Perimeter and Area

Multiplication Tables: the product of two numbers can be represented by the area of a rectangle where the factors are represented as the dimensions of the rectangle. Notice each square in the area grid is a unit square.

 

Area of a rectangle is represented by a grid of unit squares. Perimeter is the distance around the outside rectangle.

 

Area of a square is represented by a grid of unit squares. Perimeter is the distance around the outside square.

 

Area of a parallelogram is represented by a grid of unit squares. Perimeter is the distance around outside of the parallelogram.

 

Area of a rhombus is represented by a grid of unit squares. Perimeter is the distance around outside of the rhombus.

 

Area of Triangles is represented by a grid of unit squares. Perimeter is the distance around outside of the triangle.

 

Area of Trapezoids is represented by a grid of unit squares. Perimeter is the distance around outside of the trapezoid.

 

 

 

 

 

 

Shape Explorer - perimeter and area on a grid

Maximize Area

Minimize Perimeter

Interactive sums

 

Each of the links below provide additional interactive activiites.

 Polygon/Circle

P = Perimeter

C = Circumference

distance around the polygon/circle

 

A = Area

the surface inside the boundary( perimeter)

triangle

  • a = altitude
  • b = base

 

 

P = b + c + d

or

P = s1 + s2 + s3

A = 1/2ab

Note: The area of the triangle is half the area of the corresponding parallelogram/rectangle.

quadrilateral

P = b + c + d + e

or

P = s1 + s2 + s3 + s4

don't need to memorize the formula in junior high

enrichment

trapezoid

  • a = altitude
  • b = base

 

 

P = b1 + b2 + c + d

or

P = s1 + s2 + s3 + s4

A = 1/2ab1 + 1/2ab2)

simplifies to

A = 1/2a(b1 + b2)

parallelogram

  • a = altitude
  • b = base

 

P = 2b + 2c

A = bh

rhombus

  • a = altitude
  • s = side length

 

P = 4s

A = as

rectangle

  • l = length
  • w = width

 

P = 2l + 2w

 

A = lw

square

  • s = side length

 

P = 4s

A = s2

chevron

P = 2c + 2d

 

don't need to memorize the area formulae in junior high

kite

P = 2c + 2d

 

don't need to memorize the area formulae in junior high

enrichment

regular polygon

  • a = apothem length
  • s = side length
  • n = #sides

 

pentagon = "five sides"
hexatagon = "six sides"
heptaagon = "seven sides"
octaagon = "eight sides"
nonagon = "nine sides"
decagon = "ten sides"

P = ns

A = 0.5ans = 0.5aP

circle

Circumference is the distance around the circle.

  • r = radius of circle
  • d = diameter of circle

 

C = Circumference

C = pd

A = Area

A = p r2

 

Relationships:

1. For a square, non-square rectangle and circle with equal perimeter/circumference the relative area from largest to smallest is:

2. Increasing dimensions by a factor increases the perimeter by the same factor.

  • tripling the dimensions triples the perimeter

1. For a square, non-square rectangle and circle with equal areas the relative perimeter from largest to smallest is:

2. Increasing the dimensions by a factor increases the area by the square of the factor.

  • tripling the dimensions increases the area by a factor of 9 (32).

 Polygon/Circle

Perimeter/Circumference

 

 

 

Geometric Shape

Surface Area

B = area of the base

P = perimeter of the base

Volume

B = area of the base

P = perimeter of the base

prism (general)

  • B = area of base
  • P = perimeter of base
  • h = height

 

   SA = 2B + Ph V = Bh

triangular prism

  • a = altitude
  • b = base
  • h = #height

 

   SA = 2B + Ph

SA = 2(1/2ab) + (b + c + d)h

SA = ab + (b + c + d)h

triangular_prism_sml.png (33744 bytes)

 V = Bh

V = 1/2 abh

triangular_prism.gif (1150 bytes)

rectangular prism

  • l = length
  • w = width
  • h = height

 

 SA = 2B + Ph

SA = 2(lw) + (2l + 2w)h

rectangular_prism_net.gif (1694 bytes)

V = lwh

rectangular_prism.gif (1540 bytes)

Regular square prism

  • s = side length
  • h = height

 

