ID 922

Geometry and  Measurement K12A fence around a rectangular garden has a perimeter of 14 meters.
Its length is 4 meters.

Find the length of the diagonal of this rectangular garden.



ID 924

Geometry and  Measurement K12The volume of water in a glass of V cubic centimeters (cc), varies directly as its height, H centimeters (cm).
The volume is 100 cc when the height is 5 cm.

Calculate the volume when the height is 11 cm.



ID 925

Geometry and  Measurement K12Tim travels 15 meters north of the flag in the football stadium.
He turns west and travels 8 meters.
He turns south and travels 6 meters and then comes back directly to the flag.

Calculate Tim's travel distance.



ID 927

Geometry and  Measurement K12S varies directly as the square of (H+1)
S = 121 when H = 10

Find H when S = 225



ID 930

Geometry and  Measurement K12A rectangle has width W, length L, and area A.

If integer W ≤ 9 and L is an even number between 9 and 99 which of the following CANNOT be the area of the rectangle?



ID 931

Geometry and  Measurement K12A wire is wound around a circular rod.
The wire goes exactly 5 times around the rod.
The circumference of the rod is 12 cm and its length is 25 cm.

Identify the length of the wire.



ID 932

Geometry and  Measurement K12Which is the best estimate of the area of the colored shape?



ID 934

Geometry and  Measurement K12These two shapes are similar.

What is the length of x?



ID 936

Geometry and  Measurement K12Identify the area of the rounded square.



ID 938

Geometry and  Measurement K12A boy stacked colored cubes in a square pyramid like the one shown here.
The top layer had 1 cube, the second layer had 4 cubes, and so on.

If the pyramid were 15 layers high, how many cubes would be in the fifteenth layer?



ID 940

Geometry and  Measurement K12A six-pointed regular star consists of two areas.

What is the ratio of the shaded region in the area of the star?



ID 942

Geometry and  Measurement K12The figure shows a red equilateral triangle inscribed within another equilateral triangle. The side of the bigger triangle measures 10 meters.

We want to obtain the smallest area of the red triangle. What would be the distance x in this case?



ID 947

Geometry and  Measurement K12If the volume of a cube is X cubic meters and the total surface area of the cube is X square meters, then what is the cube's edge length?



ID 948

Geometry and  Measurement K12The side of the colored square is x cm.

What is the area of the red rectangle in terms of x?



ID 949

Geometry and  Measurement K12The three circles have fixed centers, and the diameter of a circle is 10% less than that of its 'left neighbor'.

The left circle completes a hundred revolutions per minute.

How many revolutions does the right circle complete?



ID 956

Geometry and  Measurement K12L3 and L4 are two parallel lines in a plane.

If L3 has 3 points equally spaced along its length, and if L4 has 4 points also equally spaced along its length, how many different triangles can be formed by connecting the points on the two parallel lines?

A triangle must be formed by 2 points on one line and 1 point on the other.



ID 957

Geometry and  Measurement K12What is the total number of lines of tangency that are common to circle C1 and circle C2?



ID 958

Geometry and  Measurement K12For this rectangular solid, which plane(s) contain(s) B and is/are parallel to plane AEH?



ID 960

Geometry and  Measurement K12Two similar pyramids have volumes of 216 m3 and 64 m3.

What is the ratio of their surface areas?



ID 961

Geometry and  Measurement K12AB is parallel to CD.

What is the value of X?



ID 964

Geometry and  Measurement K12There are 2 identical circles. Circle A remains fixed while, Circle B makes 1 turn around the first one, touching it without slipping.

How many turns has Circle B made around its own axis?



ID 968

Geometry and  Measurement K12G is the area of the green region inside the biggest circle.
R is the total area of red regions of the two smaller circles.

Which statement is correct?



ID 969

Geometry and  Measurement K12The area of the external triangle is equal to 1.
Its sides' midpoints are connected to form a second triangle, and so forth.

What is the sum of the areas of all the triangles in this infinite series?



ID 971

Geometry and  Measurement K12The pattern of shading in one quarter of a square is shown in the diagram.

If this pattern is continued indefinitely, what fraction of the large square will eventually be shaded?



ID 974

Geometry and  Measurement K12How many squares can you make using twelve identical matches without crossing?



