2. MENSURATION
SUPPOSE YOU WANT TO PUT A BOUNDARY AROUND YOUR GARDEN OR FIELD. FOR
THAT, YOU NEED TO FIND OUT THE LENGTH OF THE BOUNDARY. ALSO, WHAT IF YOU
WANT TO FIND OUT THE AREA AND VOLUME OF DIFFERENT GEOMETRICAL SHAPES.
SO THIS IS SOMETHING WE ARE GOING TO SEE IN HIS CHAPTER. SO LET US STUDY
THE TOPIC MENSURATION AND MENSURATION FORMULAS IN DETAIL
3. DEFINITION
• THEORY OF MEASUREMENT.
• MENSURATION (MATHEMATICS), A BRANCH OF MATHEMATICS THAT DEALS
WITH MEASUREMENT OF VARIOUS PARAMETERS OF GEOMETRIC FIGURES AND
MANY MORE
• FOREST MENSURATION, A BRANCH OF FORESTRYTHAT DEALS WITH
MEASUREMENTS OF FOREST STAND.
4. MEANING
MENSURATION IS THE BRANCH OF MATHEMATICS WHICH DEALS WITH THE
STUDY OF DIFFERENT GEOMETRICAL SHAPES,THEIR AREAS AND VOLUME.IN
THE BROADEST SENSE, IT IS ALL ABOUT THE PROCESS OF MEASUREMENT. IT
IS BASED ON THE USE OF ALGEBRAIC EQUATIONS AND GEOMETRIC
CALCULATIONS TO PROVIDE MEASUREMENT DATA REGARDING THE WIDTH,
DEPTH AND VOLUME OF A GIVEN OBJECT OR GROUP OF OBJECTS. WHILE THE
MEASUREMENT RESULTS OBTAINED BY THE USE OF MENSURATION ARE
ESTIMATES RATHER THAN ACTUAL PHYSICAL MEASUREMENTS, THE
CALCULATIONS ARE USUALLY CONSIDERED VERY ACCURATE.
5. TYPES OF 2D FIGURES
• SQUARE
• RECTANGLE
• TRIANGLE
• CIRCLE
6. TYPES OF 3D DIMENSION
• CYLINDER
• SPHERE
• PARALLELOGRAM
• RHOMBUS
• CONE
• HEMISPHERE ETC..........
7. PERIMETER OF SQAURE
• THE FORMULA OF PERIMETER AND AREA OF SQUARE ARE EXPLAINED STEP-BY-
STEP WITH SOLVED EXAMPLES.
• IF 'A' DENOTES THE SIDE OF THE SQUARE, THEN, LENGTH OF EACH SIDE OF A
SQUARE IS 'A' UNITS
• PERIMETER OF SQUARE = AB + BC + CD + DA
• = (A + A + A + A) UNITS
• = 4A UNITS
8. AREA OF SQAURE
• PERIMETER OF THE SQUARE = 4A UNITS
• WE KNOW THAT THE AREA OF THE SQUARE IS GIVEN BY
• AREA = SIDE × SIDE
• A = A × A SQ. UNITS
9. AREA AND PERIMETER OF RECTANGLE
• THE AREA OF A RECTANGLE IS GIVEN BY MULTIPLYING THE WIDTH TIMES THE HEIGHT. AS A
FORMULA AREA OF RECTANGLE=L×B SQ. UNITS
• THE PERIMETER IS THE TOTAL DISTANCE AROUND THE OUTSIDE, WHICH CAN BE FOUND BY
ADDING TOGETHER THE LENGTH OF EACH SIDE. I THE CASE OF A RECTANGLE, OPPOSITE SIDES
ARE EQUAL IN LENGTH, SO THE PERIMETER IS TWICE ITS WIDTH PLUS TWICE ITS HEIGHT. OR AS A
FORMULA:(
• 2(L+B)
• WHERE:
W IS THE WIDTH OF THE RECTANGLE
H IS THE HEIGHT OF THE RECTANGLE
10. TRIANGLE
• IN EUCLIDEAN GEOMETRY ANY THREE POINTS, WHEN NON-COLLINEAR,
DETERMINE A UNIQUE TRIANGLE AND SIMULTANEOUSLY, A UNIQUE PLANE (I.E.
