ID 697

A hall is cube-shaped.

The volume of the hall is 216 cubic meters.

The door is 3 meters high.

Find the length of a side of the floor.

ID 702

ID 703

ID 704

The aquarium was full of water.

A girl has tipped it on its side,

so five liters are left.

How much water does the aquarium hold when it is full of water.

ID 706

ID 707

One face of a rectangular box has an area of 15 square cm. Another face is 20 square cm and the other face is 12 square cm.

What is the height, h, of the box?

ID 708

ID 709

This large cube is made from smaller wooden cubes.

All of the faces of the large cube are painted blue, and then it is taken apart.

What fraction of the surface is blue?

ID 713

A rectangular floor measures 3 meters by 9 meters.

It is tiled with one-square meter tiles.

Through how many tiles would the diagonal line in this rectangle pass?

ID 714

A sports club is replacing its water supply lines for a swimming pool. The current system uses two pipes, A and B, which have circular cross-sections with diameters of 6 and 8 cm, respectively. The club decides to use a single replacement pipe with the same capacity.

What is the diameter of the new pipe?

ID 715

John is building a flight of stairs.

Each stair is the same size.

What is the height of the flight of stairs X?

ID 717

ID 720

ID 721

Members of the Green Tennis Club are creating a logo.

The logo of the club is a blue square with a green shape inside.

What percentage of the logo is blue?

ID 722

Mr. Clever gave his students this pattern of gray tiles.

Develop an equation to represent the number of gray tiles, t, for any size of the inner square, n.

ID 724

ID 731

A wheel with a red mark on it is rolled along a straight line.

What is the path the mark follows during one revolution?

ID 734

ID 736

The figure consists of congruent squares.

The perimeter of the figure is 10 meters.

What is the area of the figure?

ID 739

The figure shows a regular octagon.

What is the area of the colored part as a fraction of the area of the entire octagon?

ID 750

I want to build a fence around a rectangular garden.

The garden has an area of 100 square meters.

The fence posts must be placed 1 meter apart.

What is the minimum and maximum number of fence posts needed?

ID 755

ID 760

ID 768

The bigger star is similar to the smaller one.

It is twice the height of the smaller star.

How much greater is its area?

ID 769

ID 774

ID 775

Cut 1 cube into 8 pieces. One of the pieces is a cube whose volume is three times less than that of the original cube.

What is the percentage of new surface area compared with the original cube's surface area?

ID 1038

ID 1046

The arrow top is at the midpoint of the edge of the rectangle.

What fraction of the rectangle is shaded?

ID 1072

Each diagonal of a cube is increased by 50%.

What was the percentage increase of the volume of the cube?

ID 1082

A and B are midpoints of the corresponding sides of the rectangle.

What fraction of the rectangle is the green part?

ID 1251

ID 1260

ID 1265

ID 1373

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 rectangular pieces of cheese can he get using only five cuts?

ID 1393

All triangles are inscribed in circles with a diameter of 1 meter.

Find the triangle with the largest area.

ID 1397

ID 1461

ID 1474

Two polygons are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

In how many ways can a regular hexagon be divided by a straight line into two congruent pentagons?

ID 1478

ID 1482

Fifteen billiard balls perfectly fit into a triangular rack.

What is the largest number of the balls that fit into the rack when its side lengths are increased by 20%?

ID 1551

I want to cut round bread into eight equal pieces.

What is the smallest number of straight cuts I can make to achieve this?

ID 1567

The diagram shows my square pond with a tree in each corner.

How much can I increase its area if I transform it into a right-angled triangle pond and keep the trees in the same place?

ID 1583

How many colored four-unit shapes can be placed inside the 6x6 square without intersecting each other?

You may turn the shape.

Find the greatest number.

ID 1595

A solid is a cube with three tunnels drilled through it, as you can see in the picture.

How many blocks are there in the solid?

ID 1599

Four matchsticks form a square.

How many non-overlapping squares can be formed using seven matchsticks?

ID 1756

Ten acrobats are evenly spaced at a circus circular arena.

Which acrobat is directly opposite the ninth performer?

ID 1792

It costs $24 to paint a cube.

Before painting, it was cut into eight cubes.

What is the cost to paint these small cubes?

ID 1793

Six shapes can be fitted together with no gaps and no overlaps to form a rectangle.

What is the smallest dimension of the rectangle?

ID 1803

I want to cut this shape into four pieces all of precisely the same size and shape.

