Solid Figures MCQ Quiz - Objective Question with Answer for Solid Figures - Download Free PDF

Find Length and Breadth of the Rectangle

Let length= 9x, breadth= 5x.

Perimeter = 2(9x + 5x) = 280

2(14x) = 280 ⟹ 28x = 280 ⟹ x = 10

Thus,

Length = 9x = 90 cm, Breadth = 5x = 50 cm

Area of rectangle = 90 × 50 = 4500 cm².

The top surface of the cylinder is a circle with area πr²

Given:

πr2 = 4500 ⟹ r2 = 4500π

Given: Radius r = 120% of height h, so:

r = 1.2h ⟹ h = r / 1.2 = 5r / 6

Now, let's find the Curved Surface Area of Cylinder:

Curved surface area = 2πrh 

Substituteh = 5r / 6:

CSA = 2πr(5r/6) = 10πr2 / 6 = 5πr2 / 3

Since πr2 = 4500:

CSA = 5 x 4500 / 3 = 7500 cm2

Thus, the correct answer is 7500 cm2.

Given:

Height of cone = 2 × Radius of base circle.

Radius of base circle = r.

Total Surface Area (TSA) of cone = Area of base + Curved Surface Area.

Formula Used:

Area of base = πr².

Curved Surface Area = πr × l, where l is the slant height.

Slant height (l) = √(r² + h²).

Total Surface Area = πr² + πr × l.

Ratio = Area of base / Total Surface Area.

Calculation:

Height (h) = 2r.

Slant height (l) = √(r² + (2r)²)

⇒ l = √(r² + 4r²)

⇒ l = √(5r²)

⇒ l = r√5.

Curved Surface Area = πr × r√5

⇒ Curved Surface Area = πr²√5.

Total Surface Area = πr² + πr²√5.

⇒ Total Surface Area = πr²(1 + √5).

Area of base = πr².

Ratio = Area of base / Total Surface Area

⇒ Ratio = πr² / [πr²(1 + √5)]

⇒ Ratio = 1 / (1 + √5).

Given:

Radius of cylinder (r_c) = r

Radius of cone (r_cone) = 2r

Height of cylinder = Height of cone = h

Formula used:

Volume of cylinder (V_c) = πr²h

Volume of cone (V_cone) = (1/3)πr²h

Calculation:

Volume of cylinder = πr²h

Volume of cone = (1/3)π(2r)²h

⇒ Volume of cone = (1/3)π(4r²)h

Ratio of volumes = Volume of cylinder : Volume of cone

⇒ Ratio = πr²h : (1/3)π(4r²)h

⇒ Ratio = 1 : (4/3)

⇒ Ratio = 3 : 4

∴ The correct answer is option (3).

Given:

Length of rectangular sheet = 31.4 cm

Breadth of rectangular sheet = 10 cm

Formula used:

Circumference of the base of the cylinder = Length of the sheet

Height of the cylinder = Breadth of the sheet

Volume of the cylinder = π × r² × h

Where, r = radius of the base, h = height

Calculation:

Length of the sheet = Circumference of the base = 2πr

⇒ 31.4 = 2 × 3.14 × r

⇒ r = 31.4 / (2 × 3.14)

⇒ r = 5 cm

Height of the cylinder = Breadth of the sheet = 10 cm

Volume of the cylinder = π × r² × h

⇒ Volume = 3.14 × (5)² × 10

⇒ Volume = 3.14 × 25 × 10

⇒ Volume = 785 cm³

∴ The correct answer is option (1).

Given:

Volume of cone (V) = 1232 m³

Radius of the base (r) = 7 m

π = 22/7

Formula used:

Volume of cone: V = (1/3) × π × r² × h

Slant height (l): l = √(r² + h²)

Calculation:

1232 = (1/3) × (22/7) × 7² × h

⇒ 1232 = (1/3) × (22/7) × 49 × h

⇒ 1232 = (1078/21) × h

⇒ h = 1232 × (21/1078)

⇒ h = 24 m

Slant height (l):

l = √(r² + h²)

⇒ l = √(7² + 24²)

⇒ l = √(49 + 576)

⇒ l = √625

⇒ l = 25 m

∴ The correct answer is option (3).

Given:

The radius of a solid hemisphere is 21 cm.

The ratio of the cylinder's curved surface area to its Total surface area is 2/5.

Formula used:

The curved surface area of the cylinder = 2πRh

The total surface area of cylinder = 2πR(R + h)

The volume of the cylinder = πR²h

The volume of the solid hemisphere = 2/3πr³ 

(where r is the radius of a solid hemisphere and R is the radius of a cylinder)

Calculations:

According to the question,

CSA/TSA = 2/5

⇒ [2πRh]/[2πR(R + h)] = 2/5

⇒ h/(R + h) = 2/5

⇒ 5h = 2R + 2h

⇒ h = (2/3)R .......(1)

The cylinder's volume and the volume of a solid hemisphere are equal.

