Mensuration Introduction

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1 Mensuration Introduction Mensuration is the process of measuring and calculating with measurements. Mensuration deals with the determination of length, area, or volume Measurement Types The basic measurement types include: distances (including perimeters and circumferences) areas volumes Quantities such as area and volume are often calculated from measured distances such as length, width, height, diameter, circumference, perimeter or other distances. Measured Shapes Measured shapes can be classified in various ways, including: two-dimensional (flat) surfaces, such as rectangles, triangles, trapezoids and other polygons circles, sectors and segments of circles. three-dimensional (solid) shapes, such as prisms and cylinders spheres cones and pyramids Another classification of shapes is: regular shapes (such as the shapes identified above) irregular shapes, which are shapes bound by irregular edges such as a pond, or trench. Two Dimensional Shapes - Perimeter and Area Many problems in architecture, construction, surveying, mechanical design and civil engineering involve two dimensional shapes. Mensuration calculations for two dimensional shapes include: distances and areas. Distance - Units Distance calculations result in linear units. Typical units for distances include: cm km m ft centimetres kilometres metres feet 71 Mensuration Notes OH1 1 16/01/010

2 Area - Units Area calculations result in square units. Typical units for areas include: cm m in ft square centimetres square metres square inches square feet This section considers perimeter and area for: triangles, quadrilaterals, and circular shapes Polygons Polygons are closed plane figures bound by three or more line segments. Common polygons include those with: Three sides Triangle Four sides Quadrilateral Five sides: Pentagon Six sides: Hexagon Eight sides: Octagon Twelve sides: Dodecagon 71 Mensuration Notes OH1 16/01/010

3 Do not copy into your notes Source: regular polygons - Wikipedia For regular polygons with side s=1 this produces the following table: Sides Name Approximate area 3 equilateral triangle square regular pentagon regular hexagon regular heptagon regular octagon regular nonagon regular decagon regular hendecagon regular dodecagon regular triskaidecagon regular tetradecagon regular pentadecagon regular hexadecagon regular heptadecagon regular octadecagon regular enneadecagon regular icosagon regular hectagon regular chiliagon regular myriagon ,000,000 regular megagon 79,577,471, Mensuration Notes OH1 3 16/01/010

4 Definition A general polygon may have unequal side lengths and unequal angles. Definition A regular polygon has equal side lengths and equal angles. This section considers mainly triangles and quadrilaterals. Triangles regular polygons Definition A triangle is a closed three sided figure, in which each side is a straight line segment and includes the following types: Right Triangles - contains a 90 degree angle Oblique Triangles contains no 90 degree angle Acute Triangles all angles are less than 90 degrees Equilateral Triangles all sides and angles are of equal length Isosceles Triangles two angles and two sides are equal Obtuse Triangles contains one angle greater than 90 degrees Scalene Triangles no equal sides and no equal angles 71 Mensuration Notes OH1 4 16/01/010

5 Quadrilaterals Definition A quadrilateral is a closed geometric figure (two dimensional) with four straight sides, and includes the following types: Rectangles and Squares all angles are 90 degrees Parallelograms - each pair of opposite sides are parallel Rhombuses Every rhombus has two diagonals connecting opposite pairs of vertices and two pairs of parallel sides. It follows that any rhombus has the following two properties: 1. Opposite angles of a rhombus have equal measure.. The two diagonals of a rhombus are perpendicular. Trapezoids - one pair of opposite sides are parallel General Quadrilaterals sides are of unequal length 71 Mensuration Notes OH1 5 16/01/010

6 Triangles Definition A triangle is a closed 3 sided figure, in which each side is a straight line segment. Notation a b c h s P A one side of the triangle base side of the triangle third side of the triangle height of the triangle one half the perimeter of the triangle perimeter of the triangle area of the triangle The height h is the perpendicular distance from any vertex to the opposite side. Any side may be considered the base of the triangle. Triangles are commonly labelled with the sides in lower case and each angle in upper case of the same letter as the opposite side (e.g. angle A opposite side a ). 71 Mensuration Notes OH1 6 16/01/010

7 Formulas The perimeter of a triangle is the sum of the lengths of the sides. P a b c The area of a triangle is one half the base times the height: A 1 bh Knowing the length of the 3 sides of the triangle, the area can be obtained from Heron's Formula: A ss as bs c where s 1 a b c Knowing the length of sides a and b and the included angle C of a triangle, the area is: A 1 absinc 71 Mensuration Notes OH1 7 16/01/010

