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# Mensuration Formulas and concepts

IACE

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## 1. Definition and Important terms of Mensuration

Mensuration is a mathematical concept which contains measurements of different geometric shapes. Such as measurement of length, area, height, breath, volume, surface area, lateral surface area etc. The geometric shapes can be expressed in either 2D and 3D.

2D objects, such as squares, rectangles, triangles, circles, and so on, can be drawn in a plane, however 3D shapes, such as Cube, cones, spheres, Cylinder and so on, cannot. Mensuration encompasses the use of mathematical formulas and algebraic equations in calculation.

## 2. Types of Geometrical Shapes

There are two geometrical shapes such as

**2D Shape :**2D things have two axis(X and Y axis), It does not have thickness. For example, a square, rectangle, triangle, or circle. These 2D items do not exist in the actual world and can only be represented by utilising simple surfaces because they have only two axes and no thickness.**3D Shape:**They are called solid shapes. 3D shapes have three axes i.e. (X, Y, and Z axis). 3-dimensional shapes have more perimeters to cover as compared with 2 2-dimensional shapes. Some examples of 3 3-dimensional shapes are a Cube, cuboid, Cylinder, Cone, Sphere, Pyramid etc.

## 3. Difference between 2D and 3D Shapes in Mensuration.

2 Dimensional shapes |
3 Dimensional Shapes |

It has X-axis and a Y-axis | It has an X-axis, Y-axis and Z-axis |

2D shapes do have depth and Height | It has height and depth |

The area and perimeter are measured with 2D shapes | curved surface area, Volume, Lateral surface, Total surface area can be measured with 3d shapes. |

## 4. Mensuration formulas of 2D Shapes

Shapes | Perimeter | Area |

Square | 4a | a2 |

Rectangle | 2 ( l + b) | l × b |

Circle | 2 π r | πr2 |

Scalene Triangle | a+b+c | √[s(s−a)(s−b)(s−c)],
Where, s = (a+b+c)/2 |

Isosceles Triangle | 2a + b | ½ × b × h |

Equilateral triangle | 3a | (√3/4) × a2 |

Right Angle Triangle | b + h+ p | ½ × b × h |

Rhombus | 4 × side | ½ × d1 × d2 |

Parallelogram | 2(l+b) | b × h |

Trapezium | a+b+c+d | ½ h(a+c) |

## 5. Mensuration formulas of 3D shapes

Shapes |
Volume |
Curved surface area |
Total surface area |

Cube | a3 | LSA = 4 a2 | 6 a2 |

Cuboid | l × b × h | LSA = 2h(l + b) | 2 (lb +bh +hl) |

Sphere | (4/3) π r3 | 4 π r2 | 4 π r2 |

Hemisphere | (⅔) π r3 | 2 π r2 | 3 π r2 |

Cylinder | π r2 h | 2π r h | 2πrh + 2πr2 |

Cone | (⅓) π r2 h | π r l | πr (r + l) |

## 6. Mensuration problems with solutions

**Q1 If the side length of a square is 10 cm then, find the area and perimeter of a square. **

**Ans.** Side length = a = 10 cm

the area of a square is a2 square units.

Area of the square =

A = 10 × 10 = 100 cm

perimeter of a square is 4a units.

P = 4 × 10 = 40 cm

the perimeter of the square is 20 cm.

**Q2. If the length of the Cuboid is 6 units, the width of 4 units, and the height of 8 units Find the surface area of a cuboid**

**Ans. **Length of the cuboid = 6 units

Breadth of the cuboid = 4 units

Height of the cuboid = 8 units

Surface Area of cuboid = 2 × (lb + bh + lh) square units

= 2[(6 × 4) + (4 × 8) + (6 × 8)]

= 2(24+32+48)

= 2(104)

= 208 units

Therefore, the surface area of the cuboid is 208 square units.

**Q3. If the radius is 10 cm, Find the area and circumference of a circle.**

** Ans.** Radius of circle = 10 cm

Area of a circle = π × r2 (π = 22/7)

= 22/7 × 10 × 10

= 314.28 cm2

Therefore, the Area of the circle = 314.28 square cm

## 7. FAQ's

### 1. What is the Formula of curved surface and total surface area of the Cube?

**Ans. **The curved surface area of Cube is LSA = 4 a2 and the Total surface area of Cube is 6 a2.

### 2. What is the Formula for the curved surface and total surface area of the Cuboid?

**Ans. **The curved surface area of the Cuboid is LSA = 2h(l + b) and the total surface area cuboid is 2 (lb +bh +hl).

### 3. What is the formula for the volume of a sphere?

**Ans.** The formula for the volume of a sphere is (4/3) π r3

### 4. What is the formula for the Curved surface area, total surface area and volume of a Hemisphere?

**Ans. **The formula for the Curved surface area of the hemisphere is 2 π r2 and total surface area of the hemisphere is 3 π r2 and the volume of a Hemisphere is (⅔) π r3

### 5. What is the Formula for the area of the circle?

**Ans. ** The formula for the area of a circle is πr2

### 6. What is the Formula for the area and perimeter of the Trapezium?

**Ans. **The Formula for the area of a trapezium is ½ h(a+c) and perimeter of the Trapezium is a+b+c+d

### 7. What is the Formula for the area and perimeter of the square?

**Ans. **The Formula for the area of a square is a2 and the perimeter of the square is 4a

### 8. What is the Formula for the area and perimeter of the rectangle?

**Ans. **The Formula for the area of a rectangle is l × b and the perimeter of the rectangle is 2 ( l + b)

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