Surface Area & Volume
From DEP-SSA WiKi
SURFACE AREA & VOLUME OF 3-D OBJECTS
--At the end of the module learners will be able to explain/tell about
- Vertices's, edges and faces of Cube,Cuboid,Cylinder,Cone,Sphere
- Surface,Surface area and volume of above 3D objects
Here with the help of the Written text ,images, Audio and Video an effort is made to explain the above said topics
Surfaces,Edges & vertices
Surface is a plane that can not be carried.
Vertices It is a terminal point of intersection of atleast two Edges .
Area:-space encircled by an enclosed figure on a surface is called area of the figure.
A cuboid is a rectangular solid which has six rectangular faces. A cube is a rectangular solid which has six square faces.
In the figure alongside of the cuboid, length = AB = CD = EF = GH. In the figure alongside of the cuboid, breadth = AD = BC = EH = FG. In the figure alongside of the cuboid, height = AE = BF = CG = DH.
Total surface area of a cuboid = 2 (Length × Breadth + Breadth × Height + Length × Height) In the figure alongside of the cuboid, total surface area = 2 (AB × BC + BC × BF + AB × BF).
ExampleFind the total surface area (in m2) of a cuboid 8 m long, 4 m broad and 2 m high. HTML: <NOSCRIPT>HTML: </NOSCRIPT>Total surface area of a cuboid = 2 (Length × Breadth + Breadth × Height + Length × Height)= 2 (8 × 4 + 4 × 2 + 8 × 2) = 112 m2.
Lateral surface area of a cuboid = 2 (Length + Breadth) × Height In the figure alongside of the cuboid, lateral surface area = 2 (AB + BC) × BF.
ExampleFind the lateral surface area (in m2) of a cuboid 10 m long, 4 m broad and 2 m high
Surface area of cuboid = 2 (Length × Breadth + Breadth × Height + Length × Height)
=2(10x4 + 4x2 + 2x10 ) = 2(40 + 8 + 20 ) = 2(68) = 136 sq. m.
Surface Area of a Cone
The equation for finding the surface area of a right circular cone:
Lateral Surface Area = (π * R) * L Where "L" is called Slant Hight so L = [sqrt(R² + H²)]
Base Surface Area = π * R2
Total Surface Area = Lateral SA + Base SA
Total Surface Area = (π * R2) + (π * R * L )
- π = Pi ~ 3.14159
- R = Radius
- H = Height
Currently when solving for the radius or height the slant height must be entered, not the total surface area.
Note: "ab" means "a" multiplied by "b". a" means "a squared", which is the same as "a" times "a".
|Surface Area of a Cube = 6 a 2|
(a is the length of the side of each edge of the cube)
In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.
|Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h|
(h is the height of the cylinder, r is the radius of the top)
Surface Area = Areas of top and bottom +Area of the side
Surface Area = 2(Area of top) + (perimeter of top)* height
Surface Area = 2(pi r 2) + (2 pi r)* h
In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.
You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r2). Since there is both a top and a bottom, that gets multiplied by two.
The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.
Add those two parts together and you have the formula for the surface area of a cylinder.
Surface Area = 2(pi r 2) + (2 pi r)* h
|Tip! Don't forget the units.|
The space within an object is termed its volume or capacity. It is calculated by multiplying
the objects three dimensions together (height, width and depth). The dimensions must be
measured in the same units, i.e. they must all be in cm or metres or feet or inches etc.. Since
the volume is three measurements multiplied together it is always measured in cubic
measurements e.g. metres3, feet3, miles3 or kilometres3.
You should find this process fairly easy for cubes and cuboids, but other shapes are slightly
Volume of Cubes and Cuboids.
A cube is a three-dimensional shape with its sides the same length, and all its angles being
A cuboid is similar in that all its angles are 900 but its sides do not have to be the same length.
Their volumes are calculated using the formula volume = w x d x h where w = width; d =
depth; h = height.
1. A hall is 25 metres long, 17.6 metres wide and 3.15 metres high, what is its volume?
The volume is the length x width x height which is 25 x 17.6 x 3.15 = 1386 m3.
2. A box measures 3 feet long, 1 foot 3 inches wide and 2 foot 6 inches high, what is its
The dimensions have to be turned to feet from feet and inches, so they become
3 x 1 3/12 x 2 6/12 = 3 x 1.25 x 2.5 = 9.375 ft3.
1. A computer case has the following dimensions 58cm (l), 45cm (w) and 28cm
(h). What is its volume?
2. A cube has sides of 50cm. What is its volume in cm3 and m3?
3. An organisation keeps all its files in rigid cardboard folders, each one 32cm by
25cm by 2.2cm. They need to store 1000 of these folders in a storeroom.
How much space would they take up?
4. A plank of wood is 2.8m long, 65 cm wide and 9cm high. What is its volume?
Volume of Cylinder
Volume of Cylinder A cylinder is a solid that has two parallel faces which are congruent circles. These faces form the bases of the cylinder. The cylinder has one curved surface. The height of the cylinder is the perpendicular distance between the two bases. The volume of a cylinder is given by the formula: Volume = Area of base × height V = r2h where r = radius of cylinder and h is the height or length of cylinder. Volume of hollow cylinder Sometimes you may be required to calculate the volume of a hollow cylinder or tube. Volume of hollow cylinder where R is the radius of the outer surface and r is the radius of the inner surface.
===Assessment – Now learners can take this exercise to evaluate themselves,what they have learned. === Exercise
- Find the surface area of a cuboid having length, breadth and height 10cm., 8 cm, and 12cm, respectively .
- Find the curved surface area of a cone having radius 14cm.and slant height 26cm.
- Find the total surface area of a right circular cone having radius 8cm. and height 6cm.
- Radius of a sphere is 7cm. , find the area of the same.
- A sphere and a hemisphere are of the same radius what will be the ratio of their surface areas.
- A hemisphere has radius 14 cm, find its curved surface area.
- Prove that the volume of the cylinder is three times the volume of the cone having same radius and height..
- Find the total surface area of the cylinder having radius 21cm and 24 cm.
Say true of false
1- Cube is the 3 dimensional body having all sides equa. (T/ F)
2-Cuboid is 2 dimensional body having sides unequal. (T/F)
3-Volume of a cone is one third of the cylinder having same radius and height. (T/F)
4- Shape of the “hair “ is cylinder. (T/F)
5- Sphere has only one surface whereas hemisphere has two. (T/F)
6- If surface area of a cone and a cylinder having equal radii are equal ,then height of cone will be greater than its radius. (T/F)
Think and try to explain
1- Why cone is generally used to give ice cream instead of cylinder?
2- In nature most of the things are of about spherical shape.
Developed by Manjinder Sing(Ptiyala PANJAB),Shara,Ravindra ku.Sharma(Ajmer,Rajasthan) and U.C.Pandey(Almoda Uttrakhand)