Mensuration is the branch of mathematics that deals with the measurement of geometric shapes/figures and their parameters like length, area, volume, lateral surface area, etc. In this post, we will explore all the mensuration formulas for 2D and 3D shapes, Mensuration Definition, providing you with a clear understanding of their calculations and applications.
Definition of Mensuration
Mensuration is a branch of mathematics that deals with the measurement of geometric figures and their parameters like length, area, volume, lateral surface area, etc. These Mensuration shapes exist either in 2D and 3D. Let’s learn the difference between 2D and 3D shapes.
Differences Between 2D and 3D shapes
| 2D Shape | 3D Shape |
|---|---|
| 2D Shapes exist only in two dimensions, namely length and width. They are flat and have no depth or thickness. Examples of 2D shapes include squares, circles, triangles, and rectangles. | 3D Shapes exist in three dimensions, encompassing length, width, and height. They have depth and occupy space. Examples of 3D shapes include cubes, spheres, cylinders, and pyramids. |
| They are represented on a flat surface, such as a piece of paper or a computer screen. When drawn or depicted, 2D shapes appear as outlines or figures with no thickness. | They are represented as solid objects that can be held or visualized in physical or digital space. 3D shapes have volume, and their representation includes depth and thickness. |
| They have properties such as area and perimeter. Area represents the amount of space enclosed by the shape, while perimeter represents the total length of its boundary. | They have properties such as volume, surface area, and capacity. Volume represents the amount of space occupied by the shape, surface area represents the total area of its outer surface, and capacity refers to the amount of liquid or material it can hold. |
| These shapes are easier to visualize and understand since they are represented on a flat surface. They can be easily drawn or sketched. | They require a more advanced level of visualization as they exist in three dimensions. Understanding the spatial relationships and perspectives of 3D shapes can be more challenging. |
| They are commonly used to represent and analyze flat objects and designs in fields such as graphic design, architecture, and engineering blueprints. | They are used to represent and analyze objects in the physical world, such as buildings, sculptures, and everyday objects. They are also essential in fields like 3D modeling, computer graphics, and manufacturing. |
Mensuration Formulas PDF
Mensuration in Math: Exploring Important Terminologies
| Terms | Abbreviation | Unit | Definition |
|---|---|---|---|
| Area | A | m2 or cm2 | The area of a shape is the measure of the extent of its surface or region. It quantifies the amount of space occupied by a shape. Calculating the area allows us to compare and analyze the sizes of different shapes. Various mensuration formulas are used to calculate the area of different shapes, including triangles, rectangles, circles, and more. |
| Volume | V | cm3 or m3 | Volume is a measure of the space occupied by a three-dimensional shape. It is the total capacity or amount of space enclosed by the shape. Understanding volume is crucial when dealing with objects like containers, tanks, or three-dimensional structures. Mensuration formulas for shapes like cubes, cylinders, spheres, and cones help us calculate their volumes accurately. |
| Cube Unit | – | m3 or cm3 | Cube unit, on the other hand, refers to the unit of measurement used to quantify the volume of a three-dimensional shape called a cube. It represents the volume of a cube with side length equal to one unit. |
| Perimeter | P | cm or m | The perimeter of a shape is the distance around its boundary. It represents the total length of the shape’s sides or edges. Perimeter is commonly used in determining the amount of fencing required for a given area or the distance around a closed figure. Mensuration formulas for different shapes, such as squares, rectangles, triangles, and polygons, enable us to calculate their perimeters accurately. |
| Square Unit | – | m2 or cm2 | In mensuration, a square unit refers to the unit of measurement used to quantify the area of a two-dimensional shape. It represents the area of a square with side length equal to one unit. |
| Lateral Surface area | LSA | m2 or cm2 | The lateral surface area is the total area of all the faces of a 3D shape. It is calculated by summing up the areas of the side faces only. The lateral surface area is typically used for shapes such as cylinders and prisms. For example, in a cylinder, the lateral surface area would be the sum of the areas of the curved side faces. |
| Total Surface Area | TSA | m2 or cm2 | The total surface area is the sum of the areas of all the faces of a 3D or Three- Dimensional shape. It includes both the lateral surfaces and the base or top and bottom faces. The total surface area provides a measure of the overall area covered by the shape. For example, in a rectangular prism, the total surface area would include the sum of the areas of all six faces. |
| Curved Surface Area | CSA | m2 or cm2 | Curved surface area is a specific the type of surface area that refers the area of the curved surface of a 3D shape. It is used for shapes that have curved surfaces, such as cones or spheres. The curved surface area excludes the areas of any flat or base surfaces and focuses solely on the curved part of the shape. |
Mensuration Formulas for 2D and 3D Shapes
Here below we will provide all the all the mensuration formulas for 2D and 3D shapes.