ID 976

Geometry and  Measurement K12A gardener has to reach the island in the middle of a pond without getting wet.
The gardener has two planks each X meters long.

What is the smallest length of the planks?



ID 979

Geometry and  Measurement K12Three reservoirs have the same height h=1.

Which reservoir is bigger?



ID 980

Geometry and  Measurement K12What is the largest semi-circle that can be inscribed in a square with a side of length a=2?



ID 1004

Geometry and  Measurement K12Twenty matchsticks form five squares (one 3x3 and four 1x1).

How many matchsticks do I need to move to make seven squares?

Find the minimum number.



ID 1041

Geometry and  Measurement K12A ladder leans against a vertical wall. The top of the ladder is 4m above the ground. When the bottom of the ladder is moved 1m closer to the wall, the top of the ladder rests 1m higher than the original position.

How far from the wall was the bottom of the ladder in the initial position?



ID 1043

Geometry and  Measurement K12If the length of the hour and minute hand of a clock are 3cm and 6cm respectively, what is the angle shown on the picture at two o'clock?



ID 1050

Geometry and  Measurement K12All inner lines connect the corners of the big square and the midpoints of the opposite sides.

What fraction of the big square is red?



ID 1054

Geometry and  Measurement K12A square is inscribed in a right triangle.

Find the greatest possible ratio of the area of the square to the area of the triangle.



ID 1058

Geometry and  Measurement K12What value of X gives the greatest ratio of red area to the area of the big rectangle?



ID 1076

Geometry and  Measurement K12Among the following shapes of equal area, which has the smallest perimeter?



ID 1083

Geometry and  Measurement K12The sizes of the sealed bottle with water are shown in the figure.
Find the height of the water when the bottle is right side up.



ID 1086

Geometry and  Measurement K12How many equilateral triangles can you make using six identical line segments?



ID 1109

Geometry and  Measurement K12The following pattern is cut and folded to a square-based pyramid.

What size does the base have to be to maximize the surface area of the resulting pyramid?



ID 1190

Geometry and  Measurement K12John is using the spinner shown here to define the direction and how many steps to move.
Blue means one step up and green means one step down.
The spinner is moved randomly.

If he starts at point 0 and makes 360 moves, where will he most likely be now?



ID 1192

Geometry and  Measurement K12How much wrapping paper is needed to entirely cover a box that is 40 cm by 30 cm by 12 cm?



ID 1264

Geometry and  Measurement K12What is the ratio of the green area to the red area?



ID 1280

Geometry and  Measurement K12Four squares have dimensions as indicated in the picture.

What is the area of the green shape?



ID 1340

Geometry and  Measurement K12What is the maximum number of possible points of intersection of N different circles?

The picture shows four circles.



ID 1348

Geometry and  Measurement K12The diameter of one hair is 0.2 mm.

Estimate the diameter of a plait of 1000 hairs.



ID 1399

Geometry and  Measurement K12Two lines trisect (divide into three equal parts) each side of the polygon ABCD.

Which polygon has the largest area?



ID 1451

Geometry and  Measurement K12The shaded rhombus is formed by joining vertices of the square to the midpoints of its sides.

What is the area of the shaded rhombus?



ID 1479

Geometry and  Measurement K12The diagram shows 15 billiard balls that fit exactly inside a triangular rack.
The rigid rack prevents the balls from sliding.

What is the largest number of balls that can be removed so that the remaining balls are theoretically unable to move?



ID 1518

Geometry and  Measurement K12The picture shows the areas of three rectangles.

What is the area of the fourth rectangle?



ID 1587

Geometry and  Measurement K12Two triangles form seven separate regions.

What is the greatest number of such regions that can be formed by three triangles?



ID 1613

Geometry and  Measurement K12There are six ways to travel from point S to point F on a small cube if only right, forward, and up moves are permitted.

Find the number of different pathways available for a 2x2x2 cube.



ID 1614

Geometry and  Measurement K12Estimate the area of the smallest square that can enclose the regular hexagon shown on the right.



ID 1876

Geometry and  Measurement K12A, B, and C are squares with sides of length 1;
D, E, and F are isosceles right triangles;
and G is an equilateral triangle.
The net can be folded to form a shape.