Which is such a piece?

ID 1808

ID 1812

A rectangle floor 12 x 16 is covered by square tiles of sides 40 cm.

A chalk line is drawn from one corner to the diagonally opposite corner.

How many tiles have a chalk line segment on them?

ID 1838

ID 1847

The colored figure in the picture consists of identical isosceles right triangles.

What is the area of the blue shape?

ID 1849

ID 1868

A guard walks completely around the walls of a rectangular castle with a perimeter of 10 km.

From any point on his path, he can see exactly 1 km horizontally in all directions, except the wall.

What is the area of the region that contains all points that the guard can see during his walk?

ID 1924

In a triangle, the sum of two of the angles is equal to the third, and the lengths of the two longer sides are 17 and 15.

What is the length of the shortest side?

ID 1938

If the coordinates of one end point of a line segment are (1 , 2 , -3) and the coordinates of the midpoint are (-3 , 2, 1), what are the coordinates of the other endpoint?

ID 1951

What is the probability that a point chosen randomly from the interior of a rectangle is closer to the rectangle's center than it is to any of the rectangle's vertices?

ID 1976

Eight poles are evenly spaced on the circumference of a big circle.

What is the largest number of ropes that can be stretched between two poles so that no two ropes intersect at any point?

ID 1993

A quadrilateral with one pair of parallel sides is referred to as a trapezoid.

A trapezoid has mutually perpendicular diagonals of lengths 4 and 6.

What is the area of the trapezoid?

ID 2043

ID 2208

ID 2280

ID 2288

ID 2290

An analog wristwatch says that the time is 14:20.

The minute hand moves 300 degrees.

How many degrees has the hour hand turned?

ID 3009

ID 3010

ID 3016

ID 3017

A curtain covers a window as shown on the right.

We approximate the area of the curtain by 2 triangles.

The window is 24 by 36 inches.

How much of the window area is covered by the curtain?

ID 3023

ID 3025

ID 3030

ID 3031

ID 3032

Which figure can be rotated 72° about its center and have its final orientation appear the same as the original orientation?

ID 3033

ID 3034

In the diagram, the radii of the two concentric circles are 4.5 and 10, respectively.

What fraction of the bigger circle is shaded?

ID 3035

If two sides of the triangle have lengths of 22 and 33, which of the following could be the perimeter of the triangle?

ID 3036

ID 3041

If G is the total area of the green regular octagon and R is the total area of the red regular octagon, which is correct?

ID 3078

ID 3088

ID 3102

ID 3107

ID 3116

ID 3120

ID 3121

ID 3135

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 3193

ID 3212

A telephone company places round cables in round ducts.

What position of the cables is the worst and demands using the round duct with the larger diameter?

ID 3298

ID 3357

ID 3390

Anna takes a rope that is 24 ft 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 3663

ID 3688

ID 3716

Cut the shape into two pieces and create a square from them.

How many small squares are in the smaller of these two shapes?

ID 3770

What is the absolute difference between the largest and smallest possible areas of two rectangles that each has a perimeter of 100 units and integer side lengths?

ID 3782

Three equilateral triangles form seven unique regions.

What is the maximum number of regions that can be formed by three congruent, overlapping triangles?

ID 3936

I increase the cube so that the diagonal AB doubles its original size.

How many times does the volume increase?

ID 3990

ID 4017

I have a cube with a volume of 1000 cm^{3}.

I make the length 20% longer, the width 10% shorter and the height 10% shorter.

What is the new volume?

ID 4080

ID 4333

If we add all these vectors and place the start of the resulting vector at the origin, where does the resulting vector end?

ID 4485

ID 4502

ID 4561

If I use 30 g of batter to make a crêpe of 30 cm in diameter, how much batter do I need for a 45-cm pancake of the same thickness?

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

ID 4784

ID 4874

ID 4967

James the Mathematician just measured the angle of his piece of pizza, which is 39.6°.

What part of the pizza did James get?

ID 5125

A solar power station includes 17,000 mirrors, each 60m^{2}.

What is the total area of the mirrors?

1 km = 1000m

The photograph courtesy of Roland Sauter

ID 5172

Dr Heidi has been called in as an expert to solve a problem. There is a device which is spherical, air-tight when the two halves are glued together, and hollow; it is designed to be thrown into the sea and sit on the sea bed. When she asked them why they said it was hush-hush, top secret, need-to-know, ... and then they looked at her sternly. She hasn’t even seen it for herself. They said its diameter was &π (=3.1415926…) units, but they then said the units were Classified!