Mensuration Formulas for 2D Shape
| Shape | Area (Square units) | Perimeter (units) |
|---|---|---|
| Square | a2 | 4a |
| Rectangle | l × b | 2 ( l + b) |
| Circle | πr2 | 2 π r |
| Scalene Triangle | √[s(s−a)(s−b)(s−c)],Where, s = (a+b+c)/2 | a+b+c |
| Isosceles Triangle | ½ × b × h | 2a + b |
| Equilateral triangle | (√3/4) × a2 | 3a |
| Right Angle Triangle | ½ × b × h | b + hypotenuse + h |
| Rhombus | ½ × d1 × d2 | 4 × side |
| Parallelogram | b × h | 2(l+b) |
| Trapezium | ½ h(a+c) | a+b+c+d |
2D Mensuration Formula Chart
Mensuration Formulas for 3D Shape
| Shape | Volume (Cubic units) | Curved Surface Area (CSA) / Lateral Surface Area (LSA) (Square units) | Total Surface Area (TSA) (Square units) |
|---|---|---|---|
| Cube | a3 | LSA = 4 a2 | 6 a2 |
| Cone | (⅓) π r2 h | π r l | πr (r + l) |
| 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 r 2 | 3 π r 2 |
| Cylinder | π r 2 h | 2π r h | 2πrh + 2πr2 |
3D Mensuration Formula Chart
All the Maths Formulas for Mensuration
Mensuration Formula for Rectangle :
- Area = l×b
- Perimeter = 2(l+b)
Mensuration Formula for Square :
- Area = a×a
- Perimeter = 4a
Mensuration Formula for Parallelogram :
- Area = l×h
- Perimeter = 2(l+b)
Mensuration Formula for Triangle :
- Area =12×b×h or √s(s−a)(s−b)(s−c),where s=a+b+c2
Mensuration Formula for Right angle Triangle :
- Area =12×b×h
- Perimeter = Sum of all sides
Mensuration Formula for Isosceles right angle triangle :
- Area = 12×b×h
- Perimeter = 2a+d, where d=a√2
Mensuration Formula for Equilateral Triangle :
- Area = √34a2
- Perimeter = 3a
Mensuration Formula for Trapezium :
- Area =12h×(sum of parallel sides)$
- Perimeter = Sum of all sides
Mensuration Formula for Rhombus :
- Area = 12×d1×d2 where d1,d2 are diagonals
- Perimeter = 4l
Mensuration Formula for Quadrilateral :
- Area = 12×b×h
Mensuration Formula for Kite :
- Area = 12×d1×d2, where d1,d2 are diagonals
- Perimeter = 2× Sum on non-adjacent sides
Mensuration Formula for Circle :
- Area = πr2
- Circumference = 2πr
- Area of sector of a circle = θπr2360∘
Mensuration Formula for Frustum :
- Curved surface area = πh(r1+r2)
- Surface area = π(r21+h(r1+r2)+r22)
Mensuration Formula for Cube :
- Volume: V = a3
- Lateral surface area = 4a2
- Surface Area: S = 6a2
- Diagonal (d) = √3a
Mensuration Formula for Cuboid :
- Volume of cuboid: lbh
- Total surface area = 2(lb+bh+hl)
- Length of diagonal = √(l2+b2+h2)
Mensuration Formula for Right Circular Cylinder :
- Volume of Cylinder = πr2h
- Lateral Surface Area (LSA or CSA) = 2πrh
- Total Surface Area = TSA = 2πr(r+h)
- Volume of hollow cylinder = πrh(R2–r2)
Mensuration Formula for Right Circular cone :
- Volume = 13πr2h
- Curved surface area: CSA= πrl
- Total surface area = TSA = πr(r+l)
Mensuration Formula for Sphere:
- Volume: V = 43πr3
- Surface Area: S = 4πr2
Mensuration Formula for Hemisphere :
- Volume = 23πr3
- Curved surface area(CSA) = 2πr2
- Total surface area = TSA = 3πr2
Mensuration Formula for Prism :
- Volume = Base area × h
- Lateral Surface area = perimeter of the base × h
Mensuration Formula for Pyramid:
- Volume of a right pyramid = 13 × area of the base × height.
