8th Grade Mathematics Notes, Lecture notes of Mathematics

Notes I wrote in 8th grade it is filled with: Graphing linear equations, Laws of Exponents, Linear Equations and Inequalities, Understanding variables and expressions

Typology: Lecture notes

2023/2024

Available from 05/30/2024

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8th Grade Mathematics
Volume and Surface Area of Cylinders, Cones, and Spheres
#### Cylinders
**Volume of a Cylinder**
- **Formula**: \( V = \pi r^2 h \)
- **Explanation**:
- \( r \) is the radius of the base of the cylinder.
- \( h \) is the height of the cylinder.
- The volume is the area of the base (\(\pi r^2\)) multiplied by the height.
**Surface Area of a Cylinder**
- **Formula**: \( SA = 2\pi r(h + r) \)
- **Explanation**:
- The surface area consists of two parts: the area of the two circular bases and the area of the
rectangular side (the label of the can).
- \( 2\pi r^2 \) accounts for the area of the two bases.
- \( 2\pi rh \) accounts for the area of the side (the circumference of the base times the height).
#### Cones
**Volume of a Cone**
- **Formula**: \( V = \frac{1}{3}\pi r^2 h \)
- **Explanation**:
- \( r \) is the radius of the base of the cone.
- \( h \) is the height of the cone (the perpendicular distance from the base to the apex).
- The volume is one-third the volume of a cylinder with the same base and height.
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Grade Mathematics

Volume and Surface Area of Cylinders, Cones, and Spheres

Cylinders

Volume of a Cylinder

  • Formula: ( V = \pi r^2 h )
  • Explanation:
    • ( r ) is the radius of the base of the cylinder.
    • ( h ) is the height of the cylinder.
    • The volume is the area of the base ((\pi r^2)) multiplied by the height. Surface Area of a Cylinder
  • Formula: ( SA = 2\pi r(h + r) )
  • Explanation:
    • The surface area consists of two parts: the area of the two circular bases and the area of the rectangular side (the label of the can).
    • ( 2\pi r^2 ) accounts for the area of the two bases.
    • ( 2\pi rh ) accounts for the area of the side (the circumference of the base times the height).

Cones

Volume of a Cone

  • Formula: ( V = \frac{1}{3}\pi r^2 h )
  • Explanation:
    • ( r ) is the radius of the base of the cone.
    • ( h ) is the height of the cone (the perpendicular distance from the base to the apex).
    • The volume is one-third the volume of a cylinder with the same base and height.

Surface Area of a Cone

  • Formula: ( SA = \pi r(l + r) )
  • Explanation:
    • ( r ) is the radius of the base.
    • ( l ) is the slant height (the distance from the edge of the base to the apex along the side).
    • The surface area includes the base ((\pi r^2)) and the lateral surface area ((\pi rl)).

Spheres

Volume of a Sphere

  • Formula: ( V = \frac{4}{3}\pi r^3 )
  • Explanation:
    • ( r ) is the radius of the sphere.
    • The volume is derived from integrating the surface area of infinitesimally small spherical shells. Surface Area of a Sphere
  • Formula: ( SA = 4\pi r^2 )
  • Explanation:
    • ( r ) is the radius of the sphere.
    • The surface area is four times the area of a great circle (the largest possible circle that can be drawn on the sphere).

Practice Problems

  1. Cylinder
    • Find the volume and surface area of a cylinder with a radius of 3 cm and a height of 5 cm.
    • Volume: ( V = \pi \times 3^2 \times 5 = 45\pi \approx 141.37 , \text{cm}^3 )
    • Surface Area: ( SA = 2\pi \times 3 \times (5 + 3) = 48\pi \approx 150.80 , \text{cm}^2 )