Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 6 Geometry
Problem
Solution
1) Lamar wants a circular pool in his back yard. He maps out the circle and decides it is far too small. If Lamar increases the diameter by 6 feet, it doubles the circumference. What is the circumference of the bigger circle?
You may leave your answer in terms of
or use = 3.14 and round to the nearest tenth. 		1. If you increase the circumference of a circle by a factor of X, then the diameter is increased by that same factor.
2. Compute the diameter of the original pool = 6 ft.
3. Compute the diameter of the larger pool = 12 ft.
4. Compute the circumference of the larger pool =
    12 or 37.7 sq. ft

2) Some of the vertices of a polygon are located at the coordinates
(-2, 1), (2, 2), and (4, 4). The polygon has a vertical line of symmetry at x = 4 which meets the opposite edge at the midpoint (4, 1). Name the polygon that has the fewest number of vertices satisfying these conditions, and give the coordinates for one its other vertices.
 First, this is not a regular polygon.
1. The fact that the line of symmetry goes through one of the points means the polygon has an odd number of points (the one on the line is not replicated, but the other 2 are). This means the polygon has at least 5 sides and is a pentagon.
2. From the line of symmetry, find the two points that mirror-image the other 2 given points.
They are (10,1) and (6,2).
The figure to the left shows this pentagon and its line of symmetry.

Problem
Solution
3) In the pyramid of cubes shown, what is the minimum number of cubes that would need to be added to complete it to a solid cube? 1. The bottom layer has 5 cubes along a side and so is the smallest dimension to make this a solid cube.
2. Fill in each layer and count them up.
    bottom: add 0 cubes
    layer 2: add 9
    layer 3: add 16
    layer 4: add 21
    layer 5: add 24
The total number of added cubes is 70.
4) Write the letters M, T, and H on figure 2 on the answer sheet so that when the figure is folded into a box, it will spell MATH around the sides of the box. Figure 1 shows the correct placement of the letters.
 1. The placement of M and T are pretty obvious. The question is where and in what orientation does the H go?
2. The top and the bottom of the cube are the squares above and below the T.
3. Draw in the H. Be careful with it's orientation.
This is the solution:
5) An animator creates the idea of a rolling eye by having a circle roll around the inside of a larger circle. If the radius of the larger circle is 2 cm and the radius of the small circle is 3/4 cm, what is the area of the large circle that is never touched by the small circle?
You may leave your answer in terms of
or use = 3.14. 		1. As the small circle rolls around the edge of the large circle, the only area that is not touched is the area around the center of the large circle.
2. Subtract the diameter of the small circle from the radius of the large circle:
    2 - 1.5 = .5 cm
3. Compute the area of a circle with that radius = 1/4 or .785 sq. cm.