Category : 8th Class
Mensuration
Learning Objectives
Mensuration
Mensuration is the branch of mathematics which deal with the study of geometric shapes, their area, volume and related parameters. Some important formulae of area and volume are listed below.
Area of some plane figures:
Figure |
Shape |
Area |
Rectangle |
\[a\times b\] |
|
Square |
\[\begin{align} & {{a}^{2}} \\ & \frac{1}{2}{{d}_{1}}{{d}_{2}} \\ \end{align}\] Where \[{{d}_{1}}={{d}_{2}}=\sqrt{2}a\] |
|
Triangle |
\[\frac{1}{2}\times b\times h\] |
|
Parallelogram |
\[b\times h\]
|
|
Rhombus |
\[a\times h\] \[\frac{1}{2}{{d}_{1}}{{d}_{2}}\] where \[d_{1}^{2}+d_{2}^{2}=4{{a}^{2}}\] |
|
Trapezium |
\[\frac{1}{2}h\left( a+b \right)\] |
|
Circle |
\[\pi {{r}^{2}}\] |
Area and volume of some solid figures:
Figure |
Shape |
Total Surface area |
Volume |
Cube |
\[6{{a}^{2}}\] |
\[{{a}^{3}}\] |
|
Cuboid |
\[2\left( lb+bh+hl \right)\] |
\[lbh\] |
|
Cylinder |
\[2\pi r\left( r+h \right)\] where \[\pi {{r}^{2}}\] = Surface area of top \[\pi {{r}^{2}}\] = Surface area of bottom \[2\pi rh\] = Curved surface area |
\[\pi {{r}^{2}}h\] |
|
Cone |
\[\pi {{r}^{2}}+\pi rl\] where \[\pi {{r}^{2}}\] = surface area of base \[\pi rl\] = curved surface Here |
\[\frac{1}{3}\pi {{r}^{2}}h\] |
Some important conversions should also keep in mind.
\[1\,\,c{{m}^{2}}\] = 1 mL
1 L = \[1000\,c{{m}^{3}}\]
\[1\,{{m}^{2}}\] = \[1000000\,c{{m}^{3}}=1000\,L\]
Visualizing Solid Shapes
Commonly Asked Questions
1. A 'T' shaped letter is made by sticking together 2 cuboids as shown in the diagram. What is the total volume of the letter 'T'?
(a) \[57\,c{{m}^{3}}\]
(b) \[64\,\,c{{m}^{3}}\]
(c) \[48\,\,c{{m}^{3}}\]
(d) \[25\,\,c{{m}^{3}}\]
(e) None of these
Answer: (c)
Explanation: Volume of upper part \[=7\times 2\times 2=28\,c{{m}^{3}}\].
Volume of lower part \[=5\times 2\times 2=20\,c{{m}^{3}}.\]
Total volume \[=20+28=48\,c{{m}^{3}}.\]
2. A polyhedron has 12 faces and 20 vertices. It contains how many edges?
(a) 12 (b) 32
(c) 30 (d) 4
(e) None of these
Answer: (c)
Explanation: Using Euler's Formula, \[F+V-E=2\]
\[\Rightarrow \,12+20-E=2\Rightarrow E=32-2=30\]
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