M5: Applications of area and volume

M5: Further applications of area and volume Areas of ellipses, annuluses and parts of a circle Calculating areas of composite figures Applying Simpson’s Rule Surface area of Cylinders Surface area of spheres Volume of composite solids Errors in calculations

 Pythagoras theorem  Circumference of circle  Area of circle  Area of triangle  Area of rectangle  Area of parallelogram  Area of trapezium  Area of rhombus  Volume of Prism

 Pythagoras c²=a²+b²(The square on the hypotenuse is equal to the sum of the squares on the other two sides.)  Circumference of a circleC=2Πr  Area of a circleA = Πr²  Area of a triangleA = ½bh ORA =bh/2  Area of a rectangle A =bh  Area of a parallelogram A = bh  Area of a trapezium A=h/2 (a+b)  Area of a rhombus A=Dd/2  Volume of a prism V=Ah (Area of the cross-section x height)

 Determining appropriate units to use  Conversion between commonly used units  Accuracy in measurement  Error in measurement  Significant figures, scientific notation  Rates and ratios  Area of triangles and quadrilaterals  Field diagrams  Classifying polyhedra  Surface area  Volume and capacity

 Area of annulus=area of big circle – area of small circle= Π(R² – r²)  Area of an ellipse = Πab ◦ Where a=length of semi-major axis ◦ And b=length of semi-minor axis  Area of a sector = Θ/360 x Πr²  Arc length ℓ= Θ/360 x 2Πr

 Remember that more than one method may be used, adding or subtracting are both acceptable.

 Simpson’s rule is used to find the area of an uneven field where one side is a curved boundary.  A=h/3(d₁ + 4.d₂ + d₃) where h= width of strip (between successive measurements) d₁ = first distance d₂ = middle distance d ₃ = last distance

 You can either do two or more separate applications or you can put the first and last in brackets and then 4 times the even slotted distances and 2 times the odd slotted distances.

 Read the question carefully to determine whether it is open or closed and whether it is open both top and bottom  If open and asking for the surface area of the curved surface only, then  SA = 2Πrh (if you cut longways through the cylinder you would have a rectangle with the breadth being the circumference of the circle, thus 2Πr, and the height of the cylinder being h.)  If a closed cylinder then you have to find the area of the circular base and add that in.  i.e. A closed cylinder with top and bottom is  SA = 2Πrh + 2 x Πr²

 Surface area of a sphere  SA = 4Πr²  Volume of a sphere  V = 4/3Πr³

 Volume of composite figures can be found by adding the volume of multiple different solids or subtracting the volume of one solid from another if looking for remaining spaces.

 The accuracy of measurement is ± the smallest unit of the measuring instrument.  (If it is a ruler measuring in cm’s then the error is 0.5cm or 5 mm. If it is a set of scales measuring in 100gram increments then the error is ±50 gram.)  Percentage error  % error = (difference ÷ original) x 100%  Or  % error = (½ the smallest unit ÷ actual measurement) x 100%

 When answering questions do not round off during a calculation. Continue to use full calculator display and write down this as your solution before writing a concluding statement with a rounded answer. Rounding off too early causes significant differences in the final result. You can obtain marks for correct rounding even if your answer is incorrect.

 On attached sheets