Surface Area and Volume

Shapes, Vertices, Edges, and Faces

Platonic Solid: a 3D shape formed by the same regular polygons

  • Ex. tetrahedron, cube, octahedron, dodecahedron, icosahedron —> These are the only platonic shapes

  • (They are named platonic solids because these were discovered by Pluto!)

Polyhedron: any 3D shape created by polygons as faces

  • Ex. prism, pyramid

  • Ex of non-polyhedra. cylinder, cone, sphere

Prism: A shape that has two bases which are the same polygon

Solid

Vertices, V

Edges, E

Faces, F

Tetrahedron

4

6

4

Cube

8

12

6

Octahedron

6

12

8

Dodecahedron

20

30

12

Icosahedron

12

30

20

The relationship between the numbers of vertices, edges, and faces of a polyhedron is as follows: (discovered by Euler)

F+V-2=E

Cross Sections

Cross Section: the shape formed when a non-empty three dimensional space is intersected with a plane


Circle


Rectangle


Trapezoid

Surface Area

Surface Area: the area of the net of a 3D shape

The general formula for a prism is: SA=Ph+2B, where

  • SA is the surface area

  • P is the perimeter of the base

  • h is the height of the prism

  • B is the area of the base


  1. SA = Ph+2B

  2. SA = (16)(10)+2[(1/2)(6)(4)]

  3. SA = 160 +24

  4. SA = 184 ft²

**Don’t forget that surface area is a type of area, so the unit will be squared!

The formula for an oblique prism is: SA=2B+2AParallelogram+2ARectangle, where

  • SA is surface area

  • B is the area of the base

  • AParallelogram is the area of the parallelogram side

  • ARectangle is the area of the rectangle side

  1. SA = 2B + 2AParallelogram + 2ARectangle

  2. SA = 2(11)(7) + 2(11)(15) + 2(17)(7)

  3. SA = 722 m²

Regular Pyramid: a pyramid with a regular base and congruent isosceles lateral faces

The formula for a regular pyramid is: SA=½Pl + B, where

  • SA is the surface area

  • P is the perimeter of the base

  • l is the slant height

  • B is the area of the base

  1. SA = ½Pl + B

  2. SA = ½(30)(√93.75) + [½(2.5√3)(30)]

  3. SA = 210.2 ft²

The formula for a cone is: SA=πrl+πr², where

  • SA is the surface area

  • r is the radius

  • l is the slant height

**Write the answer in terms of π

  1. SA = πrl + πr²

  2. SA = π(8.4)(√106.56) + π(8.4)²

  3. SA = [8.4√(106.56)π+106.56π] cm²

The formula for a cylinder is: SA=2πrh+2πr², where

  • SA is the surface area

  • r is the radius

  • h is the height

The formula for a sphere is: SA=4πr³, where

  • SA is the surface area

  • r is the radius

**Write the answer in terms of π

  1. SA = 4πr³

  2. SA = 4π(8)³

  3. SA = 256π in²

Volume

The formula of a prism is: V=Bh, where

  • B is the area of the base

  • h is the height of the prism

The formula of a pyramid is: V=⅓Bh, where

  • B is the area of the base

  • h is the height of the pyramid

    1. V = ⅓Bh

    2. V = ⅓(½aP)h

    3. V = ⅙(7.5/tan36°)(75)(22)

    4. V = 2838.8 mm³

**Don’t forget the units for volume are cubed!

The formula for a sphere is: V=⁴⁄₃πr³, where

  • r is the radius

  1. V = ⁴⁄₃πr³

  2. V = ⁴⁄₃π(4.5)r³

  3. V = 121.5π in³