A TWO-DIMENSIONAL EUCLIDEAN SPACE). IN OTHER WORDS, THERE IS ONLY
ONE PLANE THAT CONTAINS THAT TRIANGLE, AND EVERY TRIANGLE IS
CONTAINED IN SOME PLANE. IF THE ENTIRE GEOMETRY IS ONLY
THE EUCLIDEAN PLANE, THERE IS ONLY ONE PLANE AND ALL TRIANGLES ARE
CONTAINED IN IT; HOWEVER, IN HIGHER-DIMENSIONAL EUCLIDEAN SPACES,
THIS IS NO LONGER TRUE.
• A TRIANGLE IS HALF AS BIG AS THE RECTANGLE THAT SURROUNDS IT, WHICH
IS WHY THE AREA OF A TRIANGLE IS ONE-HALF BASE TIMES HEIGHT.
• AREA OF TRIANGLE =1/2 BH
11. SPHERE
• A SPHERE IS A 3D LOCUS OF POINTS WHICH ARE ALL EQUIDISTANT FROM THE
CENTRE OF THE SPHERE.
•
(4/3) * Π * RADIUS3
4 * Π * RADIUS2
12. HEMISPHERE
• MENSURATION FOR A HEMISPHERE
•
• LET'S CUT THE SPHERE INTO TWO HEMISPHERES. WHAT IS THE VOLUME AND TOTAL
SURFACE AREA, FOR EITHER OF THE HEMISPHERES?
• VOLUME OF THE HEMISPHERE = (2/3) * Π * RADIUS3
• SURFACE AREA OF THE HEMISPHERE = CURVED SURFACE AREA + AREA OF BASE = 2 * Π *
RADIUS2 Π * RADIUS2 = 3 Π RADIUS2
13. PARALLELOGRAM
• PARALLELOGRAM
• WHAT DOES A PARALLELOGRAM MEAN? PARALLEL LINES ARE THE TWO LINES THAT NEVER MEET AND A PARALLELOGRAM IS A
SLANTED RECTANGLE WITH THE LENGTH OF THE OPPOSITE SIDES BEING EQUAL JUST LIKE A RECTANGLE. BECAUSE OF THE
PARALLEL LINES, OPPOSITE SIDES ARE EQUAL AND PARALLEL. SUPPOSE IF EVERY PAIR OF OPPOSITE SIDES OF A
QUADRILATERAL IS EQUAL, THEN IT BECOMES A PARALLELOGRAM.
• DIAGONALS OF A PARALLELOGRAM BISECT EACH OTHER. SO, WHEN THE DIAGONALS OF A PARALLELOGRAM BISECT EACH
OTHER, IT DIVIDES IT INTO TWO CONGRUENT TRIANGLES. IN A PARALLELOGRAM, THE ANGLES ARE NOT RIGHT ANGLES. THE
SUM OF THE ANGLES OF A PARALLELOGRAM IS 360°.
• AREA OF PARALLELOGRAM= B×H
• PROPERTIES OF PARALLELOGRAM
• OPPOSITE SIDES ARE CONGRUENT, AB = DC
• OPPOSITE ANGLES ARE CONGRUENT D = B
• IF ONE ANGLE IS RIGHT, THEN ALL ANGLES ARE RIGHT.
• THE DIAGONALS OF A PARALLELOGRAM BISECT EACH OTHER.
14. RHOMBUS
• RHOMBUS IS A QUADRILATERAL WHICH MEANS IT HAS FOUR SIDES. SO, JUST LIKE A SQUARE WITH CONGRUENT
OR EQUAL SIDES. THE OPPOSITE SIDES OF THE RHOMBUS ARE PARALLEL TO EACH OTHER AND ITS ANGLES ARE
CONGRUENT TO EACH OTHER. IN A RHOMBUS THE DIAGONALS ARE PERPENDICULAR AND THEY BISECT EACH
OTHER AT RIGHT ANGLES AND CUT EACH OTHER IN HALF.
