Please refer to Surface Areas and Volumes Class 9 Mathematics notes and questions with solutions below. These revision notes and important examination questions have been prepared based on the latest Mathematics books for Class 9. You can go through the questions and solutions below which will help you to get better marks in your examinations.
Class 9 Mathematics Surface Areas and Volumes Notes and Questions
Surface Areas of Cubes and Cuboids
Surface Areas of a Cube and a Cuboid
We give gifts to our friends and relatives at one time or another. We usually wrap our gifts in nice and colourful wrapping papers. Look, for example, at the nicely wrapped and tied gift shown below.
Clearly, the gift is packed in box that is cubical or shaped like a cube. Suppose you have a gift packed in a similar box. How would you determine the amount of wrapping paper needed to wrap the gift? You could do so by making an estimate of the surface area of the box. In this case, the total area of all the faces of the box will tell us the area of the wrapping paper needed to cover the box.
Knowledge of surface areas of the different solid figures proves useful in many real-life situations where we have to deal with them. In this lesson, we will learn the formulae for the surface areas of a cube and a cuboid. We will also solve some examples using these formulae.
Did You Know?
♦ The word ‘cuboid’ is made up of ‘cube’ and ‘-oid’ (which means ‘similar to’). So, a cuboid indicates something that is similar to a cube.
♦ A cuboid is also called a ‘rectangular prism’ or a ‘rectangular parallelepiped’.
Formulae for the Surface Area of a Cuboid
Consider a cuboid of length l, breadth b and height h.
The formulae for the surface area of this cuboid are given as follows:
Lateral surface area of the cuboid = 2h (l + b)
Total surface area of the cuboid = 2 (lb + bh + hl)
Here, lateral surface area refers to the area of the solid excluding the areas of its top and bottom surfaces, i.e., the areas of only its four standing faces are included. Total surface area refers to the sum of the areas of all the faces.
Did You Know?
Two mathematicians named Henri Lebesgue and Hermann Minkowski sought the definition of surface area at around the twentieth century.
Did You Know?
The concept of surface area is widely used in chemical kinetics, regulation of digestion, regulation of body temperature, etc.
Formulae for the Surface Area of a Cube
Consider a cube with edge a.
The formulae for the surface area of this cube are given as follows:
Lateral surface area of the cube = 4a2
Total surface area of the cube = 6a2
Here, lateral surface area refers to the area of the solid excluding the areas of its top and bottom surfaces, i.e., the areas of only its four standing faces are included. Total surface
area refers to the sum of the areas of all the faces.
Did You Know?
♦ A cube can have 11 different nets.
♦ Cubes and cuboids are convex polygons that satisfy Euler’s formula, i.e., F + V − E = 2.
Know More
Length of the diagonal in a cube and in a cuboid
A cuboid has four diagonals (say AE, BF, CG and DH). The four diagonals are equal in length.
Let us consider the diagonal AE.
In rectangle ABCD, length of diagonal AC = √l2 + b2
Now, ACEG is a rectangle with length AC and breadth CE or h.
So, length of diagonal AE = √AC2 + CE2
= √(√l2+b2) + h2
= √l2+b2+h2
∴ Length of the diagonal of a cuboid = √l2+b2+h2
A cube is a particular case of cuboid in which the length, breadth and height are equal to a.
∴ Length of the diagonal of a cube = √a2+a2+a2 = √3a2 = √3a
Surface Area of Right Circular Cylinders
Surface Area of a Right Circular Cylinder
We come across many objects in our surroundings which are cylindrical, i.e., shaped like a cylinder, for example, pillars, rollers, water pipes, tube lights, cold-drink cans and LPG cylinders. This three-dimensional figure is found almost everywhere.
We can easily make cylindrical containers using metal sheets of any length and breadth.
Say we have to make an open metallic cylinder (as shown below) of radius 14 cm and height 40 cm. How will we calculate the dimensions of the metal required for making this specific cylinder?
We will do so by calculating the surface area of the required cylinder. This surface area will be equal to the area of metal sheet required to make the cylinder.
Knowledge of surface areas of three-dimensional figures is important in finding solutions to several real-life problems involving them. In this lesson, we will learn the formulae for the surface area of a right circular cylinder. We will also solve examples using these formulae.
Features of a right circular cylinder
1. A right circular cylinder has two plane surfaces circular in shape.
2. The curved surface joining the plane surfaces is the lateral surface of the cylinder.
3. The two circular planes are parallel to each other and also congruent.
4. The line joining the centers of the circular planes is the axis of the cylinder.
5. All the points on the lateral surface of the right circular cylinder are equidistant from the axis.
6. Radius of circular plane is the radius of the cylinder.
Two types of cylinders are given below.
