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SURFACE AREAS AND VOLUMES – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 9 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
Curved Surface Area of a Cone: A_c = πrl
A_c represents the curved surface area, r is the base radius, and l is the slant height. This formula is used to calculate the outer area of a cone excluding its base.
Total Surface Area of a Cone: A_t = πr(l + r)
A_t represents the total surface area. This formula includes both the curved surface area and the circular base area.
Surface Area of a Sphere: A_s = 4πr²
A_s is the surface area of the sphere and r is the radius. This formula is vital for calculating areas of spherical objects.
Curved Surface Area of a Hemisphere: A_h = 2πr²
A_h is the curved surface area of a hemisphere where r is the radius. Used in various real-world geometrical applications.
Total Surface Area of a Hemisphere: A_h_total = 3πr²
A_h_total includes both the curved surface area and the flat circular base area of a hemisphere.
Volume of a Cone: V = (1/3)πr²h
V is the volume, r is the radius of the base, and h is the height. Important for calculating space inside conical objects.
Volume of a Sphere: V = (4/3)πr³
V represents the volume of a sphere. This formula is crucial in finding the capacity of spherical containers.
Volume of a Hemisphere: V_h = (2/3)πr³
V_h gives the volume of a hemisphere. Useful for determining capacity in semi-spherical shapes.
Slant Height of a Cone: l = √(r² + h²)
l is the slant height, r is the radius, and h is the height. This derived relation helps in calculations involving cones.
Circumference of the Base of a Cone: C = 2πr
C represents the circumference of the circular base. This is essential for conversions between linear and area measurements.
Equations
Curved Surface Area of Cone Example: A_c = π × 7 cm × 10 cm = 220 cm²
This example calculates the curved surface area for a cone with radius 7 cm and slant height 10 cm.
Total Surface Area of Cone Example: A_t = π × 12 cm × (20 cm + 12 cm) = 1205.76 cm²
Example calculation for the total surface area of a cone with radius 12 cm and slant height derived as 20 cm.
Volume of Cone Example: V = (1/3)π × 7 cm² × 24 cm = 1232 cm³
Demonstrates how to compute the volume of a cone with base radius 7 cm and height 24 cm.
Surface Area of Sphere Example: A_s = 4π × 7 cm² = 616 cm²
This is the example of calculating the surface area for a sphere with radius 7 cm.
Volume of Sphere Example: V = (4/3)π × (4.9 cm)³ ≈ 493 cm³
Find the volume of a sphere with radius 4.9 cm. Useful for applications in physics and engineering.
Curved Surface Area of Hemisphere: A_h = 2π × (3.5 cm)² = 77 cm²
This equation shows calculation for the curved surface area of a hemisphere with radius 3.5 cm.
Total Surface Area of Hemisphere Example: A_h_total = 3π × (21 cm)² = 4158 cm²
Covers the total surface area of a hemisphere with radius 21 cm.
Slant Height Calculation Example: l = √[(12 cm)² + (16 cm)²] = 20 cm
This calculates the slant height of a cone using the Pythagorean theorem.
Radius from Volume Example: r = ((3 × V) / (4π))^(1/3)
Reverse calculation to find the radius of a sphere based on its volume.
Grains on Corn Cob Example: Total grains = Curved Surface Area × Average grains/cm²
Explains how to find total grains on a corn cob based on its curved surface area. Useful in agricultural studies.
