Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) If the slant height of a cone is 13 cm and radius is 5 cm, what is its height? h = cm

Perfect! The height, radius, and slant height form a right triangle.

(2) A cube has surface area 600 cm². Find the area of one face. Surface area = cm2

Correct! A cube has 6 equal square faces.

(3) A sphere is melted and recast into a cone. What remains constant during the process?

Excellent! Volume is conserved when matter is melted and recast.

(4) Write the formula for the total surface area of a frustum of a cone. TSA = πR+rl2πR+rl + (πR22πr2 + πr24πr) where R, r are radii and l is slant height

Perfect! This includes both circular bases and the curved surface.

(5) State whether: "Volume of hemisphere = 23πr3" is true or false.

Correct! Hemisphere volume is half of sphere volume: 12 × 43πr3 = 23πr3.

(6) What is the surface area of a cube of side √3 cm? Surface area = cm2

Excellent calculation!

(7) A hemisphere is fitted onto the top of a cylinder. Name the solid formed. CapsuleCone

Good! This is a common composite solid in mensuration.

(8) If radius of a sphere is tripled, by what factor does the volume increase? times

Perfect understanding of cubic scaling!

Short Answer Questions (2 Marks Each)

(1) A metallic cone of radius 3.5 cm and height 12 cm is melted and cast into a sphere. Find the radius of the sphere. R = cm

Excellent application of volume conservation!

(2) A tent is in the shape of a cylinder surmounted by a cone. Radius is 2.8 m, height of cylinder is 5 m, and slant height of cone is 3.5 m. Find the total surface area of the tent. Total surface area = m2 (Upto one decimal place)

Perfect! Remember to include the base but not the junction between cylinder and cone.

(3) A hemisphere of radius 7 cm is cut out from the top of a cylinder of radius 7 cm and height 10 cm. Find the surface area of the remaining solid. Total surface area = π cm2

Good! The hemispheric cut creates an internal curved surface.

(4) A frustum of a cone has lower radius 7 cm, upper radius 4 cm and height 10 cm. Find its volume. Volume of frustum = π cm3

Excellent use of the frustum volume formula!

(5) A hollow metallic cylinder has an outer radius 8 cm and inner radius 6 cm. If the height is 20 cm, find the volume of metal used. Volume of metal = π cm3

Perfect! This is the standard formula for hollow cylinder volume.

(6) The total surface area of a sphere is 154 cm2. Find its volume. Volume = π cm3 (Upto two decimal places)

Excellent calculation connecting surface area to volume!

(7) A cylindrical hole is bored through a sphere of radius 10 cm. The hole has radius 4 cm. Find the volume of the remaining solid. Volume of remaining solid = π cm3 (Enter in fraction form)

Good! The hole goes through the entire diameter of the sphere.

Long Answer Questions (4 Marks Each)

(1) A metallic spherical ball of radius 4.2 cm is melted and recast into smaller cones of radius 1.4 cm and height 2.1 cm. How many cones can be formed? Number of cones =

Excellent! 72 small cones can be formed from the sphere.

(2) A cone is divided into two parts — a smaller cone and a frustum — by cutting it parallel to the base at mid-height. Prove that the volumes of the smaller cone and frustum are in the ratio 1:7.

Volume of smaller cone = πR2H247πR2H24πR2H7

Volume of frustum = 7πR2H2449πR2H24πR2H49

Perfect proof using similar triangles and volume calculation!

(3) A solid is in the form of a right circular cone mounted on a cylinder. The radius of the base of the cone and cylinder is 5 cm. Height of the cone is 12 cm and that of the cylinder is 10 cm. Find the total surface area and volume of the solid.

Total surface area = π cm2

Total volume = π cm3

Excellent complete solution for composite solid!

(4) Water is poured into a hemispherical bowl of radius 10 cm at the rate of 100 cm³ per second. How much time will it take to fill the bowl completely?

Time = seconds (Upto two decimal places)

Perfect application of hemisphere volume and rate calculation!

(5) A wooden toy is in the form of a cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm. Find the total surface area and volume of the toy.

Total surface area = π cm2 (Upto two decimal places)

Total volume = π cm3 (Upto two decimal places)

Excellent comprehensive solution for the composite toy!

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The volume of metal used in making a hollow cylinder is:

(a) πr2h (b) πR2−r2h (c) πr2h−πRh2 (d) 2πrh

Correct! Volume = Outer volume - Inner volume = πR2h - πr2h = πR2−r2h.

(2) A cylindrical vessel of radius 7 cm is filled with water to a height of 20 cm. The water is poured into a hemispherical bowl. The radius of the bowl is 14 cm. The height of water in the bowl will be:

(a) 3.5 cm (b) 5 cm (c) 7 cm (d) 10 cm

Correct! The radius is 3.5 cm.

(3) Volume of a frustum is:

(a) πhR2+r2+Rr3 (b) πr2h (c) πh(R – r) (d) πr2+πrl

Correct! This is the standard formula for frustum volume.

(4) The surface area of a cube is 384 cm2. What is its volume?

(a) 64 cm3 (b) 216 cm3 (c) 512 cm3 (d) 729 cm3

Correct! If surface area = 6a2 = 384, then a2 = 64, so a = 8. Volume = a3 = 512.

(5) A solid cone has radius r and height h. Its volume is:

(a) πr2h (b) 12πr2h (c) 13πr2h (d) πr2h2

Correct! Cone volume is one-third of the corresponding cylinder volume.

(6) A tent is shaped as a cone over a cylinder. If both have same radius, the total surface area includes:

(a) Only cone CSA (b) Cone CSA + cylinder CSA (c) Cone CSA + cylinder TSA (d) Cone CSA + cylinder CSA + base

Correct! We need the curved surfaces plus the base (ground), but not the junction.

(7) If radius of a sphere is doubled, surface area becomes:

(a) 2 times (b) 4 times (c) 6 times (d) 8 times

Correct! Surface area ∝ r2, so doubling radius gives 4× the area.

(8) A solid sphere of radius 7 cm is melted and recast into smaller spheres each of radius 3.5 cm. The number of smaller spheres formed is:

(a) 8 (b) 4 (c) 16 (d) 12

Correct! We get a total of 8 spheres.

(9) If surface area of a cube is 96 cm2, its edge is:

(a) 4 cm (b) 6 cm (c) 3 cm (d) 2 cm

Correct! Surface area = 6a2 = 96, so a2 = 16, therefore a = 4 cm.

(10) A toy is made by attaching a cone to a hemisphere. The formula for total surface area is:

(a) πr2 + πrl (b) 2πr2 + πrl (c) πr2 + 2πrl (d) πr2 + πr2

Correct! Hemisphere CSA (2πr2) + Cone CSA (πrl) = 2πr2 + πrl.

Mensuration Challenge

Determine whether these statements about 3D shapes are True or False:

Mensuration Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) 3D Shape Formulas: Volume and surface area for cones, spheres, cylinders, and composite solids

(2) Volume Conservation: Understanding that volume remains constant during melting and recasting

(3) Composite Solids: Calculating properties of shapes formed by combining basic 3D shapes

(4) Frustum Calculations: Using the special formula for truncated cones

(5) Hollow Shapes: Finding volume of material in hollow cylinders and spheres

(6) Scaling Effects: How doubling dimensions affects volume and surface area

(7) Practical Applications: Solving real-world problems involving water filling, material usage

(8) Problem-solving Strategies: Breaking complex shapes into simpler components

Excellent work mastering advanced mensuration concepts and their practical applications!