Surface Areas and Volumes – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Surface Areas and Volumes chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
TSA of Cuboid = 2(lb + bh + hl)
Where l = length, b = breadth, h = height. This formula calculates the total surface area to paint or cover the cuboid.
CSA of Cylinder = 2πrh
Where r = radius, h = height. This gives the curved surface area of a cylinder, useful in tasks such as wrapping a cylindrical object.
TSA of Cylinder = 2πr(r + h)
This is the total surface area of a cylinder, including its bases. It's important for knowing the area available for decoration.
Volume of Cylinder = πr²h
Where r = radius, h = height. Used for calculating the capacity of cylindrical containers.
CSA of Cone = πrl
Where r = radius, l = slant height. This formula is essential for calculating the area to cover a conical shape.
TSA of Cone = πr(l + r)
This represents the total surface area of a cone, which is helpful in design and manufacturing.
Volume of Cone = (1/3)πr²h
Used to find the capacity of a conical container, where r = radius and h = height.
CSA of Sphere = 4πr²
Where r = radius. This formula helps in calculating the area required to paint spherical objects.
Volume of Sphere = (4/3)πr³
This is used to determine the volume of spherical objects, vital for capacity calculations.
TSA of Hemisphere = 3πr²
Where r = radius. It provides the total surface area of a hemisphere, often used in geometric designs.
Equations
π ≈ 3.14 or π = 22/7
Pi is the ratio of the circumference of a circle to its diameter, widely used in calculations involving circles and spheres.
l = √(r² + h²)
The slant height 'l' of a cone can be derived using the radius and height, important in calculating CSA of cones.
TSA of Combination of Solid = CSA of Hemisphere + CSA of Cylinder + CSA of Hemisphere
This formula is for finding total surface areas of solids made from two or more basic shapes.
Volume of Composite Shape = Volume of Cylinder + Volume of Hemisphere
Used when calculating the total volume of a composite solid formed by basic shapes.
Area to be Coloured = CSA of Cone + (base area of Cone - base area of Cylinder)
Calculates the surface area on which paint is applied when cone and cylinder intersect.
H = h₁ + h₂
Total height of composite shapes, where h₁ and h₂ are the heights of individual solids.
TSA of Cube = 6a²
Where a = length of side. This gives the surface area for cubes, essential for determining paint coverage.
Surface Area of a Block = TSA of Cube + TSA of Hemisphere - base area of Hemisphere
Used for calculating the surface area of a solid shape made from a cube and a hemisphere.
V_total = V_cylinder + V_cone
The combined volume of a cylinder and a cone, useful for total capacity assessments.
r' = r - a
This relationship provides the adjusted radius when subtracting from a larger shape's radius.
