math

Geometric Formulas for Prisms, Cylinders, Pyramids, Cones, and Spheres

Prisms

• Lateral Area of a Prism

  • Formula: $l = p imes h$

  • Where:

    • $l$ = lateral area

    • $p$ = perimeter of the base

    • $h$ = height of the prism

• Surface Area of a Prism

  • Formula: $s = l + 2b$

  • Where:

    • $s$ = surface area

    • $l$ = lateral area

    • $b$ = area of the base

Cylinders

• Lateral Area of a Cylinder

  • Formula: $l = 2 ext{π}r imes h$

  • Where:

    • $l$ = lateral area

    • $r$ = radius of the base

    • $h$ = height of the cylinder

• Surface Area of a Cylinder

  • Formula:
    $s = 2 ext{π}r imes h + 2 ext{π}r^2$

  • Where:

    • $s$ = surface area

    • The first term, $2 ext{π}r imes h$, is the lateral area.

    • The second term, $2 ext{π}r^2$, accounts for the areas of the two circular bases.

Regular Pyramids

• Lateral Area of a Regular Pyramid

  • Formula: $l = rac{1}{2}p imes l$

  • Where:

    • $l$ = lateral area

    • $p$ = perimeter of the base

    • The lowercase $l$ is typically used for the slant height.

• Surface Area of a Regular Pyramid

  • Formula: $s = rac{1}{2}p imes l + b$

  • Where:

    • $s$ = surface area

    • The first term is the lateral area, and the second term $b$ is the area of the base.

Cones

• Lateral Area of a Cone

  • Formula: $l = ext{π}r l$

  • Where:

    • $l$ = lateral area

    • $r$ = radius of the base

    • The lowercase $l$ is typically used for the slant height.

• Surface Area of a Cone

  • Formula: $s = ext{π}r l + ext{π}r^2$

  • Where:

    • $s$ = surface area

    • The first term is the lateral area, and the second term $ext{π}r^2$ is the area of the circular base.

Volume Formulas

• Volume of a Prism

  • Formula: $V = b imes h$

  • Where:

    • $V$ = volume

    • $b$ = area of the base

    • $h$ = height of the prism.

• Volume of a Cylinder

  • Formula: $V = ext{π}r^2 imes h$

  • Where:

    • $V$ = volume

    • $r$ = radius of the base

    • $h$ = height of the cylinder.

• Volume of a Pyramid

  • Formula: $V = rac{1}{3}b imes h$

  • Where:

    • $V$ = volume

    • $b$ = area of the base

    • $h$ = height of the pyramid.

• Volume of a Cone

  • Formula: $V = rac{1}{3} ext{π}r^2 imes h$

  • Where:

    • $V$ = volume

    • $r$ = radius of the base

    • $h$ = height of the cone.

• Surface Area of a Sphere

  • Formula: $s = 4 ext{π}r^2$

  • Where:

    • $s$ = surface area

    • $r$ = radius of the sphere.

• Volume of a Sphere

  • Formula: $V = rac{4}{3} ext{π}r^3$

  • Where:

    • $V$ = volume

    • $r$ = radius of the sphere.