1. Hollow cylinder: It is formed by the lateral surface only. Example: A pipe
2. Solid cylinder: It is the region bounded by two circular plane surfaces with the lateral surface. Example: A garden roller
Formulae for the Surface Area of a Right Circular Cylinder
Consider a cylinder with base radius r and height h.
The formulae for the surface area of this cylinder are given as follows:
Curved surface area of the cylinder = 2πrh
Area of two circular faces of cylinder = 2πr2
Total surface area of the cylinder = 2πr (r + h)
Note: We take π as a constant and its value as 22/7 or 3.14.
Here, curved (or lateral) surface area refers to the area of the curved surface excluding the top and bottom surfaces. Total surface area refers to the sum of the areas of the top and bottom surfaces and the area of the curved surface.
Did You Know?
Pi
♦ Pi is a mathematical constant which is equal to the ratio of the circumference of a circle to its diameter.
♦ It is an irrational number represented by the Greek letter ‘π’ and its value is approximately equal to 3.14159.
♦ William Jones (1706) was the first to use the Greek letter to represent this number.
♦ Pi is also called ‘Archimedes’ constant’ or ‘Ludolph’s constant’.
♦ Pi is a ‘transcendental number’, which means that it is not the solution of any finite polynomial with whole numbers as coefficients.
♦ Suppose a circle fits exactly inside a square; then, pi = 4 x Area of the circle /Area of the Square
Whiz Kid
There are many types of cylinders—right circular cylinder (whose base is circular), elliptic cylinder (whose base is an ellipsis or oval), parabolic cylinder, hyperbolic cylinder, imaginary elliptic cylinder, oblique cylinder (whose top and bottom surfaces are displaced from each other), etc.
Formulae for the Surface Area of a Right Circular Hollow Cylinder
Consider a hollow cylinder of height h with external and internal radii R and r respectively,
Here, curved surface area, CSA = External surface area + Internal surface area
Here, thickness of the hollow cylinder = R − r.
Surface Areas of Cones
Surface Area of a Right Circular Cone
Traffic cones, conical tents, party hats, ice cream cones are some examples of objects shaped like a cone. The knowledge of the surface area of a cone is essential in the manufacture of such conical objects. Take, for example, the following case.
X Ltd. is a company that organizes adventures trips. It has a contract with Y Ltd., a company that manufactures tents. Y Ltd. uses canvas to make the specific conical tents ordered by X Ltd. Now, the area of canvas required to make one such conical tent is exactly equal to the surface area of the conical tent. Thus, Y Ltd is able to order the required amount of canvas from the market to prepare the tents according to the specifications.
This is just one of the many examples from real life involving the concept of surface area. In this lesson, we will learn the formulae for the surface area of a right circular cone. We will also apply the formulae in solving a few examples.
Formulae for the Surface Area of a Right Circular Cone
Consider a cone with a base radius r, height h and slant height l.
The fixed point V is the vertex of the cone and the fixed line VO is the axis of the cone.
The length of line segment joining the vertex to the centre O of the base is called the height of the base and the length of the line segment joining the vertex to any point on the circular edge of the base is called the slant height of the cone.
The relation between the height, radius and slant height of the cone is: l2 = r2 + h2.
The formulae for the surface area of the given cone are given as follows:
Curved surface area of the cone = πrl
Total surface area of the cone = πr (l + r)
Here, curved (or lateral) surface area refers to the area of the curved surface excluding the base, and total surface area refers to the sum of the area of the base and the area of the curved surface.
Did You Know?
A cone is the shape obtained by rotating a right triangle around one of its two shorter sides.
Did You Know?
A cone is a three-dimensional geometric figure that does not have uniform or congruent cross-sections.
Largest Cone Cut Out from a Cylinder
Surface Areas of Sphere and Hemisphere
Surface Areas of a Sphere and a Hemisphere
What images come to your mind when the word ‘sphere’ is mentioned? The light hollow ball used in table tennis, the leather ball used in cricket, the inflatable balls used in the games of football and basketball and the heavy metallic shots used for shot-putting are all examples of the perfectly round three-dimensional shape called sphere. Now, if you were to cut each of the objects mentioned above along its diameter, then you would obtain the three-dimensional figure called hemisphere. As its name indicates, a hemisphere is the half of a sphere. Any sphere when cut along the diameter yields two equal hemispheres.
