1. CLAST GEOMETRY and MEASUREMENT
CLAST Geometry by Dick Pilsetnieks /99

2. Round Measurement: Conversions
Length:
1 foot = 12 inches (in)
1 yard = 3 feet (ft)
1 yard = 36 inches (in)
1 mile = 5280 feet (ft)
Weight:
1 pound (lb) = 16 ounces (oz)
1 ton (T) = 2000 pounds (lb)
Capacity:
1 Cup (c ) = 8 fluid (oz)
1 pint (pt) = 2 Cups (c)
1 quart (qt) = 2 pints (pt)
1 gallon (gal) = 4 quarts (qt)
Time:
1 minute (min) = 60 seconds (sec)
1 hour (hr) = 60 minutes (min)
1 day (d) = 24 hours (hr)
1 week (wk) = 7 days (d)
1 year (yr) = 365 days (d)
CLAST Geometry by Dick Pilsetnieks /99

3. Round Measurement: Conversion Example
Convert 3 hours, 24 minutes into hours.
From the chart, 1 hr = 60 min. Note that = 1
24 minutes multiplied by 1 becomes
3 hr 24 min = 3 hr hr, or 3.4 hr
______
1 hr
60 min /
______
24 min hr hr or 0.4 hr
______ __ = = /
min
CLAST Geometry by Dick Pilsetnieks /99

4. Rounding a Measurement
Express the measurement in decimal form. Make any necessary
conversions.
Underline the digit to be rounded  
If the digit to the right of the underlined digit is 5 or more, add 1
to the underlined digit. If the digit to the right of the underlined
digit is less than 5, leave the underlined digit as is.
Write the rounded measurement by dropping all digits to the right
of the underlined digit. If necessary, insert zeros as placeholders.
CLAST Geometry by Dick Pilsetnieks /99

5. Pythagorean Theorem Add : B c a c-x D (Congruent Triangles) x A C b
The equation c2 = a2 + b2 , known as the Pythagorean Theorem, describes the relationship among the three sides of a right triangle. It is used to find the length of a third side when the lengths of the two other sides are known. B c a
c-x D
(Congruent Triangles) x C A b
Add :
A Greek mathematician, Pythagoras, is believed to have the first proof of this theorem in about 500 B.C.
CLAST Geometry by Dick Pilsetnieks /99

6. Metric Conversion Chart
Kilometer (km) m King
Hectometer (hm) m Henry
Dekameter (dam) m Died
Meter (m) m Monday
Decimeter (dm) m Drinking
Centimeter (cm) m Chocolate
Millimeter (mm) m Milk
CLAST Geometry by Dick Pilsetnieks /99

7. Examples of Metric Conversion
Example 1: Convert 5.2 km to meters
From the conversion chart, 1 km = 1000 m \ \
Example 2: Convert 80 cm to hectometers
From the conversion chart, 1 cm = 0.01 m and
1 hm = 100 m \ \ \ \
CLAST Geometry by Dick Pilsetnieks /99

8. CLAST Geometry by Dick Pilsetnieks 2/99
Calculate Distance
Sides polygon name
triangle
quadrilateral
pentagon
hexagon
Sides polygon name
heptagon
octagon
nonagon
decagon
To find the perimeter of a polygon (distance around) :
Change all lengths to the same unit of measure.
If necessary, draw the figure and label each side.
Add the lengths of all sides.
CLAST Geometry by Dick Pilsetnieks /99

9. Circumference of a Circle
The distance around a circle is called its circumference
The radius, r , of a circle is the length of a line segment from the center of the circle to a point on the circle. r
The diameter, d , of a circle is the length of a line segment through the center of the circle with endpoints on the circle. d
The formula for the circumference of (distance around) a circular region is
CLAST Geometry by Dick Pilsetnieks /99

10. CLAST Geometry by Dick Pilsetnieks 2/99
Area of a Circle
The radius, r , of a circle is the length of a line segment from the center of the circle to a point on the circle. r
The diameter, d , of a circle is the length of a line segment through the center of the circle with endpoints on the circle. d
The formula for the area of a circle is
Or equivalently,
CLAST Geometry by Dick Pilsetnieks /99

11. Infer Formulas for Measuring Geometric Figures
3 sides
1 triangle
S = 180o
4 sides
2 triangles
S = 360o
6 sides
4 triangles
S = 720o
There is a formula that states that the sum (S) of the measures of the interior angles
of a convex polygon is (n - 2) x 180o , where n represents the number of sides.
For an eight sided polygon, then, we would infer that the sum is
S = (8-2) x 180o = 1080o
CLAST Geometry by Dick Pilsetnieks /99

12. Identify Relationships between Angle Measures
Two lines that intersect form vertical angles. Vertical angles have equal measure. 3 1 2
m 1 = m 2 m 3 = m 4 4
Special angle relationships are true when two parallel lines are cut by a third line (a transversal)
A parallelogram is a four-sided polygon that has two pairs of parallel sides 1 1 2 2
m 1 = m 5 m 2 = m 6 m 3 = m 7 m 4 = m 8
m 1 = m 4 m 3 = m 2 3 4 5 6 3 4 7 8
CLAST Geometry by Dick Pilsetnieks /99

13. Names of Plane Angles
Angles are classified according to their measure:
An acute angle is an angle of measure greater than 0o and less than 90o )
A right angle is an angle of measure 90o )
An obtuse angle is an angle of measure greater than 90o and less than 180o
A straight angle is an angle of measure 180o 
CLAST Geometry by Dick Pilsetnieks /99

14. Triangles Classified by Measure of their Angles
A right triangle has one right angle
An obtuse triangle has one obtuse angle
A scalene triangle has three sides of unequal length
60o )
60o
60o )
An equilateral triangle has three sides of equal length and all three angles measure 60o
An acute triangle has three acute angles
An Isosceles triangle has two congruent sides and the angles opposite these sides are of equal measure
CLAST Geometry by Dick Pilsetnieks /99

15. Quadrilaterals A trapezoid has one pair of parallel sides.
A rhombus is a parallelogram with all sides congruent.
Diagonals are perpendicular to each other.
A parallelogram has two pairs of parallel sides. Opposite sides are equal in length. Opposite angles are equal in measure. Diagonals bisect each other.
A square is a parallelogram with all sides equal in length
and all four right angles 90o. Diagonals are perpendicular
and equal in length.
CLAST Geometry by Dick Pilsetnieks /99

16. Calculate Area Area of a rectangle height (h) width (W)
A = b h or A = L W
height (h) width (W)
base (b) length (L)
Area of a parallelogram
A = b h h b
Area of a triangle
A = b h h b
CLAST Geometry by Dick Pilsetnieks /99

17. Calculate Surface Area and Volume
Surface area of a rectangular solid
SA = 2(LW) + 2(LH) + 2(HW) H
Volume of a rectangular solid (prism)
V = LWH W L
Surface area of a right circular cylinder
SA = (2 r)(h) + 2( r2) r h
Volume of a right circular cylinder
V = ( r2 ) h
CLAST Geometry by Dick Pilsetnieks /99

18. Surface Area and Volume of a Right Circular Cone
CLAST Geometry by Dick Pilsetnieks /99

19. CLAST Geometry by Dick Pilsetnieks 2/99
Volume of a Sphere r
CLAST Geometry by Dick Pilsetnieks /99