Discover the magic of 3D shapes through interactive problems!
Volume and Surface Area Challenges
1
Hemispherical Bowl with Cylinder
A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter is 14 cm and the height of the vessel is 13 cm. Find the capacity of the vessel.
Step 1: Understand the shape - a hemisphere topped by a cylinder
Step 2: Find dimensions:
Diameter = 14 cm → Radius (r) = 7 cm
Total height = 13 cm
Height of hemisphere = radius = 7 cm
So height of cylinder (h) = 13 - 7 = 6 cm
Nathan, an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of the model that Nathan made.
Step 1: Understand the shape - cylinder with cones at both ends
Step 2: Find dimensions:
Diameter = 3 cm → Radius (r) = 1.5 cm
Total length = 12 cm
Each cone height (hcone) = 2 cm
So cylinder height (hcyl) = 12 - (2+2) = 8 cm
Step 3: Calculate volume of two cones:
Vcone = (1/3)πr²h
= 2 × (1/3) × (22/7) × 1.5 × 1.5 × 2 = 9.43 cm³
Step 4: Calculate volume of cylinder:
Vcylinder = πr²h
= (22/7) × 1.5 × 1.5 × 8 = 56.57 cm³
Step 5: Add volumes for total model volume
Total Volume = 9.43 + 56.57 = 66 cm³
3
Cylinder with Cone Carved Out
From a solid cylinder whose height is 2.4 cm and the diameter 1.4 cm, a cone of the same height and same diameter is carved out. Find the volume of the remaining solid to the nearest cm³.
Step 1: Understand we have a cylinder with a cone removed
Step 2: Find dimensions:
Diameter = 1.4 cm → Radius (r) = 0.7 cm
Height (h) = 2.4 cm (same for both)
Step 3: Calculate volume of cylinder:
Vcylinder = πr²h
= (22/7) × 0.7 × 0.7 × 2.4 = 3.696 cm³
Step 4: Calculate volume of cone:
Vcone = (1/3)πr²h
= (1/3) × (22/7) × 0.7 × 0.7 × 2.4 = 1.232 cm³
Step 5: Subtract cone volume from cylinder
Remaining Volume = 3.696 - 1.232 = 2.464 ≈ 2 cm³
4
Cone on Hemisphere in Water
A solid consisting of a right circular cone of height 12 cm and radius 6 cm standing on a hemisphere of radius 6 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of the water displaced out of the cylinder, if the radius of the cylinder is 6 cm and height is 18 cm.
Step 1: Understand the scenario - cone+hemisphere submerged in water
Step 2: Water displaced equals volume of submerged object
A capsule is in the shape of a cylinder with two hemisphere stuck to each of its ends. If the length of the entire capsule is 12 mm and the diameter of the capsule is 3 mm, how much medicine it can hold?
Step 1: Understand the shape - cylinder with hemispherical ends
Step 2: Find dimensions:
Diameter = 3 mm → Radius (r) = 1.5 mm
Total length = 12 mm
Two hemispheres make a full sphere with radius 1.5 mm
So cylinder length = 12 - (1.5 + 1.5) = 9 mm
Step 3: Calculate volume of two hemispheres (which is a full sphere):
A cubical block of side 7 cm is surmounted by a hemisphere. Find the surface area of the solid.
Step 1: Find dimensions
Cube side = 7 cm
Hemisphere diameter = 7 cm → Radius (r) = 3.5 cm
Step 2: Surface area of cube (excluding top face)
SA = 5 × side²
= 5 × 7² = 245 cm²
Step 3: Curved surface area of hemisphere
SA = 2πr²
= 2 × (22/7) × 3.5² = 77 cm²
Step 4: Total surface area
Total SA = 245 + 77 = 322 cm²
7
Sphere Enclosed in Cylinder
A right circular cylinder just encloses a sphere of radius r units. Calculate:
(i) the surface area of the sphere
(ii) the curved surface area of the cylinder
(iii) the ratio of the areas obtained in (i) and (ii).
Step 1: Find dimensions
Sphere radius = r
Cylinder radius = r
Cylinder height = 2r (since sphere fits perfectly)