Geometry Forum - Problem of the Week Solutions - Inscribed Tombstone, May 30-June 3, 1994 Tony Aiello ____________________ When the area of the sguare is given as four the area of the triangle is 2 and the area of the circle is 3.14. To find the area of the triangle you take the side of the square, which is 2, as the base, and that is also the height. The equation to find the area is 1/2 the base times the height. (1/2 x 2 x 2). To get the area of the circle you take pi r squared. r=1 When the area of the square is x squared you use the same equations that you did before. One side of the square is x; you just put that in where you used 2 before. For the circle the radius is 1/2 x. The area of the triangle is 1/2 x squared. The area of the circle is 1/2 x squared times pi. Ian Ross ____________________ When the area of the square is 4 units sq., then the area of the circle will be 3.14 units sq. and the area of the triangle is 2 units sq. When the area of the square is x^2, then the area of the triangle will be .5x^2 units sq. and the area of the circle will be .5x^2 pi. Brandon Verdream ____________________ If the area of the square is 4 units sq. then the area of the circle is 3.14 units sq. and the area of the triangle is 2 units sq. If the area of the square is x^2 then the area of the circle is .5x^2 pi and the area of the triangle is .5x^2. Paul Curcio ____________________ The ratio of square:circle:triangle is 4:pi:2, in the unit circle. This is because the radius of the circle is equal to half a side (2) of the square. The triangle follows, as its base and height are equal to sides of the square. Mark Berneburg ____________________ I first constructed the square and then put in the circle. I then realized that when you take the midpoint of the square and use that for one vertex of your triangle with the other two being on the square's base, the triangle must be an isosceles triangle. I then put in 4 as the area of the square, which means that each side is 2 so the base of the triangle is also two and the diameter of the circle is also 2. Then I used the formulas I know for triangles and circles and found their area. The area of the triangle was 1/2bh so since 2 is the base and the height is two the area is 2. The area of the circle is pi times the radius squared. I then realized that the radius is half of the diameter so the area of the circle is pi. If the square's area is x^2 then the sides are x. In the triangle again the side and base are the same as the square so they are x also. The formula you could use would be (1/2)x^2. The circle would have diameter x so the radius would be x/2 so the formula would be pi times (x/2)^2. This would work for the given scenario. Chris Taormina ____________________ All of the areas are related. Since the circle is inscribed in the square its diameter would be the same as a side of the square. So the area would be one half the side of the square squared multiplied by pi. So if the area of the square is x then the area of the circle is pi*(1/2*the sqr root of x)^2. With the triangle it is much easier, because the base and the height are the sides of the triangle. Since the area of a triangle is 1/2*b*h then the are of the triangle is just half the area of the square. With the area of the square being 4, the area of the triangle would be 2, and the area of the circle would be 3.14 or pi. This is figured out by the above paragraph. With area of the square being x^2, the area of the triangle would be (1/2)*x^2, and the area of the circle would be 3.14((x^2)/4). This is also figured out by the above paragraph. Dan Di Girolamo ____________________ For the Problem of the Week for May 30 to June 3 I drew the picture to find a solution. First I used your hint of making the square have an area of four. The formula for finding the area of a square is length*width or X^2. The area of the square is 4. That makes x=2. Then I drew in the diameter of the circle which I also made the altitude of the triangle. Then I found that the altitude/diameter is equal to each side of the square. Therefore, the area is equal to 2 by the equation 1/2x^2. Finally, I found the circle's area to be 3.14 by the equation of 1/2x^2pie. Therefore the area of the square is twice as big as the triangle's area, and the circle's area is 1.27 times smaller than the square. Hannah Guhm ____________________ The ratio of the areas is 4:*:2 in order of square, circle,triangle. Percy Rosario ____________________ Areas are x^2, 1/2x^2 and 1/4*x^2 when side of square is x. The ratio of the areas is 4:*:2 in order of square, circle, triangle. Patrick McGinley ____________________ The areas are in the ratio: square (4): circle (*): triangle (2). Karen Brown ____________________ If side of square is 10, areas are: square is 100; circle is 25*; and triangle is 50. So areas are in the ratio 4:*:2 with square largest and triangle smallest. Susan Quan ____________________ If the area of the square is x^2 then the area of the circle is 1/4*x^2 and the area of the triangle is 1/2 x^2. Therefore the ratio of the areas is:4:*:2. The square is the largest, then the circle and the area of the triangle is the smallest. Keith Monteleone ____________________ Using the pythagorean theorem you will obtain these measurements The area of the square is 4 The area of the triangle is 2 The area of the circle is 3.14 Using the pathagorean theorem you will find that The area of the square is x^2 The area of the triangle is 1/2x^2 The area of the circle is 1/4pi (x^2) Jenn Strong ____________________ The area of the square is always the square of one of the sides. In this case, x^2. If we take this number to be 4, then x=2. The area of the triangle would then be (1/2 b h = A) .5 times 2 times 2= 2 units^2 for the area of the triangle. Since the height of the square is 2, the radius of the circle is 1. pi r^2 = A of circle. this means that the area of the circle is pi. Taking this into the general form, where one of the sides of the square is x, the square's area is x^2. Using proportions (or using the same method as above) one finds: area square/area triangle =area square/area triangle 4 / 2 = x^2 / ? here ?= x^2 by solving the proportion. the same type of proportion can be set up for the area of the circle. 4 / pi = x^2 / ? ?= (pi times x^2)/4 (both the triangle and circle where x = one side of the square) As mentioned before, this can also be found be 1/2 base times height. for the triangle equals .5 x times x or one half x^2 or x^2/ 2. Also, 1/2 x = the radius of the circle. pi r^2 So, pi (.5 x) ^2 This means that (pi times x^2)/ 4. These results confirm the results from the proportions. Rebecca Naughton ____________________ First, one must construct a square. To do that, one has to use a given length for all sides. Make four right angles by constructing perpendiculars. To inscribe a circle inside the square, one must find the incenter. You can do this by finding the intersection of all the angle bisectors. From the incenter to the perpendicular bisector of one of the sides is the radius. To find the midpoint of the top edge, you must find the perpendicular bisector. To construct the triangle, one must connect the point just constructed to the two endpoints of the base of the square. If the area of the square is 4, then the area of the triangle is 2. I know this because the formula for the square is A=bh. In the triangle, the formula is A=.5bh=.5(4)=2. The area of the circle would be pi. I know this because the square has all sides congruent, so the base and height must be the same. The only positive number that when multiplied by itself equals 4 is 2. The center point of the circle is the incenter so it is equidistant from all sides. Therefore, the radius would have to be 1. The formula for circles is A=pi*r^2=pi*1^2=pi. If the area of the square is x^2. then the area of the triangle would be .5x^2, and the area of the circle would be .25*pi*x^2 or A=pi(.5x)^2. Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page