Surface Areas and Volumes

Surface Areas and Volumes

Lateral surface area = 4 × (edge)²

Total surface area = 6 × (edge)²

Volume = (edge)³

Diagonal of a cube =$\sqrt{3}\times edge$  

Surface Area and Volume of Cylinder

A cylinder is a solid shape that has two circular bases, connected with each other, through a lateral surface. Thus, there are three faces, two circular and one lateral, of a cylinder. Based on these dimensions, we can find the surface area and volume of a cylinder.

Surface Area of Cylinder

Take a cylinder of base radius r and height h units. The curved surface of this cylinder, if opened along the diameter (d = 2r) of the circular base can be transformed into a rectangle of length 2πr and height h units. Thus,

Transformation of a Cylinder into a rectangle.

CSA of a cylinder of base radius r and height h = 2π × r × h

TSA of a cylinder of base radius r and height h = 2π × r × h + area of two circular bases

=2π × r × h + 2πr²

=2πr (h + r)

Volume of a Cylinder

Volume of a cylinder = Base area × height = (πr²) × h = πr²h

Cylinder with height h and base radius r

Surface area ${=\pi r}^2$

Volume ${=\frac{4}{3}\pi r}^3$

Area of curved surface ${=2\pi r}^2$

Total surface Area = Area of curved surface + Area of base

${=2\pi r}^2{+\pi r}^2{=3\pi r}^2$

${Volume=\frac{2}{3}\pi r}^3$

Surface Area of Combined Figures

Areas of complex figures can be broken down and analysed as simpler known shapes. By finding the areas of these known shapes, we can find out the required area of the unknown figure.

Example: 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.

Length of each cube = 64^(1/3) = 4cm

Since these cubes are joined adjacently, they form a cuboid whose length l = 8 cm. But height and breadth will remain the same = 4 cm.

Combination of 2 equal cubes

∴ The new surface area, TSA = 2(lb + bh + lh)

TSA = 2 (8 x 4 + 4 x 4 + 8 x 4)

= 2(32 + 16 + 32)

= 2 (80)

TSA = 160 cm²

Volume of Combined Solids

The volume of complex objects can be simplified by visualising them as a combination of shapes of known solids.

Example: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 3 cm and the height of the cone is equal to 5 cm.

This can be visualised as follows:

Volume of combined solids

V(solid) = V(Cone) + V(hemisphere)

V(solid) = (1/3)πr²h + (2/3)πr³

V(solid) = (1/3)π(9)(5) + (2/3)π(27)

V(solid) = 33π cm³

Surface area and volume of Frustum of a cone

When a solid is cut by a plane, then another form of solids is formed. One such form of solid is the frustum of a cone, which is formed when a plane cuts a cone parallelly to the base of the cone. Let us discuss its surface area and volume here.

If a right circular cone is sliced by a plane parallel to its base, then the part with the two circular bases is called a Frustum.

Frustum with radius r₁ and r₂ and height h

CSA of frustum = π(r₁ + r₂)l, where l = √[h² + (r₂ – r₁)²]

TSA of the frustum is the CSA + the areas of the two circular faces = π(r₁ + r₂)l + π(r₁² + r₂²)

Volume of a Frustum

The volume of a frustum of a cone =(1/3)πh(r₁² + r₂² + r₁r₂)

Important Questions

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Multiple Choice questions-

1. If the surface areas of two spheres are in ratio 16 : 9, then their volumes will be in the ratio:

(a) 27 : 64

(b) 64 : 27

(c) 4 : 3

(d) 3 : 4

2. A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?

(a) 3 : 1 : 2

(b) 3 : 2 : 1

(c) 1 : 2 : 3

(d) 1 : 3 : 2

3. If the area of three adjacent faces of cuboid are X, Y and Z respectively, then the volume of cuboid is:

(a) XYZ

(b) 3XYZ

(c) $\sqrt{xyz}$

(d) $\sqrt{3xyz}$

4. The volumes of two spheres are in the ratio 27 : 8. The ratio of their curved surface is:

(a) 9 : 4

(b) 4 : 9

(c) 3 : 2

(d) 2 : 3

5. The ratio of the volumes of two spheres is 8 : 27. If r and R are the radii of spheres respectively, then (R – r): r is:

(a) 1 : 2

(b) 1 : 3

(c) 2 : 3

(d) 4 : 9

6. The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. The ratio of their volumes is:

(a) 27 : 20

(b) 20 : 27

(c) 9 : 4

(d) 4 : 9

7. If the radius of base of a right circular cylinder is halved, keeping the height same, the ratio of the volume of the reduced cylinder to that of the original cylinder is:

(a) 2 : 3

(b) 3 : 4

(c) 1 : 4

(d) 4 : 1

8. If the volumes of a cube is 1728 cm³, the length of its edge is equal to:

(a) 7 cm

(b) 12 cm

(c) 18 cm

(d) 19 cm

9. The volume (in cm³) of the largest right circular cone that can be cut off from a cube of edge 4.2 cm is: .

(a) 9.7

(b) 72.6

(c) 58.2

(d) 19.4

10. The circumference of the edge of hemispherical bowl is 132 cm. When π is taken as $\frac{22}{7}$, the capacity of bowl in cm³ is:

(a) 2772

(b) 924

(c) 19404

(d) 9702

  Very Short Questions:

OR

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius on its circular face. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Long Questions :

Case Study Questions:

Assertion Reason Questions-

Assertion: If diameter of a sphere is decreased by 25%, then its curved surface area is decreased by 43.75%.

Reason: Curved surface area is increased when diameter decreases

Assertion: The external dimensions of a wooden box are 18 cm, 10 cm and 6 cm respectively and thickness of the wood is 15 mm, then the internal volume is 765 cm³.

Reason: If external dimensions of a rectangular box be l, b and h and the thickness of its sides be x , then its internal volume is (l – 2x) (b – 2x) (h – 2x).

Answer Key-

Multiple Choice questions-

Very Short Answer :

Short Answer :

Long Answer :

Case Study Answer:

Assertion Reason Answer-

(c) A is true but R is false.

(a) Both A and R are true and R is the correct explanation of A.

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