8 Perimeter Problems Example 1 Determine the perimeter of a triangular structure with sides of.5m, 4.7m, and 3.1m. Solution Analysis: Calculation: For P : P a b c P = = 10.3 Conclusion: The perimeter of a triangular structure is 10.3m. 71 Mensuration Notes OH1 8 16/01/010

9 Example Determine the perimeter of the triangular steel plate. Solution Analysis: Calculations: For a : For P : It s a right angle triangle, so use Pythagoras to solve for the length of the missing side r = x + y 54 = x + 46 x = = P = = 18.8 Conclusion: The perimeter of the steel plate is 18.8cm. 71 Mensuration Notes OH1 9 16/01/010

10 Example 3 Determine the perimeter of the fabricated truss. Solution Analysis: Calculations: For a (the third side): For P : a = b + c bccosa a = b + c bccosa ( )( ) = + = cos a = = 33.6 P = = Conclusion: The perimeter of the fabricated truss is 70.51m. 71 Mensuration Notes OH /01/010

11 Area Problems Example 1 Determine the triangular section area of an industrial building wall. Solution 1 Analysis: Calculation: Area = 1 bh 1 Area = (8.5)(6.0) = 85.5 Conclusion: The area of the wall is 85.5 m. 71 Mensuration Notes OH /01/010

12 Example Determine the area of the illustrated triangular structure. Solution Analysis: Calculations: For s : For A : s 1 = ( ) = 5.15 m ( )( )( ) Area = s s a s b s c ( )( )( ) = = Conclusion: The area of the triangular structure is m. 71 Mensuration Notes OH1 1 16/01/010

13 Example 3 Calculate the enclosed floor area. Solution 3 Analysis: Calculations: 1 Area = absin C 1 ( 163 )( 178 ) sin15 o Area = = Conclusion: The enclosed floor area is cm 71 Mensuration Notes OH /01/010

14 Example 4 Determine the floor area of a commercial display. Analysis : A 1 bh (requires the base and height) A ss as bs c (requires three sides) A 1 absinc (requires two sides and the included angle) Which formula can be used in this example? For C: = sin38.4 sin C For B: sin C = C = 1 sin C = o B = 180 ( ) = o 71 Mensuration Notes OH /01/010

15 For the area A : 1 Area = acsin B 1 = ( 4.83 )( 6.5 ) sin = Conclusion: The floor area of a commercial display is m. 71 Mensuration Notes OH /01/010

16 Quadrilaterals The next sections explore mensuration calculations involving the following quadrilateral shapes. Rectangle and Square Definition A rectangle is a quadrilateral where each interior angle is 90 sides are equal in length. Definition A square is a rectangle where all sides are equal in length. Parallelogram and Rhombus and the opposite Definition A parallelogram is a quadrilateral with two pairs of opposite sides that are equal and parallel. Definition A rhombus is a parallelogram where all sides are of equal length. The diagonals of a rhombus intersect at right angles. Trapezoid Definition A trapezoid is a quadrilateral with one pair of parallel sides. 71 Mensuration Notes OH /01/010

17 General Quadrilateral Definition A general quadrilateral is a closed geometric figure with four straight sides. Rectangles and Squares Rectangles and squares are quadrilaterals with specific characteristics. Rectangles Definition A rectangle is a quadrilateral where each interior angle is 90 and the opposite sides are equal in length. Squares Definition A square is a rectangle where all sides are equal in length. Notation l length of the rectangle w width of the rectangle or square d diagonal distance P perimeter of the square or rectangle A area of the square or rectangle Note that the length and width for a square are equal and are both labelled w. 71 Mensuration Notes OH /01/010

18 Formulas With reference to the diagram: The perimeter is the sum of the lengths of the sides. rectangle P l w P l w square P w w P 4w The area is the length times the width. rectangle square A l w A w The diagonal can be determined from the Pythagorean Theorem: rectangle square d l w d w w d l w d w 71 Mensuration Notes OH /01/010

19 Example 1 Determine the perimeter and area of a rectangular garden plot of width 1.4m and length 5.m. Solution Analysis Calculations: For P : P = = 35. For A : A = ( 1.4)( 5.) = Conclusion: The rectangular garden plot has a perimeter of 35. m and an area of m. 71 Mensuration Notes OH /01/010

