Solid figures are three-dimensional figures that include cube, cuboid, cylinder, cone, sphere and hemisphere.
A solid figure is made up of only its boundary (or outer surface).
Surface Area: Measuring the surface (or boundary) constituting the solid.
Volume: Measuring the space region enclosed by the solid figure. Volume is measured in cubic units.
1. Cuboid
A cuboid has six rectangular regions as its faces. Opposite faces are congruent and parallel to each other. The two adjacent faces meet in a line segment called an edge of the cuboid. There are in all 12 edges of a cuboid. There are 8 corners or vertices of a cuboid. In all there are four diagonals of cuboid.
At each vertex, three edges meet. One of these three edges is taken as the length, the second as the breadth and third is taken as the height (or thickness or depth) of the cuboid.
Length (l), Breadth (b), Height (h)
The surface area of the cuboid is equal to the sum of the areas of all the six rectangles.
Lateral Surface Area = 2h(l + b)
Total Surface Area = 2(lb + bh + lh)
Volume = length × breadth × height = lbh
Diagonal = √(l2 + b2 + h2)
Example: Length, breadth and height of cuboid are 4 cm, 3 cm and 12 cm respectively. Find (i) surface area (ii) volume and (iii) diagonal of the cuboid.
Example: Find the volume of a cuboidal stone slab of length 3 m, breadth 2 m and thickness 25 cm.
Example: The length and breadth of a cuboidal tank are 5 m and 4 m respectively. If it is full of water and contains 60 m3 of water, find the depth of the water in the tank.
Example: A wooden box 1.5 m long, 1.25 m broad, 65 cm deep and open at the top is to be made. Assuming the thickness of the wood negligible, find the cost of the wood required for making the box at the rate of Rs. 200 per m2.
Example: A river 10 m deep and 100 m wide is flowing at the rate of 4.5 km per hour. Find the volume of the water running into the sea per second from this river.
Example: A tank 30 m long, 20 m wide and 12 m deep is dug in a rectangular field of length 588 m and breadth 50 m. The earth so dug out is spread evenly on the remaining part of the field. Find the height of the field raised by it.
2. Cube
A cube is a special type of cuboid in which length = breadth = height
l = b = h
Side: a
Lateral Surface Area = 4a2
Total Surface Area = 6a2
Volume = a3
Diagonal = √3a
Example: Volume of a cube is 2197 cm3. Find its surface area and the diagonal.
3. Cylinder
Rotate a rectangle about one of its edges. The solid generated as a result of this rotation is called a right circular cylinder. The two ends (or bases) of a right circular cylinder are congruent circles. The surface formed by two circular ends are flat and the remaining surface is curved.
Radius (r), Height (h)
Take a hollow cylinder of radius r and height h and cut it along any line on its curved surface parallel to the line segment joining the centres of the two circular ends. You will get a rectangle of length 2πr and breadth h. Area of this rectangle is equal to the area of the curved surface of the cylinder.
Curved or Lateral Surface Area = 2πrh
In case the cylinder is closed at both the ends, then the total surface area of the cylinder:
Total Surface Area = = 2πrh + 2πr2 = 2πr(r + h)
Volume = Area of the base × height = πr2 × h = πr2h
Example: The radius and height of a right circular cylinder are 7 cm and 10 cm respectively. Find its (i) curved surface area (ii) total surface area, and (iii) volume.
Example: A hollow cylindrical metallic pipe is open at both the ends and its external diameter is 12 cm. If the length of the pipe is 70 cm and the thickness of the metal used is 1 cm, find the volume of the metal used for making the pipe.
Example: Radius of a road roller is 35 cm and it is 1 metre long. If it takes 200 revolutions to level a playground, find the cost of levelling the ground at the rate of Rs. 3 per m2.
Example: A metallic solid of volume 1 m3 is melted and drawn into the form of a wire of diameter 3.5 mm. Find the length of the wire so drawn.
4. Cone
Rotate a right triangle about one of its side containing the right angle. The solid generated as a result of this rotation is called a right circular cone. End or base of a right circular cone is a circle. The surface formed by the base of the cone is flat and the remaining surface of the cone is curved.
Radius (r), Height (h), Slant Height (l)
Slant Height = l = √(r2 + h2)
Take a hollow right circular cone of radius r and height h and cut it along its slant height. You will get a sector of a circle of radius l and its arc length is equal to 2πr.
Curved surface of the cone = Area of the sector
Curved or Lateral Surface Area = πrl
If the area of the base is added to the above, then it becomes the total surface area.
Total Surface Area = πrl + πr2 = πr(l + r)
Take a right circular cylinder and a right circular cone of the same base radius and same height. Now, fill the cone with water and pour it in to the cylinder. Repeat the process three times. You will see that the cylinder is completely filled with the water. It shows that volume of a cone with radius r and height h is one-third the volume of the cylinder with radius r and height h.
Volume = 1/3 x πr2h
Example: The base radius and height of a right circular cone is 7 cm and 24 cm. Find its curved surface area, total surface area and volume.
Example: A conical tent is 6 m high and its base radius is 8 m. Find the cost of the canvas required to make the tent at the rate of Rs. 120 per m2 (Use π = 3.14).
5. Sphere
Rotate a semicircle about its diameter. The solid so generated with this rotation is called a sphere. The locus of a point which moves in space in such a way that its distance from a fixed point remains the same is called a sphere. The fixed point is called the centre of the sphere and the same distance is called the radius of the sphere.
Radius: r
Surface area of the hemisphere = 2 × area of the circle
Lateral or Curved Surface Area = 4πr2
Take a hollow hemisphere and a hollow right circular cone of the same base radius and same height (say r). Now fill the cone with water and pour it into the hemisphere. Repeat the process two times. You will see that hemisphere is completely filled with water. It shows that volume of a hemisphere of radius r is twice the volume of the cone with same base radius and same height.
Volume = 4/3 x πr3
Example: Find the surface area and volume of a sphere of diameter 21 cm.
6. Hemisphere
If a sphere is cut into two equal parts by a plane passing through its centre, then each part is called a hemisphere.
Radius: r
Lateral Surface Area = 2πr2
Total Surface Area = 3πr2
Volume = 2/3 x πr3
Example: The volume of a hemispherical bowl is 2425.5 cm3. Find its radius and surface area.