ID 781

Geometry and  Measurement K10The scale of a map is 1:20 000.
The distance is measured as 5 centimeters on the map.

How many kilometers is this equivalent to?

Comments: 1 meter = 100 centimeters.
Comments: 1 kilometer = 1000 meters



ID 794

Geometry and  Measurement K10What is the greatest number of circles that can be placed around the central circle?

They have to touch the central circle.



ID 795

Geometry and  Measurement K10Which reservoir has the greatest volume?



ID 796

Geometry and  Measurement K10Anna has made puzzle pieces by cutting wedges from a disk.
Each wedge cut from the disk has a 50-degree angle at the center of the disk.
The weight of the uncut disk is 72 grams.

How many grams does each 50-degree wedge weigh?



ID 797

Geometry and  Measurement K10Sixty-four coins are melted down and recast as a single coin of the same thickness h.

How many times larger than the diameter of the original coin is the diameter of the new coin?



ID 798

Geometry and  Measurement K10The diameter of the rear wheel of a circus bike is 99 cm. It is 1 cm smaller than the diameter of the front wheel.
When the bike goes around the circus, the number of rotations of the smaller wheel is 1 more than the number of rotations of the larger wheel.

Find the number of rotations made by the larger wheel. 



ID 799

Geometry and  Measurement K10Rectangle ABCD contains five small congruent rectangles. The smaller dimension of one of the small rectangles is 3 cm.

What is the area of rectangle ABCD in square cm?



ID 800

Geometry and  Measurement K10A diagonal is a line joining two non-consecutive vertices of a polygon or polyhedron.

How many different diagonals are in the cube?



ID 801

Geometry and  Measurement K10ABCD is a rectangle.
E, F, G and H are midpoints of AO, BO, CO and DO respectively.

What is the fraction of EFGH to ABCD?

Compare the areas.



ID 802

Geometry and  Measurement K10Which is the front view of the 3-D shape?



ID 803

Geometry and  Measurement K10Which shape was proportionally enlarged, turned, and moved to the point O?



ID 804

Geometry and  Measurement K10 Which fraction of the design is white?



ID 807

Geometry and  Measurement K10How many lines of symmetry does a circle contain?



ID 808

Geometry and  Measurement K10Which two lines are parallel?



ID 809

Geometry and  Measurement K10What is the volume of the figure?



ID 812

Geometry and  Measurement K10Which rays on this picture appear to intersect?



ID 814

Geometry and  Measurement K10Shape A and B are congruent equilateral triangles.
Shape C is formed by superimposing shapes A and B by about their centers.

What is the perimeter of shape C if the perimeter of shape A is 36 centimeters?



ID 815

Geometry and  Measurement K10The diagram illustrates a row of three squares formed by matches.

How many matches will it take to make a row of 30 squares?



ID 817

Geometry and  Measurement K10A quadrilateral is any four-sided figure.

How many quadrilaterals are there in this shape?



ID 819

Geometry and  Measurement K10The big square has a side length of 1.
Its sides' midpoints are connected to form a second square, and so forth.

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



ID 821

Geometry and  Measurement K10A line passes through P(3,1) and Q(36,1000).

How many other points with integer coordinates are on the line and between P and Q?



ID 826

Geometry and  Measurement K10How many planes of symmetry does a brick have?



ID 828

Geometry and  Measurement K10The picture shows two identical squares with sides that have a length of 1 meter.
M is the midpoint of the corresponding sides of both squares.

What is the area of the blue section?



ID 830

Geometry and  Measurement K10Two lines and two diagonals are drawn through the center of the rectangle.
What fraction of the area of the rectangle is red?



ID 839

Geometry and  Measurement K10In a triangle, the sum of two of the angles is equal to the third.
The lengths of the sides are 12,13 and X.

Find X.



ID 840

Geometry and  Measurement K10The Earth's diameter is 12,700 km.
The horizon is 11 km from the top of a lighthouse.

Estimate the height of the lighthouse.



ID 841

Geometry and  Measurement K10How many equilateral triangles can you create using six identical matches?

The length of the side of the triangle must be equal to the length of the match.

Source: Fixx, James F Solve It!, 1978



ID 843

Geometry and  Measurement K10Four towns are situated at the corners of a square. The government decided to build a new road linking all four towns together. Engineers suggested four different designs.

Which design illustrates the shortest road?



ID 847

Geometry and  Measurement K10What is the volume of the object?



ID 851

Geometry and  Measurement K10How many squares can be placed inside the circle without intersection?



ID 1012

Geometry and  Measurement K10Which of the nets can be folded into a box with a red ribbon printed continuously all the way around it?



ID 1016

Geometry and  Measurement K10The blocks of ice are of the same height and depth.

Which assembly melts fastest?



ID 1045

Geometry and  Measurement K10Egyptian pyramids are square pyramids.

Which of the following nets can be folded to form a square pyramid?



ID 1052

Geometry and  Measurement K10The figure shows an example how a 3x4 net can be covered by L-shaped figures.

Which of the figures can be covered by the L-shaped figures in such a manner?