20 Example Determine the perimeter of the foundation layout as shown. Analysis: Calculations: For a and b : =.3m =.5m For P : P = = 51.9 Conclusion: The perimeter of the foundation layout is 51.9 m. Stopped Tuesday Jan 1 71 Mensuration Notes OH1 0 16/01/010

21 Example 3 Determine the area of the steel plate. Solution c = 15.1 c= 3.4cm A 1 1 = = 8.84cm ( )( ) A = = 3.4cm ( 4.5)( 5.) A 3 1 = = 18.7cm ( )( ) A = = Conclusion: The area of the steel plate is cm. 71 Mensuration Notes OH1 1 16/01/010

22 Example 4 Determine the perimeter and the area of the retail floor display space. The floor area is square with a diagonal of 48.5 m. Analysis: Calculation: From the Pythagorean Theorem: w + w = 48.5 w = w = = w = 34.9 perimeter = = area = = Conclusion: The retail floor display space has a perimeter 137. m of and an area of m. 71 Mensuration Notes OH1 16/01/010

23 Parallelograms and Rhombuses Parallelograms and rhombuses are quadrilaterals with specific characteristics. Parallelogram Rhombus A parallelogram is a quadrilateral with two pairs of opposite sides that are equal and parallel. A rhombus is a parallelogram where all sides are of equal length. The diagonals of a rhombus intersect at right angles. Parallelograms Rhombus a and b are the lengths of the sides. h height (the perpendicular distance between a pair of parallel sides). A area P perimeter Formulas The perimeter is the sum of the edge lengths. parallelogram P a b rhombus P 4b The area is the base length times the height. parallelogram A b h rhombus A b h 71 Mensuration Notes OH1 3 16/01/010

24 Example 1 A parallelogram shaped building lot which covers an area of 500 m. Determine the perimeter of the lot. For h : α = = 40 o o h cos 40 = 4.8 h= m For b : For P : Conclusion: P Area = bh ( ) 500 = b b= = 6.319m ( ) ( ) = = 10.4m The perimeter of the lot is 10.4 m. 71 Mensuration Notes OH1 4 16/01/010

25 Example Determine the perimeter of the rhombus shaped pendant. For b : w = = w = =.3585 For P : P = 4(.3585) = Conclusion: The perimeter of the pendant is cm. 71 Mensuration Notes OH1 5 16/01/010

26 Trapezoids Trapezoid A trapezoid is a quadrilateral with one pair of parallel sides. a and b are the two parallel sides c and d are the other sides h height of the trapezoid (perpendicular distance between the parallel sides). A area P perimeter The perimeter is the sum of the lengths of the sides. P a b c d The area is the average of the lengths of the two parallel sides times the height. A a b h 71 Mensuration Notes OH1 6 16/01/010

27 Example 1 Determine the area of the steel plate. Analysis: Calculations: A a+ b = h A = 5. = Conclusion: The area of the steel plate is cm. 71 Mensuration Notes OH1 7 16/01/010

28 Example Calculate the perimeter and area for this cross section of an industrial building roof structure. 71 Mensuration Notes OH1 8 16/01/010

29 Calculations: For e : Consider the similar triangles: 1 = 10 e e = 10 e = 5 c = c= 11.18m For f : Consider the similar triangles: 3 = 10 f f = Mensuration Notes OH1 9 16/01/010

30 d = d = 18.08m For P : b = = 8. For A : A a+ b = h A = 10 = 18 P = = Conclusion: The industrial building roof structure has a perimeter of 65.6 m and an area of 18 m. 71 Mensuration Notes OH /01/010

31 Example 3 Calculate the area of the trapezoid shaped panel with dimensions as indicated. Solution 3 Analysis: Consider combining the two triangles into a single triangle. Knowing all three sides of the triangle provides different alternatives. an algebraic alternative a trigonometric alternative Calculate A triangle from Heron's Calculate an angle by the cosine law formula Calculate h from A 1 bh Calculate h by right trigonometry Calculate A rectangle from A lw Sum the two areas Calculate A trap. from A a b h Calculations: Calculations are shown for both alternatives. 71 Mensuration Notes OH /01/010

32 The trigonometric alternative: For angle C (by the cosine law) : c = a + b abcosc ( )( ) 11.1 = cos cosc = C = o For h (by right triangle trigonometry) : C sin o h = 15.4 h = 15.4sin = cm For A trapezoid : Area = = 95.0 cm The area of the trapezoid is 95.0 cm. Stopped Tuesday Jan Mensuration Notes OH1 3 16/01/010

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