A common exam question states that the dimensions of a shape or solid have been increased in a certain way, and asks us to determine what effect such a change has on the area or volume of the shape or solid.
Think about the area formulas which we studied. Regardless of the shape, they all involved multiplying two dimensions together. They might be called length and width, or base and height, or in the case of a circle, simply the radius times itself.
What would happen if we doubled one of the dimen- sions but not the other? For example, what happens to the area of a 3 × 5 rectangle if we make it 6 × 5? The area doubles from 15 to 30. Think about why this makes
sense, not just as far as the arithmetic, but visually as well. If you double the length of your garden, but leave the width as is, you’ll have twice as much room to plant.
What would happen to the area if you doubled both of the dimensions, taking the 3 × 5 rectangle and making it 6 × 10? The area quadruples from 15 to 60. Again, think about why this makes sense. We have a doubling times a doubling, resulting in four times the area.
What would happen to the area if you doubled the length, but tripled the width? Since the area formula is length times width, we must take into account that the doubling is being multiplied by the tripling, result in the new area being six times as large as the original.
Be careful about problems of this type which involve circles. If you were to triple the radius, the area of the new circle would be nine times as big as the original. This is because the formula requires us to square the radius, which means we must square the tripling, resulting in a new area that is 32 or 9 times as big.
This pattern holds for all shapes, and for any multiplica- tive changes (e.g., doubling, tripling, etc.) made to one or
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both dimensions. Just multiply any stated changes to determine the overall change.
As expected, all of this extends to three-dimensional solids. For example, confirm that if the length, width, and depth of a rectangular solid were each tripled, it would increase the volume of the solid by 27 times (i.e., triple times triple times triple).
In the case of a sphere, any change to the radius must be cubed, per the volume formula. Confirm that multiply- ing a sphere’s radius by 4 (i.e., quadrupling it) results in a new sphere that is 43 or 64 times as big as the original.
PRACTICE EXERCISES AND REVIEW
Remember: Area is always expressed in square units,
and volume is always expressed in cubic units. You will lose points unnecessarily if you do not express your answers as such.
Remember: The area of a triangle is ½bh where h is
defined as the distance from the highest point on the triangle “straight down” to the base (or its extension).
Remember: The perimeter of a figure is the distance
Remember: The diameter of the circle is the distance
from edge to edge though the middle, and the radius is half that. The area formula of a circle is 𝜋𝑟2, and the circumference formula is 2𝜋𝑟 or 𝜋𝑑.
Remember: We usually use 3.14 as an approximate value
of 𝜋, but it’s actually non-repeating and non-terminating.
Remember: Volume formulas are derived from compu-
ting the area of a solid’s base, and multiplying it by the third dimension, usually denoted height or depth.
Remember: If a figure’s or solid’s dimensions change
(e.g., double, triple, etc.), multiply the changes to deter- mine the net effect on its area or volume.
For practice, try these exercises:
1) Find the area of a 9 cm by 7 cm rectangle. 2) Find the area of a square of side 25 yards.
3) Find the area of a right triangle with legs of 5 and 12 inches, and a hypotenuse of 13 inches.
4) Find the area of a parallelogram with a base of 8 cm, slanted sides of 5 cm each, and a height of 4 cm. 5) Find the area of a trapezoid with bases of 9 ft. and 10
ft., height 3 ft., and slanted sides of unknown length. 6) True/False: π = 3.14
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7) True/False: π is approximately 22/7 8) Find the area of a circle of diameter 16 cm. 9) Find the circumference of a circle of radius 5 ft. 10) Find the volume of a rectangular solid of dimen-
sions 3 by 8.6 by 4.7 feet.
11) Find the volume of a cube of edge 3 cm.
12) Find the volume of a cylinder with circular base of area 21 𝑐𝑚2 and height of 8.5 cm.
13) Find the volume of a triangular prism of depth 20 in. whose triangular base has a height of 4 in. and a base of 6 in.
14) A circle’s radius is tripled. How many times larger is the area of the resulting circle?
15) A rectangular solid has both its length and depth tripled, and its width quadrupled. How many times larger is the area of the resulting solid?
SO NOW WHAT?
The next chapter covers the basics of the Pythagorean Theorem and triangles, both of which are very popular topics on geometry exams. Before proceeding, review the concepts of proportion and squaring. Also review the algebra section on solving more complicated equations.