This chapter explores how to find the surface areas and volumes of various solids, including combinations of basic shapes like cubes, cones, cylinders, and spheres, essential for real-world applications.
Surface Areas and Volumes – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Surface Areas and Volumes chapter of Mathematics. 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
Volume of a Cuboid: V = l × b × h
l is length, b is breadth, h is height. This formula calculates the space inside a cuboid. Example: Finding the volume of a box.
Surface Area of a Cuboid: SA = 2(lb + bh + hl)
l is length, b is breadth, h is height. It gives the total area of all cuboid faces. Useful for painting or wrapping.
Volume of a Cylinder: V = πr²h
r is radius, h is height. Calculates the space inside a cylinder. Example: Volume of a water tank.
Curved Surface Area of a Cylinder: CSA = 2πrh
r is radius, h is height. Area of the side surface. Example: Wrapping a cylindrical gift.
Total Surface Area of a Cylinder: TSA = 2πr(r + h)
r is radius, h is height. Includes top and bottom areas. Example: Painting a closed cylinder.
Volume of a Cone: V = (1/3)πr²h
r is radius, h is height. Space inside a cone. Example: Ice cream cone volume.
Curved Surface Area of a Cone: CSA = πrl
r is radius, l is slant height. Side surface area. Example: Making a conical hat.
Total Surface Area of a Cone: TSA = πr(l + r)
r is radius, l is slant height. Includes base area. Example: Wrapping a cone-shaped gift.
Volume of a Sphere: V = (4/3)πr³
r is radius. Space inside a sphere. Example: Volume of a basketball.
Surface Area of a Sphere: SA = 4πr²
r is radius. Total outer area. Example: Painting a spherical object.
Equations
Combined Volume of Solids: V_total = V1 + V2
V1 and V2 are volumes of individual solids. Used when solids are combined without overlapping.
Combined Surface Area of Solids: SA_total = SA1 + SA2 - Overlapping Area
SA1 and SA2 are surface areas of individual solids. Subtract the area where solids join.
Volume of a Hemisphere: V = (2/3)πr³
r is radius. Half the volume of a sphere. Example: Volume of a dome.
Curved Surface Area of a Hemisphere: CSA = 2πr²
r is radius. Half the sphere's surface area. Example: Covering a hemispherical bowl.
Total Surface Area of a Hemisphere: TSA = 3πr²
r is radius. Includes the flat circular base. Example: Painting a closed hemisphere.
Slant Height of a Cone: l = √(r² + h²)
r is radius, h is height. Essential for finding cone's CSA and TSA.
Volume of a Hollow Cylinder: V = πh(R² - r²)
R is outer radius, r is inner radius, h is height. Example: Volume of a pipe.
Surface Area of a Hollow Cylinder: SA = 2πh(R + r) + 2π(R² - r²)
R is outer radius, r is inner radius, h is height. Includes both inner and outer surfaces.
Volume of a Frustum of a Cone: V = (1/3)πh(R² + Rr + r²)
R and r are radii of the two bases, h is height. Example: Volume of a bucket.
Surface Area of a Frustum of a Cone: SA = π(R + r)l + π(R² + r²)
R and r are radii, l is slant height. Includes the curved and the two base areas.
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