Use area formulas of circles and squares to solve problems
Dan’s badge problem can be solved in a number of ways. They all involve working with squares and circles to corner the required area. Hence they all require that students know the area formula for the square and the circle.
Area is an important quantity in mathematics. There are obvious applications of plane figures such as rectangles to fields and house sections. In fact the whole business of geometry may well have started in Egypt when fields had to be surveyed after the annual flood in the Nile Delta. Newton and Leibnitz, of course took the whole thing further when they introduced the integral calculus and so opened up a new range of areas to calculation. They also opened up areas other than those of plane figures that were of great use.
Students in the last two years of high school will come across the integral calculus and its applications.
A simpler version of this problem can be found by checking Bill’s Badge, Measurement, Level 5.
All the students in the class decided to make a badge. Dan did his with ruler and compass and coloured it red and white. His drawing is below. The centres of the circles lie at the corners of a square.
Construct your own version of Dan’s Badge.
Dan used a radius of 10cm for his circles. What area of his badge was shaded red?
- Plan a way to interest the students in the problem. One way is to look at the designs on some "real" badges. Ask the students to identify the shapes in the badges and to think about how they would determine the areas used of different colours.
- Pose the problem to the class. Ask them to think first about the construction.
- Give the students time to think about the problem before asking for their ideas about how to reconstruct Dan’s badge. List suggestions on the board.
- Ask the students to reconstruct Dans’ badge.
- As the students work on the construction ask questions that focus on the measurements used and the relationships that they can see.
- Share constructions and approaches used.
- Pose the second part of the problem. Remind the students to think about how they constructed the badge and how this will help in thinking about the areas.
- As the students work (in pairs) ask questions that focus on the known measurements in the problem and the relationship between these and the required area.
- Ask the students to record their solution for others to read.
- Share solutions.
Construction: First construct a square. (There is no need to be too precise on the dimensions in this problem.) This can be done by marking off two points on a line and drawing perpendicular lines from these two points. Use dividers to measure the length between the two original points and mark this length off on each of the two perpendicular lines. Then join the two points so produced.
We now know where the centres of the circles are. To draw in the circles we need to know the radius on the diagram. We note that the four circles meet at the centre of the square. So draw in the two diagonals of the square. The radius of a circle is the distance between the centre of a circle and the point where the diagonals meet. Now the circles can be drawn to complete the diagram.
Area: There are many ways to find the area of the red part of Dan’s Badge. One way is to draw in the square shown in the diagram. To check that this is indeed a square, note that the side AX is half of the diagonal of the square that Dan started with. So this is at 45o to the side AB. AB goes symmetrically through the middle of the red part and so the angle of XAY = 90o.
The square has area 10 x 10 = 100 cm2 because the length of the side is the radius of the circle which is 10 cm.
Now the sectors AXB and AYB are each 1/4 area of the circle = 1/4 π 102 = 25π cm2. Hence the area not dotted is 100 - 25π .
The area of the red part in the diagram is therefore = area square – 2 x area not dotted
= 100 – 2(100 - 25π )
= 50π - 100.
The red part of Dan’s Badge is therefore 4(50π - 100) = 200π - 400 = 228.32 cm2.