Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 7 Geometry
Problem
Solution
1) The surface area of a rectangular prism box is 288 square inches. Another box that has twice the width and twice the height has surface area 672 square inches. What is area of one side panel of the smaller box given by the product of its width and height? Use L for length, W for width and H for height of the box.
  1. The surface area of the small box is:
        2LW + 2LH + 2WH = 288 sq. in., so:
        LW + LH + WH = 144 sq. in.
  2. The surface area of the large box
    (using twice the width and height) is:
        2L(2W) + 2L(2H) + 2(2W)(2H) = 672 sq. in.
        4LW + 4LH + 8WH = 672, so, dividing through by 4:
        LW + LH + 2WH = 168 sq. in.
  3. Substituting the first expression into the second:
        144 + WH = 168
        Therefore, WH = 168 - 144 = 24 sq. in.
2) The circle shown has center point P and radius 5 units. What is the area of the shaded portion of the circle? Give your answer as an exact answer or round to the nearest hundredth.
You may use the approximation = 3.142
  1. The area of the circle is 25
  2. The angle at the center of the white sector is
    90 + 40 = 130 degrees.
  3. The shaded area is (360-130)/360 of the circle's area =
        (230/360) x 25 =
        (23/36) x 25 =     575 / 36 sq. units
3) The radius of all three semicircles is 1 unit. The height of the unshaded triangle is 3 units. What is the area of the shaded trapezoidal portion of the figure?
  1. The width of the big triangle is 4 units
  2. The height of the big triangle is also 4 units
  3. The larger and smaller triangles are similar so the width of the smaller triangle is the same as it's height which is 3 units.
  4. From here you can solve it by 2 different methods:
    1. Method 1: Area subtraction:
      Shaded area = large triangle area - small triangle area =
      8 sq. units - 4.5 sq. units = 3.5 sq. units
    2. Method 2: Trapezoid formula:
      Shaded area = (b1 + b2) h / 2 = (4+3)/2 = 3.5 sq. units

Problem
Solution
4) The coordinates of three of the vertices of a rhombus located in the first quadrant are: (2,5), (4,7) and (1,8). What are the coordinates of the fourth point? Use the coordinate grid to the right. Two rhombuses can be created from these 3 points. Actually, 3 can be created, but the 3rd one (-1,6) requires a point in Quadrant II.
  1. The given points are plotted on the graph.
  2. A rhombus can be made by creating a 4th point off point 3 by taking the offset of point 2 from point 1 (+2,+2) and applying it to point 3: (+3,+10)
  3. A rhombus can be also made by creating a 4th point off point 1 by taking the offset of point 2 from point 3 (+3,-1) and applying it to point 1: (5,4)
5) The figure shows a stained glass window design. The white triangles are congruent and equilateral and have side length 5 inches. The two triangles in the shaded region are isosceles with side lengths shown, and the central region is a rectangle. What is the area of the entire shaded region?
  1. The area of the rectangle is
    8x5 = 40 sq. in.
  2. If you make a right triangle by cutting one of the isosceles triangles in half, making a base of 4 and a hypotenuse of 5, the height becomes 3 because 32 + 42 = 52
  3. The area of one of the isosceles triangles is therefore 8x3/2 = 12 sq. in.
  4. The total shaded area is, therefore, 12 + 12 + 40 = 64 sq. in.