What is the volume of the shape?



ID 1878

Geometry and  Measurement K12I drew three lines from the center of a square that has sides with a length of 1 to form two congruent trapezoids and a pentagon.
All three shapes have the same area.

What is the length of the pentagon's shortest side?



ID 1949

Geometry and  Measurement K12What is the maximum number of pieces that an apple can be divided into with four straight planar cuts?

The pieces do not move.



ID 1974

Geometry and  Measurement K12Each of these five circles is tangent to at least 3 others.
The medium sized circles have a radius 3.

What is the radius of the smallest green circles?



ID 2031

Geometry and  Measurement K12Find the radius of the circle.



ID 2153

Geometry and  Measurement K12In a city, there were seven bridges.
There was a tradition that a newly married couple walks and crosses over each of the seven bridges only once.
If a couple starts and finishes at the same point, which city plan allows the couple to acomplish this task?



ID 2193

Geometry and  Measurement K12A sphere fits inside a cube. The maximum possible ratio of the volume of the sphere to that of the cube is pi / 6, or about 0.52.

If we put many small spheres inside a cube, then what is the largest possible ratio of the spheres' volume to that of the cube?



ID 2198

Geometry and  Measurement K12I want to cut a wooden cube that is five inches on each side into 125 one-inch cubes.

I can do this by making 4 + 4 + 4 = 12 cuts, keeping the pieces together in the cube shape.

What is the minimum number of cuts needed if rearrangement of the pieces after each cut is allowed?



ID 2238

Geometry and  Measurement K12What line divides the green polygon into two parts of equal area?



ID 3509

Geometry and  Measurement K12I need 1 + 4 + 9 + 16 + 25 = 55 cubes to build a pyramid with a height of 5 cubes.

Estimate the number of cubes for a pyramid with a height of 30 cubes.



ID 3552

Geometry and  Measurement K12I want to place N cubes so that each cube touches every other one.

What is the largest possible N?

Inspired by Martin Gardner



ID 3564

Geometry and  Measurement K12What is the volume of the second glass compared to the first one?



ID 3599

Geometry and  Measurement K12The Babylonians used a base 60 number system.

What shape inspired them to decide that a circle has 360 degrees?



ID 3608

Geometry and  Measurement K12Connect N points on the circumference of a circle.

What is the largest number of intersections for the chords?



ID 3674

Geometry and  Measurement K12I take a map of the city where I live and lay it on my table.

There is a "You are here" point on the map, which represents the same point in the city.
The point on the map coincides with its real position.

True or False?



ID 3678

Geometry and  Measurement K12What is the largest number of intersections of N lines?



ID 3696

Geometry and  Measurement K12I connected the midpoints of a polygon and constructed a new polygon that was a quadrilateral with opposite sides parallel.

What shape was the initial polygon?



ID 3712

Geometry and  Measurement K12A pyramid and a tetrahedron with edges of the same length are glued together on a triangular face.

How many faces does the resulting solid have?



ID 3718

Geometry and  Measurement K12I need 1 + 9 + 25 = 35 cubes to build a pyramid with a height of 3 cubes.

Estimate the number of cubes for a pyramid with a height of 30 cubes.



ID 3741

Geometry and  Measurement K12Inspired by Boris Kordemsky.

Four Knights problem:

Cut the chessboard into 4 congruent parts, each with a queen on it.

How many sides does a shape has?



ID 3806

Geometry and  Measurement K12Which shape is more symmetrical (i.e. has more planes of symmetries)?



ID 3909

Geometry and  Measurement K12How many 1x1 squares fit into the large square with the side length 4.8?

Find the maximum possible number.



ID 3916

Geometry and  Measurement K12What is the greatest possible number of points of intersection for 11 lines in a plane?



ID 3917

Geometry and  Measurement K12The picture shows three squares with the side lengths of 10, 8, and 4 cm.
What is the difference between the areas of the green and blue regions?



ID 3920

Geometry and  Measurement K12A trapezoid is formed by cutting off the top part of an isosceles right triangle such that the short base is 8 m.

What is the area of the blue trapezoid?



ID 3924

Geometry and  Measurement K12If regular stars are placed as shown to form a complete circle, how many stars are needed?