 SA = 2B + Ph

SA = 2(s2) + (4s)h

square_prism_sml.png (32923 bytes)

 V = Bh

V = s2h

square_prism_vol.png (28523 bytes)

regular pentagonal prism

  • a = apothem length
  • s = side length
  • n = #sides
  • h = height

 

 

SA = 2B + Ph

SA = 2(1/2ans) + nsh

SA = 2(1/2a)(5)s + 5sh

SA = 5as + 5sh

pent_net.gif (1423 bytes)

 V = Bh

V = 1/2ansh

V = 1/2a(5)sh

V = 5/2ash

pentagonal_prism.gif (1269 bytes)

regular hexagonal prism

  • a = apothem length
  • s = side length
  • n = #sides
  • h = height

 

 

SA = 2B + Ph

SA = 2(1/2ans) + nsh

SA = 2(1/2a)(6)s + 6sh

SA =6as + 6sh

hex_net.gif (1832 bytes)

 V = Bh

V = 1/2ansh

V = 1/2a(6)sh

V = 3ash

hexagonal_prism.gif (1406 bytes)

cube

  • s = side length

 

 SA = 2B + Ph

SA = 2(s2) + (4s)s = 6s2

cube_net_sml.png (28862 bytes)

 V = Bh

V = s3

cube.gif (1355 bytes)

regular pyramid (general)

  • B = area of base
  • s = side length
  • l = slant heigth

 

 SA = B + n(1/2sl)

l = slant height

 V = 1/3Bh

regular triangular pyramid

  • a = apothem length
  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

SA = B + n(1/2sl)

SA = 1/2as + (3)(1/2sl)

SA = 1/2as + 3/2sl

triangular_reg_pyramid_sml.png (40438 bytes)

l = slant height

V = 1/3Bh

V = 1/3(1/2 ab)h

V = 1/6 abh

tpyramid.gif (2377 bytes)
 

regular square pyramid

  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

  

SA = B + n(1/2sl)

SA = s2 + (4)(1/2sl)

SA = s2 + 2sl

square_pyramid_sml.png (37371 bytes)

l = slant height

 V = 1/3Bh

V = 1/3(b2)h

= 1/3b2h

spyramid2.gif (2815 bytes)

regular pentagonal pyramid

  • a = apothem length
  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

  

SA = B + n(1/2sl)

SA = 1/2a(5)s + (5)(1/2sl)

SA = 5/2as + 5/2sl

pentagonal_pyramid.gif (2204 bytes)

l = slant height

 V = 1/3Bh

V = 1/3(1/2anb)h

V = 1/6 anbh

pentagonal_pyramid_vol.png (29199 bytes)

regular hexagonal pyramid

  • a = apothem length
  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

  

SA = B + n(1/2sl)

SA = 1/2a(6)s + (6)(1/2sl)

SA = 3as + 3sl

hexagonal_pyramid_sml.png (40635 bytes)

l = slant height

 V = 1/3Bh

V = 1/3(1/2anb)h

V = 1/6 anbh

hexagonal_pyramid_vol.png (28342 bytes)

cylinder

  • B = area of base
  • P = perimeter of base
  • r = radius of circle
  • h = height

 

 

SA = 2B + Ph

SA = 2(p r2) + (2pr)h

cylinder_net_sml.png (33177 bytes)

V = Bh

V = p r2h

cylinder.GIF (1709 bytes)

cone

  • B = area of base
  • r = radius of circle
  • h = height

 

cone_sml.png (34352 bytes)

Explanation

V = 1/3 Bh

V = 1/3pr2h

cone.png (38273 bytes)

The volume of a cone is one third the volume of the corresponding cylinder.

sphere

r = radius of circle

SA = 4pr2

sphere.GIF (3975 bytes)

V = 4/3pr3

sphere_volume.gif (1432 bytes)

 

Prism

hexagonal_prism.gif (1406 bytes)

V = Bh

Pyramid

hexagonal_pyramid_vol.png (28342 bytes)

V = 1/3 Bh

The volume of a pyramid is one third the volume of the corresponding prism.

Cylinder

cylinder.GIF (1709 bytes)

V = Bh

Cone

cone.png (38273 bytes)

V = 1/3Bh

 The volume of a cone is one third the volume of the corresponding cylinder.