- Area of the lateral faces of a right pyramid = 12 × perimeter of the base × slant height.
- Area of whole surface of a right pyramid = area of the lateral faces + area of the base.
Mensuration Formula for Tetrahedron :
- Area of its slant sides = 3a2sqrt34
- Area of its whole surface = √3a2
- Volume of the tetrahedron = √212a3
Mensuration Formula for Regular Hexagon :
- Area = 3×√3a22
- Perimeter = 6a
Mensuration Formula Chart
Mensuration Solved Problems: Mensuration Formulas
Problem 1: Calculating the Area of a Triangle
Question: Find the area of a triangle with a base of 10 inches and height of 8 inches.
Solution: Need to apply formula to calculate the area of a triangle is: A = (1/2) * base * height.
Substituting the given values into the formula, we have:
A = (1/2) * 10 * 8 = 40 square inches.
Therefore, the area of the triangle is 40 square inches.
Problem 2: Determining the Perimeter of a Rectangle
Question: Find the perimeter of a rectangle with a length of 12 meters and a width of 8 meters.
Solution: Need to apply formula to calculate the perimeter of a rectangle is: P = 2 * (length + width).
Substituting the given values into the formula, we have:
P = 2 * (12 + 8) = 2 * 20 = 40 meters.
Therefore, the perimeter of the rectangle is 40 meters.
Problem 3: Calculating the Volume of a Cylinder
Question: Find the volume of a cylinder with a radius of 5 centimeters and a height of 10 centimeters.
Solution: Need to apply formula to calculate the volume of a cylinder is: V = π * radius^2 * height.
Substituting the given values into the formula, we have:
V = π * 5^2 * 10 = 250π cubic centimeters.
Therefore, the volume of the cylinder is 250π cubic centimeters.
Problem 4: Finding the Surface Area of a Sphere
Question: Find the surface area of a sphere with a radius of 7 inches.
Solution: Need to apply formula to calculate the surface area of a sphere is: SA = 4 * π * radius^2.
Substituting the given value into the formula, we have:
SA = 4 * π * 7^2 = 4 * 49π = 196π square inches.
Therefore, the surface area of the sphere is 196π square inches.
Problem 5: Determining the Volume of a Cone
Question: Find the volume of a cone with a radius of 6 centimeters and a height of 12 centimeters.
Solution: Need to apply formula to calculate the volume of a cone is: V = (1/3) * π * radius^2 * height.
Substituting the given values into the formula, we have:
V = (1/3) * π * 6^2 * 12 = 144π cubic centimeters.
Therefore, the volume of the cone is 144π cubic centimeters.
Problem 6: Calculating the Area of a Trapezoid
Question: Find the area of a trapezoid with a base of 10 inches, a top length of 6 inches, and a height of 4 inches.
Solution: Need to apply formula to calculate the area of a trapezoid is: A = (1/2) * (base1 + base2) * height.
Substituting the given values into the formula, we have:
A = (1/2) * (10 + 6) * 4 = 8 * 4 = 32 square inches.
Therefore, the area of the trapezoid is 32 square inches.
Problem 7: Determining the Circumference of a Circle
Question: Find the circumference of a circle with a diameter of 12 meters.
Solution: Need to apply formula to calculate the circumference of a circle is: C = π * diameter.
Substituting the given value into the formula, we have:
C = π * 12 = 12π meters.
Therefore, the circumference of the circle is 12π meters.
Problem 8: Calculating the Surface Area of a Cube
Question: Find the surface area of a cube with a side length of 5 centimeters.
Solution: Need to apply formula to calculate the surface area of a cube is: SA = 6 * side^2.
Substituting the given value into the formula, we have:
SA = 6 * 5^2 = 6 * 25 = 150 square centimeters.
Therefore, the surface area of the cube is 150 square centimeters.
Problem 9: Finding the Volume of a Prism
Question: Find the volume of a rectangular prism with a length of 8 inches, a width of 6 inches, and a height of 4 inches.
Solution: Need to apply formula to calculate the volume of a rectangular prism is: V = length * width * height.
Substituting the given values into the formula, we have:
V = 8 * 6 * 4 = 192 cubic inches.
Therefore, the volume of the rectangular prism is 192 cubic inches.
Problem 10: Calculating the Area of a Circle
Question: Find the area of a circle with a radius of 9 inches.
Solution: Need to apply formula to calculate the area of a circle is: A = π * radius^2.
Substituting the given value into the formula, we have:
A = π * 9^2 = 81π square inches.
Therefore, the area of the circle is 81π square inches.
