ID 781

The 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

What is the greatest number of circles that can be placed around the central circle?

They have to touch the central circle.

ID 795

ID 796

Anna 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

Sixty-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

The 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

Rectangle 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

A diagonal is a line joining two non-consecutive vertices of a polygon or polyhedron.

How many different diagonals are in the cube?

ID 801

ABCD 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

ID 803

ID 804

ID 807

ID 808

ID 809

ID 812

ID 814

Shape 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

The 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

ID 819

The 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

A 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

ID 828

The 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

Two lines and two diagonals are drawn through the center of the rectangle.

What fraction of the area of the rectangle is red?

ID 839

In 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

The 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

How 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

Four 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

ID 851

ID 1012

Which of the nets can be folded into a box with a red ribbon printed continuously all the way around it?

ID 1016

ID 1045

Egyptian pyramids are square pyramids.

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

ID 1052

The 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

ID 1057

A 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

The figure shows a regular hexagon.

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

ID 1073

ID 1074

ID 1255

A 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

ID 1339

ID 1349

The 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

I want to place together five identical shapes without overlapping them to form a figure.

What is the least perimeter of the figure?

ID 1372

The 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

Anna 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

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

ID 1541

ID 1596

ID 1747

ID 1763

Shape 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

Seven 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

The 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

ID 1931

ID 1959

A 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

I 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

The 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

What 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

This 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

I 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

ID 2267

ID 2269

What 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

ID 3140

A 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

ID 3146

ID 3148

A 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

The 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

A 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

ID 3543

A 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

Connect 7 points on the circumference of a circle.

What is the largest number of intersections for the chords?

ID 3673

All 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

ID 3705

ID 3732

ID 3769

In 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

What 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

ID 3939

An aquarium has a water surface area of 10,000 cm^{2}.

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

A 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

ID 4055

ID 4218

A 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

A 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

I 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

ID 4845

I 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

A 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

The 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

There are 17 parallels and 12 meridians on a globe.

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

ID 5109

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

ID 5110

ID 5187

Swiss 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 km^{2}.

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

The sum of the perimeters of three rectangles is 172cm.

What is the largest possible sum of their areas?

ID 5291