Test: Surface Area Sphere and Hemisphere - Class 9 MCQ

Formula for the surface area of a sphere:

Surface Area = 4πr²

Where:

• r is the radius,
• π = 3.1416.

301.84 = 12.5664 × r²
r² = 301.84 / 12.5664
r² = 24
r = √24 = 4.9 cm

The diameter of the sphere is:

Diameter = 2 × r = 2 × 4.9 = 9.8 cm

The inner radius of the hemispherical bowl is 6 cm, and the thickness is 1 cm. The outer radius is:

R = 6 + 1 = 7 cm.

The formula for the curved surface area (CSA) of a hemisphere is:

CSA = 2πR²

Solution:

Substitute R = 7 cm into the formula:

CSA = 2π(7²) = 2π(49) = 98π

Take π = 3.14:

CSA = 98 × 3.14 = 307.72 cm²

Final Answer:

Rounding off, the outer curved surface area is approximately:

(c) 308 cm².

o find the total surface area of a hemispherical bowl, we use the formula:

Total Surface Area = 2πr² + πr² = 3πr²

Given r = 77 cm and π = 3.1416:

Total Surface Area = 3 × 3.1416 × 77²
77² = 5929
Total Surface Area = 3 × 3.1416 × 5929 = 55912.32 cm²

Approximating, the total surface area is:

55900 cm²

The surface area of a sphere is given by:

Surface Area = 4πr²

Let the radius of the original sphere be r₁ = r and the radius of the second sphere be r₂ = 3r (radius is tripled).

1. Surface Area of the Original Sphere: S₁ = 4πr²

2. Surface Area of the Second Sphere: S₂ = 4π(3r)² = 4π × 9r² = 36πr²

3. Ratio of Surface Areas:

S₁ / S₂ = (4πr²) / (36πr²) = 1 / 9

The ratio of the surface area of the original sphere to the second is: 1 : 9

Using π = 22/7, let's recalculate.

The total surface area (TSA) of a hemispherical bowl is the sum of the lateral surface area (LSA) and the area of the circular base:

TSA = LSA + Area of Base

We are given:

TSA = 96 cm²

Using the formula:

TSA = 3πr²

Substitute π = 22/7:

TSA = 3 × 22/7 × r²

Simplify:

96 = 3 × 22/7 × r²

96 × 7 = 66 × r²
672 = 66 × r²

Now calculate the lateral surface area (LSA) of one hemisphere:

LSA = 2πr² / 2

Simplify:

LSA = 44 × r²

Now, solve for r²:

r² = 672 / 66 = 10.18 cm²

For 5 such bowls:

Total LSA = 5 × 64 = 320 cm²

The total surface area (TSA) of a hemisphere is given by:

TSA = LSA + Area of Base

Where:

• LSA = 2πr² (lateral surface area),
• Area of Base = πr².

Thus:

TSA = 2πr² + πr² = 3πr²

Given TSA = 525 cm², and using π = 22/7:

3πr² = 525

Substitute π = 22/7:

3 × 22/7 × r² = 525

Simplify:

66/7 × r² = 525

r² = (525 × 7) / 66 = 3675 / 66 = 55.68 cm²

Now calculate the lateral surface area (LSA):

LSA = 2πr²

Simplify:

LSA = 2 × 22/7 × 55.68 = 44 × 55.68 / 7 = 2449.92 / 7 ≈ 350 cm²

The surface area of a sphere is calculated using the formula:

Surface Area = 4πr²

Where:

• r = 7 cm (radius of the sphere),
• π = 3.1416.

Substitute the values into the formula:

Surface Area = 4 × 3.1416 × 7²

Step-by-step calculations:

1. 7² = 49
2. 4 × 3.1416 × 49 = 4 × 153.9384 = 615.7536 cm²

Approximating, the surface area is: 616 cm²

The total surface area of a hemispherical bowl is calculated using the formula:

Total Surface Area = 3πr²

Where:

• r is the radius of the hemisphere,
• π = 3.1416.

Given the diameter = 28 cm, the radius r = 28 / 2 = 14 cm.

Substituting into the formula:

Total Surface Area = 3 × 3.1416 × 14²

Step-by-step calculations:

1. 14² = 196
2. Total Surface Area = 3 × 3.1416 × 196 = 3 × 615.751 = 1847.253 cm²

Approximating, the total surface area is: 1848 cm²

The formula for the curved surface area of a sphere is:

Curved Surface Area = 4πr²

For two spheres, the ratio of their curved surface areas depends on the square of their radii:

Ratio of curved surface areas = r₁² / r₂²

1. The radius of the first sphere (r₁) is:

   r₁ = Diameter / 2 = 21 / 2 = 10.5 cm

2. The radius of the second sphere (r₂) is:

   r₂ = Diameter / 2 = 14 / 2 = 7 cm

3. The ratio of their curved surface areas is:

   r₁² / r₂² = (10.5)² / (7)² = 110.25 / 49 = 9 / 4

The ratio of their curved surface areas is 9 : 4.

The curved surface area of a hemisphere is calculated using the formula:

Curved Surface Area = 2πr²

Given the diameter d = 28 cm, the radius r = 28 / 2 = 14 cm.

Substitute into the formula:

Curved Surface Area = 2 × 3.1416 × 14²

14² = 196

Curved Surface Area = 2 × 3.1416 × 196 = 2 × 615.75104 = 1231.502 cm²

Approximating, the curved surface area is: 1232 cm²