• PROPERTIES OF RHOMBUS
• AS THE INTERNAL ANGLES DO NOT FORM A 90-DEGREE ANGLE. SO A RHOMBUS IS ALSO CALLED AS A SLANTED
SQUARE. SUM OF THE ADJACENT SIDES OF ANY RHOMBUS IS EQUAL TO 180 DEGREES. THE AREA OF RHOMBUS
CAN BE CALCULATED USING 2 DIFFERENT FORMULAE
• USING THE BASE AND HEIGHT
• AREA OF RHOMBUS= B×H
• USING THE LENGTH OF THE DIAGONALS,
• AREA OF RHOMBUS= (D1×D2)2
15. CUBOID
• SOLID ALL OF WHOSE SIDES ARE RECTANGLES ARE CUBOIDS. CUBOID IS A 3-DIMENSIONAL SHAPE OR A POLYHEDRON, HAVING 6
RECTANGULAR SIDES CALLED FACES. IT HAS 8 VERTICES AND 12 EDGES. EACH FACE IS A RECTANGLE AND ALL THE CUBOIDS CORNER ARE
VERTICES ARE 90-DEGREE ANGLES.
• HOW TO IDENTIFY A CUBOID?
• IN A CUBOID, EACH FACE IS A RECTANGLE AND THE CORNERS OR THE VERTICES ARE 90-DEGREE ANGLES. MOREOVER, THE OPPOSITE FACES
ARE ALWAYS EQUAL. THE SIMPLEST EXAMPLE IS A BOOK. IT HAS 6 SURFACES OF WHICH EACH OPPOSITE PAIR IS OF THE SAME DIMENSIONS.
LET L BE THE LENGTH, B BE THE BREADTH AND H BE THE HEIGHT OF THE CUBOID.
• AREA OF 4 SIDE FACES = 2 ( L × B ) + 2 ( B× H ) = 2 ( L × B ) × H = PERIMETER OF BASE ×HEIGHT
• AREA OF 4 SIDE FACES IS ALSO CALLED A LATERAL SURFACE AREA. FOR EXAMPLE, THE AREA OF 4 WALLS OF A ROOM. THEREFORE,
• TOTAL SURFACE AREA OF A CUBOID = 2 ( L × B ) + ( L× H ) + ( B× H )
• VOLUME OF CUBOID
• VOLUME OF CUBOID = LENGTH × BREADTH × HEIGHT
OR, VOLUME OF CUBOID (V) = L × B × H
16. CUBE
• THE CUBE IS A THREE-DIMENSIONAL STRUCTURE WHICH IS FORMED WHEN SIX IDENTICAL SQUARES BIND TO EACH
OTHER IN AN ENCLOSED FORM. A CUBE GENERALLY HAS 6 FACES, 12 EDGES, AND 8 VERTICES.
• WHEN A NUMBER IS MULTIPLIED BY ITSELF THREE TIMES, THEN THE FORMED IS A CUBE NUMBER. FOR EXAMPLE, 3 × 3
×3 = 27 IS A CUBE NUMBER.
• VOLUME OF CUBE
• THE VOLUME OF A CUBE IS LENGTH (L) × BREADTH (B) × HEIGHT (H), SINCE L = B = H IN THE CUBE, ITS SIDES CAN BE
REPRESENTED AS L = B = H = A. THEREFORE,
• VOLUME OF CUBE = A³
• WHERE A IS THE MEASUREMENT OF EACH SIDE OF THE CUBE. HENCE, THE VOLUME OF THE CUBE OF SIDE 1 CM WILL BE
EQUAL TO 1 CM × 1 CM × 1 CM = 1 CM³. ONE SHOULD ALSO BE HANDY ABOUT THE PROPERTIES OF CUBE NUMBERS, LIKE:
• CUBES OF POSITIVE NUMBERS ARE ALWAYS POSITIVE. FOR EXAMPLE: CUBE OF +4 IS = (+4) × (+4) × (+4) = +64
• CUBES OF NEGATIVE NUMBERS ARE ALWAYS NEGATIVE. FOR EXAMPLE: CUBE OF -4 IS = (-4) × (-4) × (-4) = -64
17. FORMULES
• SOME IMPORTANT MENSURATION FORMULAS ARE:
• 1. AREA OF RECTANGLE (A) = LENGTH(L) × BREATH(B)
•
• 2. PERIMETER OF A RECTANGLE (P) = 2 × (LENGTH(L) + BREATH(B))
•
• 3. AREA OF A SQUARE (A) = LENGTH (L) × LENGTH (L)
•
• 4. PERIMETER OF A SQUARE (P) = 4 × LENGTH (L)
•
• 5. AREA OF A PARALLELOGRAM(A) = LENGTH(L) × HEIGHT(H)
•
• 6. PERIMETER OF A PARALLELOGRAM (P) = 2 × (LENGTH(L) + BREADTH(B))
