Compound Shapes and Parts of Shapes (Copy)

Q1 (Non-Calculator)
Find the area of a shape made by joining a rectangle (6 cm by 4 cm) and a semicircle with diameter 4 cm.

Solution:
Area of rectangle = 6 × 4 = 24 cm²
Radius of semicircle = 4 ÷ 2 = 2 cm
Area of full circle = π × 2² = 4π cm²
Area of semicircle = (1/2) × 4π = 2π cm²
Total area = 24 + 2π cm²

Answer:
24 + 2π cm²

Q2 (Non-Calculator)
Find the perimeter of the shape in Q1. Leave your answer in terms of π.

Solution:
Perimeter = 6 + 6 + 4 + (1/2 × circumference of circle)
Circumference of full circle = 2πr = 2π × 2 = 4π
Half circumference = 2π
Perimeter = 6 + 6 + 4 + 2π
Perimeter = 16 + 2π cm

Answer:
16 + 2π cm

Q3 (Non-Calculator)
Find the area of half of a circle with radius 5 cm. Leave your answer in terms of π.

Solution:
Area of full circle = πr² = π × 5² = 25π
Area of half circle = (1/2) × 25π = 12.5π cm²

Answer:
12.5π cm²

Q4 (Non-Calculator)
Find the volume of a solid made by joining a cylinder (radius 3 cm, height 5 cm) and a hemisphere (radius 3 cm). Leave your answer in terms of π.

Solution:
Volume of cylinder = πr²h = π × 3² × 5 = π × 9 × 5 = 45π cm³
Volume of hemisphere = (1/2) × (4/3)πr³ = (2/3)π × 3³ = (2/3)π × 27 = 18π cm³
Total volume = 45π + 18π = 63π cm³

Answer:
63π cm³

Q5 (Non-Calculator)
Find the surface area of a solid consisting of a cylinder (radius 2 cm, height 7 cm) closed at one end and open at the other. Leave your answer in terms of π.

Solution:
Curved surface area = 2πrh = 2π × 2 × 7 = 28π cm²
Area of closed circular end = πr² = π × 2² = 4π cm²
Total surface area = 28π + 4π = 32π cm²

Answer:
32π cm²

Q6 (Calculator)
A shape consists of a rectangle 8 m by 5 m with a semicircular end of diameter 5 m attached. Find the total area. Give your answer in terms of π.

Solution:
Area of rectangle = 8 × 5 = 40 m²
Radius of semicircle = 5 ÷ 2 = 2.5 m
Area of full circle = π × (2.5)² = π × 6.25 = 6.25π m²
Area of semicircle = (1/2) × 6.25π = 3.125π m²
Total area = 40 + 3.125π m²

Answer:
40 + 3.125π m²

Q7 (Calculator)
Find the volume of half a sphere with radius 6 cm. Leave your answer in terms of π.

Solution:
Volume of full sphere = (4/3)πr³ = (4/3)π × 6³ = (4/3)π × 216 = 288π cm³
Volume of half sphere = (1/2) × 288π = 144π cm³

Answer:
144π cm³

Q8 (Calculator)
A cylinder has radius 4 cm and height 10 cm. A cone with the same base radius and height is removed from it. Find the volume of the remaining solid. Leave your answer in terms of π.

Solution:
Volume of cylinder = πr²h = π × 4² × 10 = π × 16 × 10 = 160π cm³
Volume of cone = (1/3)πr²h = (1/3)π × 4² × 10 = (1/3)π × 16 × 10 = (1/3)π × 160 = 53.333…π cm³
Remaining volume = 160π – 53.333…π = 106.666…π cm³

Answer:
106.666…π cm³ or (320/3)π cm³

Q9 (Calculator)
Find the surface area of a solid made of a hemisphere of radius 3 cm attached to a cylinder of radius 3 cm and height 8 cm. Leave your answer in terms of π.

Solution:
Surface area of hemisphere = 2πr² = 2π × 3² = 18π cm²
Curved surface area of cylinder = 2πrh = 2π × 3 × 8 = 48π cm²
Total surface area = 18π + 48π = 66π cm²

Answer:
66π cm²

Q10 (Calculator)
Find the volume of a quarter sphere with radius 2 m. Leave your answer in terms of π.

Solution:
Volume of full sphere = (4/3)πr³ = (4/3)π × 2³ = (4/3)π × 8 = (32/3)π m³
Volume of quarter sphere = (1/4) × (32/3)π = (8/3)π m³

Answer:
(8/3)π m³