Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the formula for total surface area of a cylinder.

Total surface area = 2πrr+h2πr22πrhπr2h

Perfect! TSA = 2πr2 + 2πrh = 2πr(r + h).

(2) If the volume of a cube is 216 cm3, what is the length of its edge?

Edge = ∛216 = cm

Excellent! Since 63 = 216, the edge length is 6 cm.

(3) The volume of a cone is one-third the volume of which solid?

Volume of cone = 13 × volume of cylinder with same base and heightspherehemispherecube

Correct! A cone's volume is 13 that of a cylinder with the same base and height.

(4) What is the slant height of a cone if its height is 12 cm and radius is 5 cm?

Using Pythagorean theorem: l = r2+h2 = 52+122 = 25+144 = 169 = cm

Perfect! Slant height = 13 cm.

(5) Write the relation between the surface area of a sphere and the surface area of a hemisphere.

Surface area of sphere = 2 × curved surface area of hemisphere3 × curved surface area of hemisphere4 × curved surface area of hemisphere

Correct! Sphere SA = 4πr2, Hemisphere CSA = 2πr2, so sphere SA = 2 × hemisphere CSA.

(6) If the diameter of a sphere is 14 cm, what is its radius?

Radius = cm

Perfect! Radius is half the diameter.

(7) Write the volume of a hemisphere of radius r.

Volume = 23πr343πr313πr3πr3

Excellent! Hemisphere volume = 12 × sphere volume = 12 × 43πr3 = 23πr3.

(8) State whether the surface area of a cone includes its base or not.

Total surface area includes the baseexcludes the base while Curved surface area excludes the baseincludes the base

Perfect! TSA includes base, CSA excludes base.

Short Answer Questions (2 Marks Each)

Answer each question in 2-3 sentences

(1) A cone has a radius of 7 cm and height of 24 cm. Find its slant height and curved surface area.

Slant height: l = r2+h2 = 72+242 = 49+576 = 625 = cm

Curved surface area: CSA = 175π180π155π225π cm2

Excellent! Slant height = 25 cm, CSA = 175π cm2.

(2) The total surface area of a cube is 150 cm2. Find its volume.

Side length of cube = a = cm

Volume = cm3

Perfect! Edge = 5 cm, Volume = 125 cm3.

(3) A solid hemisphere has a radius of 6 cm. Find its curved surface area and total surface area.

Curved surface area = 72π36π144π108π cm2

Total surface area = 108π72π144π90π cm2

Excellent! CSA = 72π cm2, TSA = 108π cm2.

(4) The radius of a sphere is 3.5 cm. Find its surface area.

Surface area = 49π42π36π56π cm2

Perfect! Surface area = 49π cm2.

(5) A cylindrical pipe has an inner radius of 2 cm and length 120 cm. Find its inner curved surface area.

Inner curved surface area = 480π240π960π360π cm2

Excellent! Inner CSA = 480π cm2.

(6) A cone and a cylinder have equal radii and heights. Find the ratio of their volumes.

Ratio = (Note: No space between numbers and ":")

Perfect! Cone : Cylinder = 1 : 3.

(7) Find the volume of a frustum of a cone with height 10 cm, lower radius 6 cm, and upper radius 3 cm. Use π = 3.14.

Volume = cm3

Excellent calculation! Volume = 659.4 cm3.

(8) A cube has volume 512 cm3. Find its surface area.

Side length (a) = cm

Surface area = cm2

Perfect! Edge = 8 cm, Surface area = 384 cm2.

Long Answer Questions (4 Marks Each)

(1) A metallic sphere of radius 4.2 cm is melted and recast into cones each of radius 1.4 cm and height 2.1 cm. Find how many cones can be formed.

Volume of sphere = 98.784π74.088π82.5π90.2π cm3

Volume of one cone = 1.372π2.744π4.116π0.686π cm3

Number of cones: =

Excellent! 72 cones can be formed.

(2) A tent is in the shape of a cylinder surmounted by a cone. The height of the cylindrical part is 2.1 m, radius is 2.5 m, and the height of the cone is 1.4 m. Find the total surface area of the tent.