ID 3941

Geometry and  Measurement K12What fraction of this regular octagon is shaded?



ID 4023

Geometry and  Measurement K12The stripes have the same width.

What part of the square is red?



ID 4064

Geometry and  Measurement K12A circle goes through two adjacent vertices of a square and it is tangent to the bottom side of the square.

Find the diameter of the circle X.



ID 4129

Geometry and  Measurement K12The figure shows adjacent squares.
Find the sum of angles X and Y.



ID 4219

Geometry and  Measurement K12A helicopter takes me from Lausanne, Switzerland to the Swiss capital Bern in 20 minutes. Bern is 36 minutes from Brig.

Which of the following could be the time of a flight from Lausanne to Brig?



ID 4324

Geometry and  Measurement K12What is the probability of breaking a stick into three pieces and forming a triangle?

The pieces must intersect at their tips to form the triangle.



ID 4342

Geometry and  Measurement K12What is the maximum number of trees that can be planted, not closer than 3 meters apart, in a square plot of 10.5 meters x 10.5 meters?



ID 4373

Geometry and  Measurement K12A rectangular yard with an area of 24 m2 has sides in the ratio 2:3.

What is the length of the fence around the yard?



ID 4382

Geometry and  Measurement K12Which triangle has the greatest area?



ID 4386

Geometry and  Measurement K12Which is a better fit?

The smaller the percentage of the ‘area left over,’ the better the fit.



ID 4388

Geometry and  Measurement K12Two perpendicular lines form four segments with lengths a, b, c, and d.

Which is correct?



ID 4447

Geometry and  Measurement K12The size of an NBA basketball court is about 29 by 16 meters.

How many courts can be planned in an 80 x 67 meter school yard, given that at least 3 meters must separate the courts?



ID 4458

Geometry and  Measurement K12How many pentagons are there?



ID 4483

Geometry and  Measurement K12The golden border of the hexagonal brooch with maximum size 60 mm includes a gemstone, whose maximum size is 40 mm.

Which area is the largest?



ID 4542

Geometry and  Measurement K12What is the maximum number of sections into which a pancake may be divided by four straight cuts through it?

(NOTE: The pieces cannot be re-arranged between cuts.)



ID 4560

Geometry and  Measurement K12If I use 30 g of batter to make a crêpe of 30 cm in diameter, how much batter do I need for a square pancake of the same thickness and with a side length of 30 cm?

"Archaeological evidence suggests that pancakes are probably the earliest and most widespread cereal food eaten in prehistoric societies." - Wikipedia



ID 4657

Geometry and  Measurement K12The numbers show the areas of corresponding triangles.

What is the area of the rectangle?



ID 4759

Geometry and  Measurement K12What is the perimeter of a right triangle whose shorter sides are 9 and 12 units?



ID 4929

Geometry and  Measurement K12The picture shows a large cube 4 x 4 x 4 that was assembled from one-unit cubes.
It will be painted on all 6 sides.

For what size of large cube will the total number of painted faces of the small cubes be equal to the number of unpainted faces?



ID 5103

Geometry and  Measurement K12What is the largest number of straight lines you can draw through 9 points, so that each line goes exactly through 3 points?

You can move points on the plane as you wish.



ID 5108

Geometry and  Measurement K12 John cuts a large piece of cheese into small pieces using straight cuts from a very sharp cheese wire.
He does not move the pieces from the original shape while he cuts the cheese.

How many small pieces of cheese can he get using only five cuts?



ID 5112

Geometry and  Measurement K12Gerry has 24 meters of fence and wants to make a rectangular garden with the largest area.
He uses the house as the fourth side of the garden.

What length should he make the long side of the garden?



ID 5116

Geometry and  Measurement K12Find X.



ID 5257

Geometry and  Measurement K12Take two sheets of A4 paper (210 x 297 millimeters or 8.3 x 11.7 inch).
Roll one into a short cylinder and the other into a tall cylinder.

Which one holds more air than the other?



ID 5260

Geometry and  Measurement K12Two right-angled triangles have integer side lengths.
Their hypotenuses are equal.

What is the smallest length of the hypotenuse?



ID 5265

Geometry and  Measurement K12Estimate the maximum number of smaller 1-inch circles that fit in a larger circle, the diameter of which is three times larger.