mathematics for appraisers

Area Calculation ToolHere is a handy little tool you can use to

find the area of common shapes.

TriangleArea = ½b × h

b = baseh = vertical

height

SquareArea = a2

a = length of side

RectangleArea = w × h

w = widthh = height

ParallelogramArea = b × h

b = baseh = vertical height

Trapezoid (US)Trapezium (UK)

Area = ½(a+b) × hh = vertical height

CircleArea = πr2

Circumference=2πrr = radius

EllipseArea = πab

SectorArea = ½r2θ

r = radiusθ = angle in radians

Volume Formulas

Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a". "b3" means "b cubed", which is the same as "b" times "b" times "b".

Be careful!! Units count. Use the same units for all measurements. Examples

Units

Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely, like blocks in a box.

Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely, like blocks in a box.

Volume of a cube = side times side times side. Since each side of a square is the same, it can simply be the length of one side cubed.

If a square has one side of 4 inches, the volume would be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic inches can also be written in3.)

NOT CORRECT .... 4 times 1 times 3 = 12 CORRECT.... 4 inches is the same as 1/3 feet. Volume is 1/3 feet times 1 foot times 3 feet = 1 cubic foot (or 1 cu. ft., or 1 ft3).

Be sure to use the same units for all measurements. You cannot multiply feet times inches times yards, it doesn't make a perfectly cubed measurement.

The volume of a rectangular prism is the length on the side times the width times the height. If the width is 4 inches, the length is 1 foot and the height is 3 feet, what is the volume?

cube = a 3

rectangular prism = a b c

irregular prism = b h

cylinder = b h = pi r 2 h

pyramid = (1/3) b h

cone = (1/3) b h = 1/3 pi r 2 h

sphere = (4/3) pi r 3

ellipsoid = (4/3) pi r1 r2 r3

 Suppose you're given the two points (–2, 1) and (1, 5), and they want you to find out how far apart they are. The points look like this:

The Distance Formula (page 1 of 2)The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry. Here's how we get from the one to the other:

You can draw in the lines that form a right-angled triangle, using these points as two of the corners:

It's easy to find the lengths of the horizontal and vertical sides of the right triangle: just subtract the x-values and the y-values:

Then use the Pythagorean Theorem to find the length of the third side (which is the hypotenuse of the right triangle):c2 = a2 + b2

...so:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

This format always holds true. Given two points, you can always plot them, draw the right triangle, and then find the length of the hypotenuse.

The length of the hypotenuse is the distance between the two points. Since this format always works, it can be turned into a formula:

Distance Formula: Given the two points (x1, y1)and (x2, y2), the

distance between these points is given by the formula:

Don't let the subscripts scare you. They only indicate that there is a "first" point and a "second" point; that is, that you have two points. Whichever one you call "first" or "second" is up to you. The distance will be the same, regardless.

•Find the distance between the points (–2, –3) and (–4, 4).I just plug the coordinates into the Distance Formula:

Then the distance is sqrt(53), or about 7.28, rounded to two decimal places.

Distance Formula 

In this lesson, the distance between two points whose coordinates are known will be found.

A general formula for this will be developed and used. Suppose it is desired to calculate the distance d from the point(1, 2) to the point (3, -2) shown on the grid below. 

We notice that the segment connecting these points is the hypotenuse of a right triangle and use the Pythagorean Theorem. The sides can be measured by counting the grids or by subtracting the coordinates. 

The vertical side of this triangle has length 4 which can be seen by subtracting the second coordinates 2 – (-2) = 4 The horizontal side has length 2 which can be seen by subtracting the first coordinates 1 – 3 = - 2 which we change to + 2 because length, or distance, is positive. 

Using the Pythagorean Theorem we have  . Therefore 

. We can summarize this as follows: 

The Distance Formula:We can generalize the method used above. The distance between any two points  is given by 

. This is known as “the distance formula.” 

Let's practice:i. What is the distance between the points (5, 6) and (– 12, 40) ? We apply the distance formula: 

which gives us two possible answers for y:y = 0 and y = 4. Consequently there are two possible points which are located at the required distance from our given point (1, 2). They are (3, 0) and (3, 4).

1.If the distance from the point (1, 2) to the point (3, y) is ,what is the value of y? We apply the distance formula:

Squaring both sides of the final equation gives us

We solve this by factoring:

                What is the distance between (–2, 7) and (4, 6)?

                What is your answer? 

                 If the distance from (x, 3) to (4, 7) is           , what is the value of x?

                 What is your answer? 

Examples