Table of contents  
Mensuration  
Mensuration 3D Formulas  
Hollow Cylinder (Hollow Right Circular Cylinder)  
Differences Between Mensuration 3D and 2D shapes 
A section of mathematics that communicates about the length, volume, or area of various geometric shapes is termed Mensuration. These shapes or patterns exist in two dimensions or three dimensions.
Let us discover a few more definitions linked to the various geometric shapes.
Formula for a Cube:
Formula for a cuboid:
Formula for a cylinder:
Formula for a cone:
With the knowledge of terms, properties and formulas of 3D shapes let us summarise some of the important formulas in the below table, these mensuration formulas are very useful while solving mensuration problems.
Here’s some more 3D shapes with their formulas and images.
A quarter sphere is specifically onefourth share of a full sphere. i.e., if we break a sphere into four equal sections, each portion is termed a quarter sphere. Therefore, the volume of a quarter sphere is onefourth of the volume of a sphere.
The frustum of a cone is the section of the cone without vertex when the given cone is split into two pieces with a plane that is parallel to the bottom of the cone.
The frustum of a cone is also called a truncated cone. Similar to any other 3D shape or object, the frustum of a cone also holds surface area and volume. The formula for the same are as follows:
The volume of a hollow cylinder is defined as the 3D space contained by it. For instance, the volume of glass indicates the available area inside it. In other words, we can say that volume represents the maximum space that can be filled by water if the water flows into the glass.
Here the inner radius of the base is ‘r’, the outer radius of the base is ‘R’ and the height of the hollow right circular cylinder is ‘h’.
Volume = πh(R^{2}−r^{2})
Curved Surface area = 2πRh+2πrh=2πh(R+r)
Total surface area = 2πH(R+r)+2π(R^{2}−r^{2})
Candidates can find different tips and tricks from below for solving the questions related to Mensuration 3D.
Question 1: If the volume of a cube is 4913 cm3, then find the total surface area.
Solution 1: Volume = a^{3}
⇒a^{3}= 4913
∴a = 17 cm
⇒Total surface area = 6a^{2}
∴Total surface area = 6 × 172 = 1734 cm^{2}
Question 2: If the sum of three sides is 45 cm and the length of diagonal is 21 cm in a cuboid, find the total surface area of this cuboid.
Solution 2: l + b + h = 45————— (1)
⇒Diagonal = l^{2}+b^{2}+h^{2}
⇒l^{2}+b^{2}+h^{2 }= 21
⇒l^{2}+b^{2}+h^{2}= 441———— (2)
∴From (1),
⇒(l+b+h)^{2} = 452
⇒l^{2}+b^{2}+h^{2} + 2(lb + bh + hl) = 2025 441 + 2(lb + bh + hl) = 2025
⇒2(lb + bh + hl) = 2025 – 441 = 1584 cm^{2}
∴Surface area of this cuboid is 1584 cm^{2}
Question 3: Find the total surface area and volume of a cylinder which has a height of 21m and a base of diagonal is 12m.
Solution 3: Diagonal = 2 × radius Radius = 12/2 = 6 m Total surface area = 2πrh
⇒Total surface area = 2 × 22/7 × 6 × 21 = 792 m^{2}
⇒Volume = πr^{2}h
∴Volume = 22/7 ×62×21=2376 m^{3}
Question 4: The base of a solid right prism is a triangle whose sides are 9 cm, 12 cm and 15 cm. The slant height of the prism is 5 cm, Find the total surface area of the prism.
Solution 4: Perimeter of base = 9 + 12 + 15 = 36 cm Area of base = ½ × 12 × 9 = 54 cm^{2}
⇒Total surface area = Perimeter of base × slant height + 2 × Area of base Total surface area = (36 × 5) + 2 × 54
∴Total surface area =288 cm^{2}.
56 videos104 docs95 tests

Problem Set on Triangles Video  09:31 min 
Important Formulae: Geometry & Mensuration Doc  8 pages 
1. What is mensuration? 
2. How is mensuration used in real life? 
3. What are the basic formulas used in mensuration? 
4. How can I calculate the area of an irregular shape? 
5. What is the difference between perimeter and area? 
56 videos104 docs95 tests

Problem Set on Triangles Video  09:31 min 
Important Formulae: Geometry & Mensuration Doc  8 pages 

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