ID 1053

Geometry and  Measurement K10A square is inscribed in a triangle.

Find the length of the side of the square.



ID 1057

Geometry and  Measurement K10A telephone company places round cables in round ducts.

What arrangement of the cables allows the engineers to use a round duct with the smallest diameter?

Remember, there may be more than one correct answer.



ID 1059

Geometry and  Measurement K10The figure shows a regular hexagon.

What is the area of the red part as a fraction of the whole hexagon?



ID 1073

Geometry and  Measurement K10A semicircle is inscribed in the right triangle.

Find the radius of the semicircle.



ID 1074

Geometry and  Measurement K10Among the following shapes of equal perimeter, which has the largest area?



ID 1255

Geometry and  Measurement K10A point of the square grid is chosen to form an isosceles triangle together with the red segment.

How many isosceles triangles can be drawn on the square grid?



ID 1309

Geometry and  Measurement K10Eugenia discovered round ink spots on her music paper.

Which music paper is the dirtiest?



ID 1339

Geometry and  Measurement K10What is the greatest number of possible points of intersection of four circles, of different radii?



ID 1349

Geometry and  Measurement K10The smallest apple weights 100 grams.
The largest apple has a perimeter 10% larger than that of the smallest apple.

Estimate the weight of the largest apple.



ID 1351

Geometry and  Measurement K10I want to place together five identical shapes without overlapping them to form a figure.

What is the least perimeter of the figure?



ID 1372

Geometry and  Measurement K10The left picture shows nine dots arranged in a 3 x 3 square.
The dots are connected using only four straight lines and without lifting the pen from the paper.

The right picture shows seven dots evenly distributed on a circle and a dot in the center.

How many straight lines connect the dots in the same way?



ID 1423

Geometry and  Measurement K10Anna takes a rope that is 16 meters long and creates a square.
Bob takes the rope and creates a rectangle that has an area 75% of the square's area.

What is the length of the rectangle?



ID 1504

Geometry and  Measurement K10Which of these diagrams could be drawn completely without lifting the pen off the paper or going over any line twice?



ID 1541

Geometry and  Measurement K10For what value of x is the ratio between the red and blue areas at its maximum?



ID 1596

Geometry and  Measurement K10What fraction of the large equilateral triangle is colored?



ID 1747

Geometry and  Measurement K10Which of the following graphs shows the dependence "square side length Y - its area X"?



ID 1763

Geometry and  Measurement K10Shape A and B are congruent equilateral triangles.
Shape C is formed by superimposing shapes A and B at their centers.

What is the ratio of area C to the area of A and B combined?



ID 1802

Geometry and  Measurement K10Seven squares with side length of 1, 2, 2, 2, 3, 4, and 5 units can be fitted together with no gaps and no overlaps, to form a rectangle.

What is the length of the shorter side of the rectangle?



ID 1806

Geometry and  Measurement K10The area of the white cross is 20% of the area of the square flag.
Five white squares form the cross.

What is the length of the side of the white square?



ID 1922

Geometry and  Measurement K10Two heights in a triangle are not less than its two sides.

Find the largest angle.



ID 1931

Geometry and  Measurement K10If the largest size of a cube is 3 times larger than the original size, what is its volume change?



ID 1959

Geometry and  Measurement K10A Heronian triangle is a triangle whose side lengths and area are all integer numbers.
It is named after Hero of Alexandria.

Find sizes of a triangle whose area is numerically the same as its perimeter.



ID 1965

Geometry and  Measurement K10I arrange 10 points so that 3 lines each go through 4 points.
I would like to rearrange these 10 points.

What is the greatest number of lines that go through 4 dots each?



ID 1972

Geometry and  Measurement K10The top of a rectangular box has area 20 square meters, and two sides have areas 12 and 15 square meters.

What is the box volume?



ID 1989

Geometry and  Measurement K10What is the probability that a point chosen randomly from the interior of an equilateral triangle is closer to a vertex of the triangle than it is to a midpoint of one of the triangle's sides?



ID 2079

Geometry and  Measurement K10This shape was formed by removing a small cube from a big cube.
The side length of the removed cube is two thirds of the side length of the original cube.

What is the volume of the new shape compared with the original volume?



ID 2199

Geometry and  Measurement K10I want to cut a wooden cube that is four inches on each side into 64 one-inch cubes.

I can do this by making 3 + 3 + 3 = 9 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 2261

Geometry and  Measurement K10What is the angle between the hour hand and the minute hand at 9:10 ?



ID 2267

Geometry and  Measurement K10I cut a pyramid from a brick as shown at the picture.

What part of the brick did I cut off?



ID 2269

Geometry and  Measurement K10What is the probability that a point chosen randomly from the interior of a circle is closer to the circle's center than it is to any point of the circle's circumference?



ID 3117

Geometry and  Measurement K10Which is the side view of the 3-D shape?



ID 3140

Geometry and  Measurement K10A rectangle has a width of 0.7x and a length of 0.4x.

Which formula is the correct one to calculate the perimeter (P) in terms of x?



