Area and Volumes of Solids

Bryan

Area: the two-dimensional size of a figure

Area of square = s2 where s is the length of a side

Area of rectangle = lw

Area of Triangle = 1/2 bh b = length of the base, h = height

Area of Circle = r2

Area of Trapezoid = 1/2h(b1 + b2)

Area of Parallelogram = bh

Pythagorean Theorem-for a right triangle: a2 + b2 = c2

1.Find the areas of the following figures. Note that d and f use the Pythagorean Theorem. Leave out if you haven’t learned it)

  1. b.

4.5cm

9 cm6 cm

7 cm

8cm

  1. 6 cm d.

4 cm

3 cm

4 cm8cm

e. f.

6 cm

5 cm

6 cm

2. A farmer in Kansas needs to irrigate his field. His field is 600 meters by 600 meters. The farmer irrigates his field by placing 4 water hoses at 4 different places. The hoses spray out water in a circular pattern and they have a maximum shooting distance of 150 meters. Find the area of the field not being irrigated if the farmer cannot have two hose irrigate the same spot. (Hint: start by drawing this situation.)

Volume: The amount of space occupied by an object

Volumes:

Volume of cube = s3 where s represents the length of a side

Volume of rectangular box = lwh

Volume of Pyramid = 1/3 Bh where B= area of the base

Volume of Cylinder = r2h

Volume of Cone = 1/3r2h

Volume of sphere = 4/3 r3

  1. The relationship between area and volume is very close. In each volume formula, one can find an area formula. Find the area formulas in each volume formula and identify the figure this formula is for. What pattern do you see?
  1. Find the volumes of the following objects:

a.

b.

2 cm 3 cm |

2 cm

2 cm 4 cm 2 cm

c. 2 cm

2 cm2 cm

d.

5.5 cm

7 cm

height=12 cm

e.f.

3m

sphere

5m height = 6m ice cream cone

  1. Assume that a tennis ball can is a perfect cylinder. Knowing that the circumference of a tennis ball equals two times the radius times pi,
  1. write a formula for the circumference of a tennis ball
  2. solve the formula for r.
  3. using a tennis ball, measure the circumference and then solve for the radius in cm.
  4. find the volume of a tennis ball in cubic centimeters
  5. using the tennis ball can, find the volume of the can.
  6. how much empty space is inside a tennis ball can before it is opened?

Surface Area: The surface are of an object equals the sum of the areas of its faces.

Find the surface areas of each of the following figures:

4m

5 m

4 m

4m 4m

10m

cut-outs are 2 x 3 meters, all the way through

8 m

3

10m h=8.7m

7 m 10m

10m

4 m

3m 6m 10 m

2 m2 m tetrahedron-3 sided, symmetrical sides

Comparison of Volumes and Surface Areas

  1. Compute the surface area and the volume of the following figures.
  1. A rectangular box: Surface area equals the sum of the areas of its faces

h=2cm w=4cm l=4cm

  1. a cylinder

surface area equals 2r2 +2rh

r=2cm h=2.09cm

  1. a sphere

surface area equals 4r2

r=2.256cm

  1. A pyramid

Surface area equals the sum of the area of its faces

s=2cm h1=15cm h2=14.97cm

h1

h2

s

s 1/2 s

2. What did you notice about the surface areas and volumes of the above figures? What generalization might you make about the volume of the sphere?