A sphere has only a curved surface; so, in its case, the total surface area is the same as the area of its curved surface. This, however, is not the case with a hemisphere. Consider the whole watermelon and the half of the same shown below.
Clearly, the whole watermelon has only a curved exterior, but what about its half? Observe how the half of the watermelon has both a curved exterior and a flat surface. So, in case of a hemisphere the total surface area is different from the area of its curved surface.
Sphere and hemisphere
A sphere is a solid described by the rotation of a semi-circle about a fixed diameter.
Properties of a sphere:
1. A sphere has a centre.
2. All the points on the surface of the sphere are equidistant from the centre.
3. The distance between the centre and any point on the surface of the sphere is the radius of the sphere.
Hemisphere: A plane through the centre of the sphere divides it into two equal parts each is called a hemisphere.
In this lesson, we will learn the formulae for the surface areas of a sphere and a hemisphere. We will also solve problems using the same.
Formula for the Surface Area of a Sphere
Consider a sphere of radius r.
The formula for the surface area (curved or total) of this sphere is given as follows:
Surface area of a sphere = 4πr2
As mentioned before, the total surface area of a sphere is the same as its curved surface area since a sphere has only a curved surface.
Did You Know?
Among all geometric shapes, a sphere has the smallest surface area for a given volume.
Take, for example, bubbles and water droplets. Their spherical shape enables them to hold as much air as possible with the least surface area.
Formulae for the Surface Area of a Hemisphere
A hemisphere is a three dimensional solid having two faces, one edge and no vertex.
Consider a hemisphere with radius r.
The formulae for the surface area of this hemisphere are given as follows:
Curved surface area of the hemisphere = 2πr2
Total surface area of the hemisphere = 3πr2
Here, curved (or lateral) surface area refers to the area of the curved surface excluding the area of the top surface, and total surface area refers to the sum of the area of the curved surface and the area of the top surface.
Formula for the Surface Area of a Hollow Hemisphere Sphere
Since a hemisphere is obtained by cutting a sphere along its diameter, the radius of a hemisphere is the same as that of the sphere from which it is cut.
Let R and r be the outer and inner radii of the hollow hemisphere.
Formula for the Surface Area of a Hollow Hemisphere Sphere
Let R and r be the outer and inner radii of the hollow hemisphere.
Here, curved surface area = Outer surface area + Inner surface area
Volumes of Cubes and Cuboids
Volumes of a Cube and a Cuboid
Abhinav’s mother gives him a container, asking him to go to the neighbouring milk booth and buy 2.5 L of milk. What does ‘2.5 L’represent? It represents the amount of milk that Abhinav needs to buy. In other words, it is the volume of milk that is to be bought.
After buying the milk, Abhinav notices that the container is full up to its brim. He says to himself, ‘This container has no capacity to hold any more milk.’ What does the word ‘capacity’ indicate? The space occupied by a substance is called its volume. The capacity of a container is the volume of a substance that can fill the container completely. In this case, the volume and the capacity of the container are the same. The standard units which are used to measure the volume are cm3 (cubic centimetre) and m3 (cubic metre).
In this lesson, we will learn the formulae for the volumes or capacities of cubic and cuboidal objects. We will also solve examples using these formulae.
Did You Know?
A cube is one among the five platonic solids. This means that it is a regular and convex polyhedron with the same number of faces meeting at each vertex.
Formulae for the Volumes of a Cube and a Cuboid
Consider a cube with an edge a.
The formula for the volume of this cube is given as follows:
Volume of the cube = a3
Now, consider a cuboid with length l, breadth b and height h.
The formula for the volume of this cuboid is given as follows:
Volume of the cuboid = l × b × h
Concept Builder
The units of capacity and volume are interrelated as follows:
♦ 1 cm3 = 1 mL
♦ 1000 cm3 = 1 L
♦ 1 m3 = 1 kL = 1000 L
Did You Know?
♦ A cube has the maximum volume among all cuboids with equal surface area.
♦ A cube has the minimum surface area among all cuboids with equal volume.
Volume of Right Circular Cylinders
Water tanks like the ones shown below are a common enough sight.
Clearly, these tanks are cylindrical or shaped like a cylinder. The choice of this shape for a water tank (and many other storage containers) is because a cylinder provides a large volume for a little surface area. Also, this shape can withstand much more pressure than a cube or a cuboid, which makes it easy to transport. Another example of a cylindrical storage container is the LPG cylinder.
The amount of space occupied by a water tank is the same as the volume of the tank. So, to find the capacity or the amount of space occupied by a tank, we need to find the volume of the tank. In this lesson, we will learn the formula to calculate the volume of a right circular cylinder and solve some examples using the same.
