Mathematical Olympiads for Elementary and Middle Schools
AREA – NO PROBLEM?
Dennis Mulhearn
1. A square has an area of 144 square inches. Suppose the square is partitioned into six
congruent rectangles as shown at the right. How many inches are there in the perimeter
of one of the six rectangles?
2. Square ABCD and rectangle AEFG each have an area of 36 square meters. E
is the midpoint of AB . What is the perimeter of rectangle AEFG?
3. As shown, a square of area 100 sq cm is split into four separate smaller regions: A, B, C,
and D. Regions B and C are squares. Find the sum of the perimeters of regions A, B, C,
and D, in cm.
4. ABCD is a rectangle with area equal to 36 square units. Points E, F, and G are
midpoints of the sides on which they are located. How many square units are
there in the area of triangle EFG?
5. ABCD is a square with area 16 sq. meters. E and F are midpoints of sides AB and
BC, respectively. What is the area of trapezoid AEFC, the shaded region?
6. ABCD is a rectangle as shown. E lies on side AD and <BEC is a right angle of
triangle BEC. CE = 6, BE = 8, and BC = 10. Find the area of rectangle ABCD.
Copyright © 2012 by Mathematical Olympiads for Elementary and Middle Schools, Inc. All rights reserved.
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7. As shown, the length of each segment in the overlapping rectangles is given, in
cm. Find the sum of the areas of the shaded regions, in sq cm.
8. A rectangle is divided into four smaller rectangles whose areas in sq cm are 35, 42,
10, and N, as shown. The length of each side of every rectangle is a whole number.
What is the value of N, in sq cm?
9. Two congruent isosceles triangles, ABC and DEF, overlap so that their bases are
parallel and the vertex of each is the midpoint of the base of the other as shown. If the
area of the overlap is 12 sq cm, how many sq cm are in the area of triangle ABC?
10. As shown, ABCD and AFED are squares with a common side AD of
length 10 cm. Arc BD and arc DF are quarter-circles. How many
square cm. are in the area of the shaded region?
11. Two congruent circles and two parallel lines intersect in a total of
five points as shown. The inner segment of each line is 6 cm long.
Using   3.14 ,find the area of the shaded region to the nearest
sq cm.
12. A square with an area of 18 square cm is inscribed in a circle as shown. Using the
approximation   3 .14 , find the area of the shaded region to the nearest sq cm.
Copyright © 2012 by Mathematical Olympiads for Elementary and Middle Schools, Inc. All rights reserved.
www.moems.org
Toll-free phone (866) 781-2411
[email protected]
Mathematical Olympiads for Elementary and Middle Schools
AREA – NO PROBLEM?
***BONUS PROBLEMS***
1. The perimeter of a rectangle is 22 inches and the inch-measure of each side is a counting number. How
many different areas in square inches can the rectangle have?
2. Each small region in the figure shown is a square. The area of the entire
figure is 320 sq cm. What is the number of cm in the perimeter of the
entire figure?
3. In rectangle ABCD, AC , BD , and FG all intersect at E. AD is 12 cm
long and AB is 18 cm long. What is the total area of the shaded regions in
sq cm?
4. ABCD is a rectangle whose area is 12 square units. How many square units
are contained in the area of trapezoid EFBA?
5. A walkway (the shaded region) is built around a rectangular pool, as shown. The
pool is 20 feet by 30 feet. The walkway is completely tiled with whole square tiles
2 feet by 2 feet. What is the fewest number of tiles that can be used?
6. Two semicircles are inscribed in a square with side 8 m, as shown. Approximate
the area of the shaded region to the nearest tenth of a sq cm. Use the
approximation   3.14
Copyright © 2012 by Mathematical Olympiads for Elementary and Middle Schools, Inc. All rights reserved.
www.moems.org
Toll-free phone (866) 781-2411
[email protected]