18. CONTI......
• AREA OF A TRIANGLE (A) = (BASE(B) × HEIGHT(B)) / 2
• AND FOR A TRIANGLE WITH SIDES MEASURING “A” , “B” AND “C” , PERIMETER = A+B+C
• AND S = SEMI PERIMETER = PERIMETER / 2 = (A+B+C)/2
• AND ALSO: AREA OF TRIANGLE =
• THIS FORMULAS IS ALSO KNOWS AS “HERON’S FORMULA”.
•
• 8. AREA OF TRIANGLE(A) =
• WHERE A, B AND C ARE THE VERTEX AND ANGLE A , B , C ARE RESPECTIVE ANGLES OF TRIANGLES AND A , B , C ARE THE RESPECTIVE OPPOSITE SIDES OF THE ANGLES AS SHOWN IN FIGURE BELOW:
• AREA OF TRIANGLE - MENSURATION
•
• 9. AREA OF ISOSCELES TRIANGLE =
• WHERE A = LENGTH OF TWO EQUAL SIDE , B= LENGTH OF BASE OF ISOSCELES TRIANGLE.
•
• 10. AREA OF TRAPEZIUM (A) =
• WHERE “A” AND “B” ARE THE LENGTH OF PARALLEL SIDES AND “H” IS THE PERPENDICULAR DISTANCE BETWEEN “A” AND “B” .
•
• 11. PERIMETER OF A TRAPEZIUM (P) = SUM OF ALL SIDES
•
• 12. AREA OF RHOMBUS (A) = PRODUCT OF DIAGONALS / 2
•
19. CONTI......
•
13. PERIMETER OF A RHOMBUS (P) = 4 × L
• WHERE L = LENGTH OF A SIDE
•
• 14. AREA OF QUADRILATERAL (A) = 1/2 × DIAGONAL × (SUM OF OFFSETS)
•
• 15. AREA OF A KITE (A) = 1/2 × PRODUCT OF IT’S DIAGONALS
•
• 16. PERIMETER OF A KITE (A) = 2 × SUM ON NON-ADJACENT SIDES
•
• 17. AREA OF A CIRCLE (A) =
• WHERE R = RADIUS OF THE CIRCLE AND D = DIAMETER OF THE CIRCLE.
•
• 18. CIRCUMFERENCE OF A CIRCLE =
20. APPLICATION
• HOW IS MENSURATION USED IN OUR DAILY LIVES?
• IN COOKING, WE USE MEASUREMENTS FOR VOLUME WHEN FOLLOWING RECIPE BOOKS. TOOLS SUCH AS
MEASURING JUGS MAY BE USED TO DETERMINE VOLUMES. WHEN MAKING A CAKE FOR EXAMPLE WE MAY NEED
TINS OF A SPECIFIC WIDTH AND LENGTH FOR A PARTICULAR RECIPE. WE MAY ALSO NEED TO WEIGHT OUT THE
WEIGHT OF THE DRY INGREDIENTS USING A MEASURING SCALE.
• BEFORE LEAVING THE DOOR, WE MAY CHECK THE WEATHER FORECAST. THE TEMPERATURES MEASURED
USING THERMOMETERS WILL HELP DETERMINE WHAT WE CHOOSE TO WEAR.