Did You Know?
LPG tanks are cylinder-shaped so that they can withstand the pressure inside them. If these tanks were square or rectangular in shape, then an increase in pressure inside them would cause the tanks to reform themselves so as to gain a rounded shape. This, in turn, could result in leakage at the corners. Actual LPG tanks are designed to have no corners.
Formula for the Volume of a Right Circular Cylinder
Consider a solid cylinder with r as the radius of the circular base and h as the height.
The formula for the volume of this right circular solid cylinder is given as follows:
Volume of the solid cylinder = Area of base × Height
Volume of the solid cylinder = πr2h
Consider a hollow cylinder with internal and external radii as r and R respectively, and height as h.
The formula for the volume of this right circular hollow cylinder is given as follows:
Volume of the hollow cylinder = π (R2 − r2) h
In right prisms, top and base surfaces are congruent and parallel while lateral faces are perpendicular to the base. Thus, their volumes can also be calculated in the same manner as that of right cylinders.
Volume of the right prism = Area of base × Height
Did You Know?
The volume of a pizza (which is always cylindrical in shape) is hidden in its name itself. If we take the radius of a pizza to be ‘z’ and its thickness to be ‘a’, then its volume is or ‘pi.z.z.a’.
Volume of Cone
Ice creams are loved by one and all. Take a look at the one shown.
Clearly, what is shown above is an ice cream cone, i.e., ice cream inside a crisp conical wafer. The amount of ice cream present in the cone is equal to the volume of the cone. In other words, the number of cubic units of ice cream that will exactly fill the cone is the volume of the cone.
In this lesson, we will learn the formula for the volume of a right circular cone and solve examples using the same.
Did You Know?
A waffle maker named Ernest Hamwi is credited by a few to be the inventor of the ice cream cone. He is said to have come up with the idea in 1904 to help an ice cream vendor
who had run out of dishes to serve ice cream.
Formula for the Volume of a Right Circular Cone
Consider a cone of radius r and height h.
The formula for the volume of this right circular cone is given as follows:
Volume of the cone = 1/3 πr2h
Using the above formula, we can find the cubic units of ice cream that exactly fill a cone.
Let us say the radius and height of an ice cream cone are 3.5 cm and 9 cm respectively.
Then,
Using the above formula, we can find the cubic units of ice cream that exactly fill a cone.
Let us say the radius and height of an ice cream cone are 3.5 cm and 9 cm respectively.
Then,
Volume of the cone = 1/3 πr2h = 1/3 x 22/7 x 3.5 x 3.5 x 9 cm3 = 115.5 cm3
Thus, the amount of ice-cream that exactly fills the cone is 115.5 cm3.
Did You Know?
For a cone and a cylinder with the same base radius and height, the volume of the cone is one-third that of the cylinder.
Hard
Volumes of Spheres and Hemispheres
Consider the basketball shown.
Clearly, the basketball, (or, for that matter, any ball) is spherical or shaped like a sphere.
Being inflatable, a basketball acquires its shape on being filled with air. The amount of air inside a basketball filled to its capacity helps us ascertain the volume of the ball.
In this lesson, we will learn the formulae for the volumes of spheres and hemispheres, and solve problems using the same.
Formula for the Volume of a Sphere
Consider a solid sphere of radius r.
The formula for the volume of this solid sphere is given as follows:
Volume of the solid sphere = 4/3 πr3
Using the above formula, we can calculate the amount of air in a basketball filled to its capacity.
Suppose we have a basketball with radius 18 cm. Then,
Volume of the basketball = 4/3 πr3 x 4/3 x 22/7 x 18 x 18 x 18 cm3 = 24438.86 cm3
Thus, the amount of air filled inside the basketball is 24438.86 cm3.
Formula for the Volume of a Hemisphere
On cutting a solid spherical object into two equal parts, we obtain two solid hemispheres.
The radius of each hemisphere so obtained is the same as that of the sphere.
Consider a hemisphere of radius r.
Since hemispheres are obtained by cutting a sphere in half, the volume of each resultant hemisphere is equal to half of that of the sphere.
The formula for the volume of this solid hemisphere is arrived at as follows:
Volume of the solid hemisphere = 1/2 x 4/3 πr3
= 2/3 πr3
Formula for the Volume of a Hollow Hemisphere
Let R and r be the outer and inner radii of the hollow hemisphere.
Volume of a hollow hemisphere = Volume of outer hemisphere − Volume of inner hemisphere
= 2/3π R3 – 2/3πr3
= 2/3π(R3 – r3)
