• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Moss's Egg. Task -1- Find the area of the shaded region inside the two circles shown below. The two large circles have a radius of 6cm.

Extracts from this document...


Moss’s Egg

The following formulas were used in the solving of the questions of this assessment piece:

Area of a Circle        


Circumference of a Circle


Area of a Sector


Arc Length


Working Out and Explanation

Task -1- Find the area of the shaded region inside the two circles shown below. The two large circles have a radius of 6cm. Their centres are A and B.



From the information given above, we know that the radii of the two larger circles are 6 cm in length. We define the radius of the circle as a straight line extending from the centre of a circle to its circumference. Since we know that points A and B are the centres of the two large circles, we can conclude that this is the length of the two points from A to B is 6 cm also, since point B is along the circumference of the top larger circle, and vice versa. From lengths A to B is the therefore the diameter of the smaller circle between the two larger ones, and thus we can conclude that the radius of the smaller circle is 3 cm. The area of the small circle can therefore be calculated using the formula indicated:                  image06.pngimage06.png, where A equals the area and r is the radius. Thus: A = image27.pngimage27.png (32) = 9 image62.pngimage62.png

 28.3 cm2


Task -2- The same circles are shown below.

...read more.


image52.png= image53.pngimage53.png= image54.pngimage54.png= 450

Thus angle ACE is 900 and therefore is right angled. As ACB and DCE are vertically opposing angles, ∠ DCE is also 900. We can now substitute the information we have to determine the area of the sector: image50.pngimage50.png where image53.pngimage53.png is 900 and r is 6- image45.pngimage45.png.


Task -4- Find the area of the sector BAE in the diagram below.

To find sector BAE in the circle, we also must apply the formula: image56.pngimage56.png . In the diagram displayed below, two internal angles of the two right angled triangles have been found. Therefore, the quickest method to determine ∠BAE is to subtract 1800 from the two known angles, as it is known that all internal angles of any triangle is equal to 1800.

∴ ∠BAE = 180 – 90 – 45

                = 45 0

The second value needed in calculating this sector is the radius. ∠BAE falls into both larger circles of the diagram. The length AB is the radius of these two circles, which we know to be 6cm in length. Therefore, the radius for determining the area of this sector is 6 cm. Thus, we simply substitute into the formula. image04.png


                ≈ 14.14 cm2

Task -5- Look for the shape of an egg in the enclosed areas of the circles shown below. This is called Moss’s Egg. Find the area and perimeter of Moss’s Egg.


...read more.



Sector DCE = image12.pngimage12.pngimage11.pngimage11.png = image13.pngimage13.pngr

Thus, in order to find a formula for determining the area of Moss’s Egg, the following equation must be used, with substituted figures in terms of r.

A Moss’s Egg = image14.pngimage14.png A small  AB + A sector BAE+ A sector ABD + A sector DCE – ΔABC

 A Moss’s Egg   = image15.pngimage15.png2 + image16.pngimage16.pngr2 + image16.pngimage16.pngr2 +                   image17.pngimage17.png2image18.pngimage18.pngimage19.pngimage19.png

= image20.pngimage20.png2 +image21.pngimage21.png2 + image22.pngimage22.pngimage23.pngimage23.png2 - image24.pngimage24.png2

= r2 (image25.pngimage25.png +image26.pngimage26.png + image22.pngimage22.pngimage27.pngimage27.png-image28.pngimage28.png)

= r2 (image29.pngimage29.pngimage30.pngimage30.pngimage31.pngimage31.png)

                ≈ 0.995r2

To test this formula, the radius 6cm can be substituted into r, which allows the answer calculated by this formula to be compared with that of question 5a.

0.995(6)2 = 35.82 cm

Therefore, the suggested formula has been supported.

  1. Much like section A of this question, finding the perimeter of the egg consists of substituting the calculated figures in the previous questions into their relationships in terms of r. Therefore, a formula for the perimeter of Moss’s Egg can be calculated using the method displayed below.

C semicircle + Arc AD + Arc BE + Arc DE

                = image32.pngimage32.pngimage33.pngimage33.png2image34.pngimage34.png + image35.pngimage35.png + image37.pngimage37.png(image38.pngimage38.png)

                = image39.pngimage39.png x image40.pngimage40.png

                = image41.pngimage41.png                                = image42.pngimage42.png

                ≈ 3.6r

To test this formula, substitute r to 6 and compare with answer calculated from question 5b.

3.6 x 6 = 21.6 cm

Therefore the formula above has been supported.


Haese, S et al. 2006, Mathematics For The International Student. Haese & Harris         Publications, Australia

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. El copo de nieve de Koch

    iteraci�n, con el primer sistema surgi� que con cada iteraci�n el per�metro aumentaba , y conociendo que la fase inicial ten�a 3 lados fue como a partir de un desarrollo surgi� el siguiente modelo matem�tico : Con la aplicaci�n de este modelo obtuvimos los siguientes resultados: Que al comparar los

  2. Math SL Circle Portfolio. The aim of this task is to investigate positions ...

    still worked when r=1, = 10, . In the following graph, r=1, = 10, . The following graph is a close up of the graph above, showing the value of = = 0.1 A greater length of, such as 75 was calculated, and it still showed the result derived from the general statement, , (n= ).

  1. In this task, we are going to show how any two vectors are at ...

    To determine the parametric equation for this vector, we split the vector equation into the x and y components. and By knowing the value of t, this can be substituted into the 2 parametric equation and the coordinate for (x, y)

  2. Stellar Numbers. In this task geometric shapes which lead to special numbers ...

    I.e.: 6Sn = 6Sn-1 + 12n To prove my equation I will use an existing example from above and then prove it with one stage further. For Stage 3: 6Sn = 6Sn-1 + 12n 6S4= 6S3 + (12x3) = 37 + 36 =73 For stage 8: 6Sn = 6Sn-1 + 12n 6S8 = 6S7 + (12x7)

  1. The segments of a polygon

    I do again same procedure as in first task and program gives me relation: (c) Prove the conjecture. To prove the conjecture we have to calculate the side of the inner square . For calculations I will use the first square (1:2).

  2. Math IA Type 1 Circles. The aim of this task is to investigate ...

    In this case, r is the constant, OP is the independent variable and OP? is the dependent variable. The Cosine and Sine Laws will be used to find OP?. Cosine Law: c2 = a2 + b2 ? 2abcosC Sine Law: = The Cosine Law will be used to find ?.

  1. Investigating Slopes Assessment

    To help me discover this, I will utilise, again three cases that will give me the possibility to work out the link of this function. Since this time we have two factors that are not constant: ?a? and ?n?, for every case, I will need to also find out three different answers.

  2. MATH IA: investigate the position of points in intersecting circles

    When OP = 2 First a graph is drawn and AP? is linked by a line and point G is introduced to make line AG which is perpendicular to OP. AP?=OA ?OAP is an isosceles triangle. AG is perpendicular to OP so as a result of this, AOP=AP?O.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work