⇒ πR²h = (2/3)πr³

⇒ R2 × (2/3)R = (2/3) × (21)³

⇒ R³ = (21)3

⇒ R = 21 cm

∴ The radius (in cm) of its base is 21 cm.

The surface area of three faces of a cuboid sharing a vertex are 20 m², 32 m² and 40 m²,

⇒ L × B = 20 sq. Mt

⇒ B × H = 32 sq. Mt

⇒ L × H = 40 sq. Mt

⇒ L × B × B × H × L × H = 20 × 32 × 40

⇒ L²B²H² = 25600

⇒ LBH = 160

∴ Volume = LBH = 160 m³

Given:

Each side of the cube = 8 cm

The rectangular container has a length of 16 cm, breadth of 8 cm, and height of 15 cm

Formula used:

The volume of cube = (Edge)3

The volume of a cuboid = Length × Breadth × Height

Calculation:

The volume of cube = The volume of the rectangular container with a length of 16 cm, breadth of 8 cm, and height of the water level rise

Let, the height of the water level will rise = x cm

So, 8³ = 16 × 8 × x

⇒ 512 = 128 × x

⇒ x = 512/128 = 4

∴ The rise of water level (in cm) is 4 cm

Given:

Sum of length,, breadth and height of a cuboid = 21 cm

Length of the diagonal(d) = 13 cm

Formula used:

d² = l² + b² + h²

T.S.A of cuboid = 2(lb + hb +lh)

Calculation:

⇒ l² + b² + h² = 13² = 169

According to question,

⇒ (l + b + h)² = 441

⇒ l² + b² + h² + 2(lb + hb +lh) = 441

⇒ 2(lb + hb +lh) = 441 - 169 = 272

∴ The answer is 272 cm² .

Given:

Three cubes with sides in the ratio of 3 ∶ 4 ∶  5 are melted to form a single cube whose diagonal is 18√3 cm.

Concept used:

The diagonal of a cube = √3a (where a is the sides)

Calculation: 

Let the s sides of the cubes will be 3x cm , 4x cm, and 5x cm

As per the question,

Volume of the new cube is 

(3x)³ +( 4x)³ +( 5x)³ = 216 x³.

⇒ side is = 6x

diagonal is 6x√3

⇒  6x√3 = 18√3

⇒ x = 3

The sides of the cubes will be 9 cm , 12 cm, and 15 cm

∴ The correct option is 2

GIVEN:

The surface area of a sphere = 1386  

FORMULA USED:

The surface area of a sphere = 4πr²where r is the radius of the sphere.

CALCULATION:

The surface area of a sphere =4πr2 = 1386 

⇒  4 × (22/7) × r² = 1386      ....(value of   is )

⇒ r2 = 110.25 

⇒ r2 =   

⇒ r =  =  = 10.5 cm.

∴ The radius of the sphere is 10.5 cm.

Given:

The curved surface area of the cone = 2 × base area of cone

Concepts used:

Formula used

Slant height (l) of cone = √r² + h²

CSA of cone = πrl

Calculation:

Let the radius of the cone be r units.

⇒ πrl = 2πr2

⇒ l = 2r

⇒ r = 6√3/2

⇒ r = 3√3

Slant height (l) of cone = √r2 + h2

⇒ 6√3² = 3√3² + h2

⇒ h2 = 108 - 27 = 81

⇒ h = 9 cm

∴ The answer is 9 cm.

Given:

Radius of Sphere = 42 cm

Radius of wire = 21 cm

Formula:

Volume of cylinder = πr²h

Volume of sphere = [4/3]πr³

Calculation:

Let length of the wire be x, then

According to the question

π × 21 × 21 × x = [4/3] × π × 42 × 42 × 42 [As volume will remain constant]

⇒ x = (4 × 42 × 42 × 42)/(21 × 21 × 3)

⇒ x = 224 cm 

∴ The length of the wire is 224 cm

GIVEN:

Cartons having length = 48 inches and breadth = 27 inches 

The volume of cartoon = 22.5 cubic feet.

FORMULA USED :

Volume of Cuboid = Length × Breadth × Height 

CALCULATION :

Volume of carton = volume of cuboid = Length × Breadth × Height 

⇒ volume of carton = 48 × 27 × Height

∵ 1 foot = 12 inches, then 22.5 cubic feet = 22.5 × 12 × 12 ×12

⇒ 22.5 × 12 × 12 × 12 = 48 × 27 × Height     

⇒ 38,880 = 1,296 × Height 

⇒ Height = 30 inches.

∴ The height of each cartoon is 30 inches.                                     

Formula Used:

Volume of sphere = πr³

Surface area of sphere = 4πr²

Calculation:

If the radius of a smaller sphere be 'r cm' then

Acoording to the question:

π(10)³ = 1000π(r)³

r = 1 cm

Surface area of the larger sphere = 4π(10)² = 400π

Total surface area of 1000 smaller spheres = 1000 × 4π(1)² = 4000π

Net increase in the surface area = 4000π − 400π = 3600π

Hence, surface area of the metal is increased by 9 times.