Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 7 Geometry
Problem
Solution

1) How many cubes with a 3 inch side would fit in a box with dimensions 12 inches by 15 inches by 9 inches?
  1. The volume of the box is 12x15x9 inches = 1620 cu. in.
  2. The volume of the 3-inch cube is 3x3x3 = 27 cu. in.
  3. 1620/27 = 60 cubes
2) What is the measure of angle x? (not drawn to scale).
 Please note: The arrows on the 2 horizontal lines indicates that they are parallel. A little tricky!
The applicable missing angles have been labeled a,b,c and d on the diagram.
  1. Angle a is 180 - 90 - 59 = 31 degrees
  2. The arrow lines are parallel, therefore
    angle d = angle a = 31 degrees
  3. Angle c is vertical with the 71 degree angle and is therefore 71 degrees
  4. Angle x, d, and c sum to 180 degrees, so
        x = 180 - 31 - 71 = 78 degrees

3) ABCD is a square with side length 20 inches. A, B, C, and D are all the centers of 4 congruent circles that touch each other as shown in the picture. Find the area of the shaded region. Round your answer to the nearest whole number.
  1. The area of the square is 20x20 = 400 sq. in.
  2. The 4 quarter circles sum to a full circle of radius 10 in, so the circle's area is 102 = 314 sq. in. (rounded)
  3. The shaded region area is the difference between these 2 areas = 400 - 314 = 86 sq. in.

Problem
Solution
4) In triangle ABC, Angle A's measure is twice angle B's measure. Angle C's measure is 41⁄2 times A's measure. What is the measure of angle A?
  1. The total measure of all angles of the triangle are:
      A + A/2 + 41⁄2(A) = 6A
  2. Since these angles add to 180, then:
      6A = 180
      A = 30 degrees
5) Which shape has the greatest volume? (not drawn to scale)
  1. Volume of a triangular prism =
        (9 x 3/2) x 7 = 94.5 cu. m.
  2. Volume of the rectangular prism =
        6 x 2.5 x 6.2 = 93 cu. m.
  3. Volume of the cylinder =
        x (3.5)2 x 4 = 153.86 cu. m.
  4. Therefore the cylinder has the greatest volume.