square has a side of 4 cm. Find the perimeter of the composite figure.

Ans:
A cross pattern has a perimeter of 12 sides of the individual squares.
Perimeter = \(12 \times 4 = 48\) cm

Q16: A rectangular field measures 50 m by 30 m. A triangular flower bed with a base of 15 m and a height of 10 m is planted in one corner of the field. Find the area of the field that is not covered by the flower bed.

Ans:
Area of rectangular field = \(50 \times 30 = 1500\) m²
Area of triangular flower bed = \(\frac{1}{2} \times 15 \times 10 = 75\) m²
Area not covered = \(1500 - 75 = 1425\) m²

Long Answer / Problem Sums

Q17: Mrs Tan wants to tile her kitchen floor. The floor is in the shape of a rectangle measuring 6 m by 4 m. There is a rectangular area measuring 2 m by 1.5 m that does not need to be tiled because there is a kitchen cabinet there. Each tile is a square of side 20 cm. How many tiles does Mrs Tan need to buy?

Ans:
Area of kitchen floor = \(6 \times 4 = 24\) m²
Area of cabinet = \(2 \times 1.5 = 3\) m²
Area to be tiled = \(24 - 3 = 21\) m²

Area of each tile = \(20 \text{ cm} \times 20 \text{ cm} = 0.2 \text{ m} \times 0.2 \text{ m} = 0.04\) m²

Number of tiles needed = \(21 \div 0.04 = 525\) tiles

Q18: A composite figure is formed by joining a square and a semicircle. The square has a side of 14 cm. The semicircle is attached to one side of the square, with the diameter of the semicircle equal to the side of the square. Find the perimeter of the composite figure. (Take \(\pi = \frac{22}{7}\))

Ans:
Perimeter consists of 3 sides of the square and the curved part of the semicircle.

3 sides of square = \(3 \times 14 = 42\) cm

Circumference of semicircle = \(\frac{1}{2} \times \pi \times d = \frac{1}{2} \times \frac{22}{7} \times 14 = \frac{1}{2} \times 44 = 22\) cm

Total perimeter = \(42 + 22 = 64\) cm

Q19: A rectangular piece of paper measures 40 cm by 28 cm. Ali cuts out 6 identical circles from the paper, each with a radius of 4 cm. What is the area of the paper that remains? (Take \(\pi = \frac{22}{7}\))

Ans:
Area of rectangular paper = \(40 \times 28 = 1120\) cm²

Area of 1 circle = \(\pi r^2 = \frac{22}{7} \times 4 \times 4 = \frac{22}{7} \times 16 = \frac{352}{7} = 50.29\) cm² (approximately)

Area of 6 circles = \(6 \times \frac{352}{7} = \frac{2112}{7} = 301.71\) cm² (approximately)

Or more precisely: \(6 \times 50\frac{2}{7} = 301\frac{5}{7}\) cm²

Remaining area = \(1120 - 301\frac{5}{7} = 818\frac{2}{7}\) cm² or 818.29 cm²

Q20: The figure shows a rectangular hall measuring 30 m by 20 m. A stage in the shape of a semicircle is built along one of the shorter sides of the hall. The diameter of the semicircular stage is equal to the width of the hall. The floor of the hall, excluding the stage, is to be carpeted at a cost of $45 per square metre. The perimeter of the stage (excluding the straight edge) is to be decorated with LED lights at a cost of $12 per metre. Find the total cost. (Take \(\pi = 3.14\))

Ans:
Step 1: Find area to be carpeted
Area of rectangular hall = \(30 \times 20 = 600\) m²

Diameter of semicircular stage = 20 m, so radius = 10 m
Area of semicircular stage = \(\frac{1}{2} \times \pi \times r^2 = \frac{1}{2} \times 3.14 \times 10 \times 10 = \frac{1}{2} \times 314 = 157\) m²

Area to be carpeted = \(600 - 157 = 443\) m²

Step 2: Calculate carpet cost
Cost of carpet = \(443 \times 45 = \$19935\)

Step 3: Find length of curved edge of stage
Curved perimeter of semicircle = \(\frac{1}{2} \times \pi \times d = \frac{1}{2} \times 3.14 \times 20 = 31.4\) m

Step 4: Calculate LED lights cost
Cost of LED lights = \(31.4 \times 12 = \$376.80\)

Step 5: Find total cost
Total cost = \(19935 + 376.80 = \$20311.80\)