Math Practice Online: MathScore.com

Math Practice Online > free > lessons > Minnesota > 9th grade > Area And Volume Proportions

If your child needs math practice, click here.

For sample problems, click here.
Here are some tips for Area And Volume Proportions, which aligns with Minnesota state standards:

Area and Volume Proportions


You will need to know area and volume formulas for this topic.
To review, see the topics below:
Area Volume
Circle πr2 Sphere (4/3)πr3
Square s2 Cube s3
Rectangle lw Rectanglar Prism lwh
Triangle (1/2)bh Triangular Prism (1/2)bhl
    Cylinder πr2h
    r   radius
    s   side
    l   length
    w   width
    h   height
    b   base


Example 1: Area

Find the area of the following shapes after the transformations have been made.
A circle has an area of 11. If the radius is increased by a factor of 3, what is the new area of the circle?  

Original area
    original Area = 11 = π (original radius)2
New area
    new Area = π (new radius)2
From the statement the radius is increased by a factor of 3, we get
    new radius = 3 × original radius

new Area = π (new radius)2
  = π (3 × original radius)2
  = 9 × π (original radius)2
  = 9 × original Area
  = 9 × 11
  = 99

Alternate solution:
The radius increased by 3. In the circle area equation, the radius is squared so we can square the factor 3 and multiply that by the original area to get 32 × 11 = 99.

The answer is


Example 2: Volume

Find the volume of the following shapes after the transformations have been made.
A triangular prism has a volume of 20. If the base height is increased by a factor of 4, what is the new volume of the triangular prism?

Original volume
    original Volume = 20 = (1/2)(base)(original height)(length)
New volume
    new Volume = (1/2)(base)(new height)(length)
From the statement the base height is increased by a factor of 4, we get
    new height = 4 × old height

new Volume = (1/2)(base)(new height)(length)
  = (1/2)(base)(4 × old height)(length)
  = 4 × (1/2)(base)(old height)(length)
  = 4 × original Volume
  = 4 × 20
  = 80

Alternate solution:
The base height increased by 4. In the triangular prism volume formula, the base height is linear so we can multiply the increased factor by the original volume to get 4 × 20 = 80.

The answer is

Copyright Accurate Learning Systems Corporation 2008.
MathScore is a registered trademark.