CLAST GEOMETRY and MEASUREMENT CLAST Geometry by Dick Pilsetnieks 2/99

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 2/99

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 + 0.4 hr, or 3.4 hr ______ 1 hr 60 min / ______ 24 min 1 hr 2 hr or 0.4 hr ______ __ = = / 1 60 min 5 CLAST Geometry by Dick Pilsetnieks 2/99

Rounding a Measurement Express the measurement in decimal form. Make any necessary conversions. Underline the digit to be rounded. 31.749  31.75 31.744  31.74 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 2/99

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 2/99

Metric Conversion Chart Kilometer (km) 1000 m King Hectometer (hm) 100 m Henry Dekameter (dam) 10 m Died Meter (m) 1 m Monday Decimeter (dm) .1 m Drinking Centimeter (cm) .01 m Chocolate Millimeter (mm) .001 m Milk CLAST Geometry by Dick Pilsetnieks 2/99

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 2/99

CLAST Geometry by Dick Pilsetnieks 2/99 Calculate Distance Sides polygon name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon Sides polygon name 7 heptagon 8 octagon 9 nonagon 10 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 2/99

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 2/99

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 2/99

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 2/99

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 2/99

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 2/99

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 2/99

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 2/99

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 2/99

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 2/99

Surface Area and Volume of a Right Circular Cone CLAST Geometry by Dick Pilsetnieks 2/99

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