• WHEN PLANNING A CAR JOURNEY, WE MAY LOOK AT A MAP TO FIND OUT THE QUICKEST WAY TO REACH A
PARTICULAR DISTANCE. WE MAY LOOK AT HOW TO CUT DOWN THE NUMBER OF MILES WE HAVE TO TRAVEL BY
TAKING SHORT CUTS. THE SPEED YOU TRAVEL AT CAN ALSO BE MEASURED AND THIS MAY CONTRIBUTE TO THE
LENGTH OF YOUR JOURNEY IF YOU TRAVEL FASTER.
• CALLING INTO THE COFFEE SHOP FOR YOUR MORNING FIX WILL ALSO SEE YOU FACED WITH A MENSURATION
DILEMMA. THE SIZE OF CUP YOU CHOOSE WILL PRESENT YOU WITH A DIFFERENT VOLUME OF LIQUID.
22. MENSURATION IN HISTORY
• HISTORY:
• WRITE A REPORT OR COMPLETE A JOURNAL PAGE ON MATHEMATICIANS (GRAB A FREE
JOURNAL PAGE AT THE END OF THIS POST).
• LEARN ABOUT THE HISTORY OF CLOCKS. PURCHASE SOME INEXPENSIVE CLOCKS AND TAKE
THEM APART AND PUT THEM BACK TOGETHER. THEN PRACTICE TELLING TIME USING THE
CLOCKS.
• LEARN HOW VARIOUS CULTURES TOLD TIME THROUGHOUT HISTORY AND WRITE A REPORT.
• CALCULATE THE NUMBER OF YEARS BETWEEN VARIOUS EVENTS.
• LEARN ABOUT THE HISTORY OF THE SCALE AND EXPERIMENT WITH DIFFERENT TYPES OF
SCALES.
• LEARN ABOUT THE HISTORY OF CURRENCY
23. IN SCIENCE
• SCIENCE:
• WRITE THE DISTANCE BETWEEN THE SUN AND EACH PLANET USING EXPONENTIAL FORM.
• EXPLORE THE HALF-LIFE OF CERTAIN RADIOACTIVE ELEMENTS OR THE SIZE OF BACTERIA
AND VIRUSES USING NEGATIVE EXPONENTS.
• EXPLORE SCIENTIFIC FACTS, SUCH AS THE BOILING AND FREEZING POINT OF LIQUIDS, THE
MELTING AND FREEZING POINT OF SOLIDS AND THE TEMPERATURE OF PLANETS.
• USE ALGEBRA TO CALCULATE HOW MUCH FORCE A GIVEN MAGNET WOULD PULL ON
ANOTHER MAGNET.
• BUILD A WEIGHT BEARING BRIDGE USING VARIOUS HOUSEHOLD ITEMS. CREATE A DESIGN
AND REDUCE IT TO SCALE, PREPARE COST ANALYSIS AND THEN BUILD AND TEST THE
BRIDGE.
24. IN ART
• ART:
• CREATE A GEOMETRIC GREETING CARD USING SHAPES THAT ARE
CONGRUENT, SIMILAR, AND EQUIVALENT.
• EXAMINE WORKS OF ART THAT INCORPORATE GEOMETRIC SHAPES.
• CREATE TESSELLATIONS.
• PLAY WITH TANGRAMS.
• CREATE A PIECE OF ARTWORK USING PERSPECTIVE AND PROPORTION.
25. IN MUSIC AND TECHNOLOGY
• MUSIC:
• LEARN HOW THESE MUSIC TERMS RHYTHM, TIME, TONE, TUNE, PITCH,
FREQUENCY, AND AMPLITUDE GO HAND IN HAND WITH MATH.
• CREATE AND EXPERIMENT WITH A MONOCHORD.
• TECHNOLOGY:
• EXAMINE THE BINARY NUMBER SYSTEM. LOOK AT THE RELATIONSHIP
BETWEEN BASE 2 NUMBERS AND HOW COMPUTER CIRCUITRY WAS
DEVELOPED.