Curved surface area of cylinder = 10.5π12.5π8.75π15π m2

Slant height of cone = m

Curved surface area of cone = 7.175π6.25π8.5π5.74π m2

Total surface area = 24.425π22.5π26.25π20.175π m2

Excellent! Total surface area = 24.425π m2.

(3) A cylindrical tank of radius 3 m and height 10 m is filled with water. Find the total surface area and volume of water it can hold.

Total surface area = 78π60π90π120π m2

Volume of water = 90π60π120π100π m3

Perfect! TSA = 78π m2, Volume = 90π m3.

(4) A cone is 24 cm high and the radius of its base is 7 cm. It is melted and recast into small spheres of radius 1 cm. How many such spheres can be made?

Volume of cone = 392π294π588π196π cm3

Volume of one sphere = 43ππ2π23π cm3

Number of spheres =

Excellent! 294 spheres can be made.

Part B: Objective Questions (1 Mark Each)

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

(1) The slant height of a cone is found using:

(a) r+h (b) r2+h2 (c) r2−h2 (d) r2 + h2

Correct! Using Pythagorean theorem: l2 = r2 + h2, so l = r2+h2.

(2) If the radius of a sphere is doubled, the volume becomes:

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

Correct! Volume ∝ r3, so when radius doubles, volume becomes 23 = 8 times.

(3) The surface area of a cube is 216 cm2. What is its edge?

(a) 6 cm (b) 12 cm (c) 9 cm (d) 18 cm

Correct! 6a2 = 216, so a2 = 36, therefore a = 6 cm.

(4) If the volume of a cone is 100 cm3 and the radius is 5 cm, what is its height? (Use π = 3.14)

(a) 3.82 cm (b) 2.55 cm (c) 7.64 cm (d) 6.37 cm

Correct! 13π52h = 100, so h = 30025π ≈ 3.82 cm.

(5) A cylinder and cone have the same base and height. The ratio of their volumes is:

(a) 1:1 (b) 2:1 (c) 3:1 (d) 1:3

Correct! Cylinder : Cone = πr2h : 13πr2h = 3:1.

(6) Volume of a frustum is given by:

(a) 13πhR2+r2+Rr (b) πr2h (c) πR2−r2h (d) 12πrh

Correct! This is the standard formula for the volume of a frustum of a cone.

(7) If the surface area of a sphere is 154 cm2, then its radius is: (Use π = 227)

(a) 7 cm (b) 14 cm (c) 3.5 cm (d) 5 cm

Yes! The radius is 3.5 cm.

(8) The lateral surface area of a cube with side a is:

(a) 4a2 (b) 6a2 (c) 3a2 (d) 2a2

Correct! Lateral surface area excludes top and bottom, so 4 sides × a2 = 4a2.

(9) The total surface area of a cone is:

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

Correct! TSA = base area + curved surface area = πr2 + πrl.

(10) A cylinder has radius 5 cm and height 10 cm. What is its volume? (Use π = 3.14)

(a) 785 cm3 (b) 550 cm3 (c) 500 cm3 (d) 250 cm3

Correct! V = πr2h = 3.14 × 25 × 10 = 785 cm3.

Mensuration Challenge

Determine whether these statements about mensuration 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 formulas for cubes, cylinders, cones, spheres, and hemispheres

(2) Slant Height Calculations: Using Pythagorean theorem to find slant heights of cones and pyramids

(3) Melting and Recasting: Converting one 3D shape into another while preserving volume

(4) Combined Shapes: Calculating dimensions of composite solids like cylinder with cone or hemisphere

(5) Frustum Properties: Understanding truncated cone volume and surface area calculations

(6) Scaling Relationships: How changes in dimensions affect volume and surface area

(7) Real-world Applications: Solving practical problems in construction, manufacturing, and storage

(8) Problem-solving Strategies: Systematic approaches to complex mensuration problems

Excellent work mastering the practical world of mensuration! 🌟