ID 3141

Geometry and  Measurement K10Which gives the largest answer?



ID 3146

Geometry and  Measurement K10An arrow rotates 372° in one second.

How many revolutions does it make in one minute?



ID 3148

Geometry and  Measurement K10A boy is making boxes from cardboard.
He is going to cut square pieces off each corner as shown in the diagram and fold the sides up.
Which size of square pieces would give a larger box in terms of volume?



ID 3159

Geometry and  Measurement K10The American flag consists of thirteen equally spaced, horizontal red and white stripes, with a blue rectangle in the canton bearing fifty small, white, five-pointed stars.

What part of the flag is white stripes?



ID 3308

Geometry and  Measurement K10A block of wood in the form of a cuboid 8 x 9 x 10 has all its six faces painted red.

If the wooden block is cut into small cubes of 1 x 1 x 1, how many of these cubes would have red paint on them?



ID 3504

Geometry and  Measurement K10A sculptor cuts a cube with a plane.

What is the maximum number of possible sides on the new face?



ID 3543

Geometry and  Measurement K10A bug walks from corner A of a room to corner B by only moving along the walls.

What is the shortest path it can take?



ID 3607

Geometry and  Measurement K10Connect 7 points on the circumference of a circle.

What is the largest number of intersections for the chords?



ID 3673

Geometry and  Measurement K10All angles are right and the lengths of the sides are given in miles in the diagram.

Find the length of the shortest path from A to B along the sides of the shape.



ID 3677

Geometry and  Measurement K10How many triangles with a perimeter 8 units have all side lengths that are integer numbers?



ID 3705

Geometry and  Measurement K10A regular polygon has some perpendicular sides and some parallel sides.

What might the shape be?



ID 3732

Geometry and  Measurement K10If a square is four and a triangle is three, how many is a regular star?



ID 3769

Geometry and  Measurement K10In a triangle, the sum of two of the angles is equal to the third, and the lengths of the two longer sides are 25 and 24.

What is the length of the shortest side?



ID 3771

Geometry and  Measurement K10What is the absolute difference between the largest and smallest possible perimeters of two rectangles that each have an area of 100 square units and integer side lengths?



ID 3931

Geometry and  Measurement K10What is the volume of a right rectangular solid with face areas of 12, 21, and 28 square meters?



ID 3939

Geometry and  Measurement K10An aquarium has a water surface area of 10,000 cm2.
I put a brick that measures 40 cm x 20 cm x 12.5 cm in the aquarium.

Estimate by how many centimeters the water rises.



ID 3994

Geometry and  Measurement K10A piece of wire 75 cm in length is cut into two parts, one of them being 30 cm long.
Each part is bent to form a square.

What is the ratio of the area of the larger square to the smaller square?



ID 4018

Geometry and  Measurement K10Estimate the ratio of the circumference c to the diameter of the circle d.



ID 4055

Geometry and  Measurement K10Find the ratio of the areas of the large and small equilateral triangles.



ID 4218

Geometry and  Measurement K10A 3 by 4 rectangle is contained within a circle.

What is the smallest possible diameter of the circle?

This is typical SAT question.



ID 4229

Geometry and  Measurement K10A photograph is placed in a frame that forms a border 2.5 cm wide on all sides of the photograph.

What is the area of the border?



ID 4326

Geometry and  Measurement K10I have 33 coins.

What is the minimum number of coins I need in order to make sure that each coin touches exactly three other coins?



ID 4583

Geometry and  Measurement K10Which regular shape cannot be used to tile a floor with no gaps between tiles?



ID 4845

Geometry and  Measurement K10I divided a 3 x 4 square into 6 squares.

What is the smallest number of squares into which you can divide a 9 x 10 rectangle?

Author: Matt Enlow



ID 4879

Geometry and  Measurement K10A wooden empty box weighs 80 pounds.

How much will another box of the same material weigh if its sides are twice as long?



ID 4903

Geometry and  Measurement K10The picture shows a polygon with 7 sides and 5 right angles.

How many interior right angles are possible in a polygon with eight sides?



ID 4987

Geometry and  Measurement K10There are 17 parallels and 12 meridians on a globe.

Into how many areas is the surface of the globe divided?



ID 5109

Geometry and  Measurement K10What is the difference between the red area and the blue area if the numbers show the side lengths of each square?



ID 5110

Geometry and  Measurement K10What is the side view of the three-dimensional object?



ID 5187

Geometry and  Measurement K10Swiss village Saas-Fee is entirely pedestrian and serviced by electric taxis and buses only. All electricity is obtained from 100% renewable hydroelectric power. The people have equipped the community's 250 wood-fired furnaces with particle filters.
Design guidelines for the village require houses to be 40% wooden, to maintain its architectural character. Its area is about 40 km2.

If the border of the village was a circle what would be the maximum distance an electric car goes to cross the entire village?



ID 5237

Geometry and  Measurement K10The sum of the perimeters of three rectangles is 172cm.

What is the largest possible sum of their areas?



ID 5291

Geometry and  Measurement K10I want to wrap 30 wooden cubes.

Which packaging uses the least package?