Surface Areas and Volumes

Chapter 13 Class 9th Mathematics chapter 13

A cuboid is a solid bounded by six rectangular plane regions. It has length, width and height.
A cuboid whose all edges are equal is called a cube.
A cylinder is a closed solid that has two parallel (usually circular) bases connected by a curved surface.
A cone is a solid that has a circular base and a single vertex.
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.
A hemisphere is half of a sphere.
Surface area of a solid is the sum of the areas of all its faces.
The total surface area of any object will be greater than its lateral surface area.
In case of a room, lateral surface area means the area of the four walls of the room, whereas total surface area means the area of four walls plus the area of the floor and the ceiling.
Volume is the space occupied by an object.
The unit of measurement of both volume and capacity is cubic unit such as cubic feet, cubic cm. cubic m etc.
If l, b, h denote respectively the length, breadth and height of a cuboid, then:

Lateral surface area or Area of four walls = 2(ℓ + b) h

Total surface area = 2(ℓb + bh + hℓ)

Volume = ℓ x b x h

Diagonal of a cuboid =l2+b2+h2

13. If the length of each edge of a cube is ‘a’ units, then:

Lateral surface area = 4 x (edge)2

Total surface area = 6 x (edge)²

Volume = (edge)3

Diagonal of a cube =3x edge

14. If r and h respectively denote the radius of the base and the height of a right circular cylinder, then: Area of each end or Base area =πr2

Area of curved surface or lateral surface area = perimeter of the base x height = 2rh

Total surface area (including both ends) = 2Πrh + 2Πr² = 2r Π(h + r)

Volume = Area of the base × height = Πr²h

15. If R and r respectively denote the external and internal radii of a right circular hollow cylinder and h denotes its height, then:

Area of each circular base = ΠR² – Πr²

Area of curved surface = 2Π(R + r)h

Total surface area = (External surface) + (Internal surface)

= (2ΠRh + 2Πrh) + 2 (ΠR² – Πr²)

Volume of = (External volume) – (Internal volume)

= (ΠR²h – Πr²h) = Πh (R² – r²)

16. If r, h and l respectively denote the radius, height and slant height of a right circular

17. If r is the radius of a sphere, then: Surface area = 4r²

Volume =4/3Πr3

18. If r is the radius of a hemisphere, then:

Area of curved surface = 2r²

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

= 2Πr² + Πr²

= 3Πr²

Volume =2/3Πr3

19. Volume of water flown in the tank in one hour = (area of cross section of the aperture) × (speed in meters per hour)

20. When an object of certain volume is recast into a cylinder, the volume of the cylinder formed will always be equal to the volume of the original object.

21. The solids having the same curved surface do not necessarily occupy the same volume.

22. When an object is dropped into a liquid, the volume of the displaced liquid is equal to the volume of the object that is dipped.

23. Of all the solids having a given volume, the sphere is the one with the smallest surface area. Of all solids having a given surface area, the sphere is the one having the greatest volume.

Cuboid

A cuboid is a three-dimensional Shape. The cuboid is made from six rectangular faces, which are placed at right angles. The total surface area of a cuboid is equal to the sum of the areas of its six rectangular faces.

Total Surface Area of a Cuboid

Consider a cuboid whose length is “l” cm, breadth is b cm and height h cm.

Area of face ABCD = Area of Face EFGH = (l × b) cm2

Area of face AEHD = Area of face BFGC = (b × h) cm2

Area of face ABFE = Area of face DHGC = (l × h) cm2

Total surface area (TSA) of cuboid = Sum of the areas of all its six faces

= 2(l × b) + 2(b × h) + 2(l × h)

TSA (cuboid)= 2(lb + bh +lh)

Lateral Surface Area of a Cuboid

Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces.

The lateral surface area of the cuboid

= Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

= 2(b × h) + 2(l × h)

LSA (cuboid) = 2h (l + b)

Cube

A cuboid whose length, breadth and height are all equal, is called a cube. It is a three-dimensional shape bounded by six equal squares. It has 12 edges and 8 vertices.

Total Surface Area of a cube

For cube, length = breadth = height

Suppose the length of an edge = a

Total surface area (TSA) of the cube = 2(a × a + a × a + a × a)

TSA (cube) = 2 × (3a2) = 6a2

Lateral Surface area of a cube

Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces.

Lateral surface area of cube = 2(a × a + a × a) = 4a2

Right Circular Cylinder

A right circular cylinder is a closed solid that has two parallel circular bases connected by a curved surface in which the two bases are exactly over each other and the axis is at right angles to the base.

Curved Surface area of a right circular 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 will be transformed into a rectangle of length 2πr and height h units. Thus,

Curved surface area (CSA) of a cylinder of base radius r and height h = 2π × r × h

Total surface area of a right circular cylinder

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

⇒ TSA = 2π × r × h + 2 × πr2

⇒ TSA = 2πr (h + r)

Right Circular Cone

A right circular cone is a circular cone whose axis is perpendicular to its base.

Relation between slant height and height of a right circular cone

The relationship between slant height(l) and height(h) of a right circular cone is:

l2 = h2 + r2 (Using Pythagoras Theorem)

Where r is the radius of the base of the cone.

Curved Surface Area of a Right Circular Cone

Consider a right circular cone with slant length l and radius r.

If a perpendicular cut is made from a point on the circumference of the base to the vertex and the cone is opened up, a sector of a circle with radius l is produced as shown in the figure below:

Label A and B and corresponding b1, b2 …bn at equal intervals, with O as the common vertex. The Curved surface area (CSA) of the cone will be the sum of areas of the small triangles: 1/2 × (b1 + b2.……..bn) × l

(b1 + b2…bn) is also equal to the circumference of base = 2πr

CSA of right circular cone = (1/2) × (2πr) × l = πrl (On substituting the values)

Total Surface Area of a Right Circular Cone

Total surface area (TSA) = Curved surface area(CSA) + area of base = πrl + πr2 = πr(l + r)

Sphere

A sphere is a closed three-dimensional solid figure, where all the points on the surface of the sphere are equidistant from the common fixed point called “centre”. The equidistant is called the “radius”.

Surface area of a Sphere

The surface area of a sphere of radius r = 4 times the area of a circle of radius r = 4 × (πr2)

For a sphere Curved surface area (CSA) = Total Surface area (TSA) = 4πr2

Surface Area Formulas

Volume of a Cuboid

The volume of a cuboid is the product of its dimensions.

Volume of a cuboid = length × breadth × height= lbh

Where l is the length of the cuboid, b is the breadth, and h is the height of the cuboid.

Volume of a Cube

The volume of a cube = base area × height.

Since all dimensions are identical, the volume of the cube = a3

Where a is the length of the edge of the cube.

Volume of a Right Circular Cylinder

The volume of a right circular cylinder is equal to base area × its height.

The volume of cylinder =πr2h

Where r is the radius of the base of the cylinder and h is the height of the cylinder.

Volume of a Right Circular Cone

The volume of a Right circular cone is 1/3 times the volume of a cylinder with the same radius and height. In other words, three cones make one cylinder of the same height and base.

The volume of right circular cone = (1/3) πr2h

Where r is the radius of the base of the cone and h is the height of the cone.

Volume of a Sphere

The volume of a sphere of radius r = (4/3)πr3

Volume Formulas

Volume and Capacity

The volume of an object is the measure of the space it occupies, and the capacity of an object is the volume of substance its interior can accommodate. The unit of measurement of either volume or capacity is a cubic unit.