They said its weight is exactly 10kg. The sphere has plenty of air space inside.

What should she recommend to make the sphere sit on the sea bed?

ID 5207

Britain’s greatest inventor of mathematical puzzles Henry Ernest Dudeney asked you to place 12 mince pies in 6 lines with an equal number of pies in each line.

What is the largest possible number of pies in a line?

ID 5394

ID 5402

ID 5411

Marek, the pan-dimensional super being, has arbitrarily defined his current location as (0,0,0,0,0,0) in 6D hyperspace. He wishes to reach location (3, 0, 2, 0, 4, 3) by one of the many shortest paths available. Despite his immense power, he can only move one hyperstep at a time, each hyperstep consisting of a unit change in exactly one of the coordinate values. Any hyperstep is of equal ‘length’.

What is the smallest number of hypersteps required for him to reach his destination?

Author: Leslie Green

ID 5414

What is the minimum number of matches that must be moved in order to obtain exactly two equilateral triangles (with no left over pieces)?

ID 5448

What is the perimeter of a regular hexagon that perfectly fits into a circle with an 8-meter diameter?

ID 5520

I can draw 2 diagonals in a square.

How many diagonals can I draw in a hexagon?

There are 6 sides in a hexagon.

ID 5580

ID 5629

The length of a soccer pitch is 110 meters and the width is 66 meters.

Before a match, ten ball boys are evenly spaced along 2 long sides of the pitch.

What is the maximum possible distance between the boys?

ID 5633

There are smaller squares inside the large rectangle.

If the green square has side length 1 and the red square has side length 3, what is the area of the large rectangle?

ID 5663

In how many ways can a square piece of paper be cut into two congruent shapes by a single straight line cut?

ID 5711

Pairs of identical rectangular strips, each measuring 5 by 1, are completely overlapped in a number of different ways to form different shapes.

What is the largest possible perimeter of the new shape?

ID 5713

A cellar floor is to be tiled in the way shown in the picture.

If the cellar measures 25 tiles x 27 tiles, how many green tiles will be needed?

ID 5798

A shape is made from 6 congruent equilateral triangles that share one, two, or three common sides.

What is the minimum possible number of sides the shape has?

ID 5946

ID 5977

ID 6021

There is a right angled triangle with a hypotenuse of unit length. Denote an angle (other than the right angle) as alpha.

Given that the side adjacent to the angle is of length cos(alpha) and the side opposite the angle is sin(alpha), evaluate the sum.

Author: Leslie Green

ID 6211

Gerry is running away from a horde of three wild animals. To escape, he always runs a dozen meters and then turns 90° to the left or to the right alternately.

Just before the seventh turn, what is his maximum displacement from the start?

ID 6308

Pretend the round red blobs are tennis balls. Pretend the blue lines are stretchy strings.

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

Author: Leslie Green

ID 6310

Pretend the round red blobs are tennis balls. Pretend the blue lines are stretchy strings.

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

Author: Leslie Green

ID 6344

ID 6442

ID 6515

A chocolatier can put only two circular chocolates (8-cm in diameter) in a circular black box which is 16.1cm in diameter.

How many moon-shaped chocolates fit in the box?

The moon-shaped chocolates have a diameter of 8cm and are formed by an arc of the same diameter that goes through the center (centre) of the circle.

Inspired by Alex van den Branhof, Jan Guichelaar, Arnout Jaspers Half a Century of Pythagoras Magazine. MAA 2011

ID 6550

The picture shows four equilateral triangles with sides of lengths
3, 6, 9 and 12.

How much greater is the green area compared to the blue area?

ID 6654

Leslie Green asks:

A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.

Which of these shapes is not a convex polygon?

ID 6658

A piece of wire is to be cut into two parts in the ratio 7 : 8.

Each part is bent to form a square.

Estimate how much larger the area of the second square is compared to the first one.

ID 6692

The two blocks have identical weights, identical densities, and identical depths (into the page).

The second block is a cube.

What is the distance X to keep the system in equilibrium?

ID 6795

ID 6881

ID 6909

Pairs of identical rectangular strips are overlapped to form different shapes.

Which shape has the smallest area?

ID 7054

The perpendicular X divides the hypotenuse of the right triangle into two segments of 18 and 32 feet respectively.

What is the length of the perpendicular?

ID 7058

ID 7259

MENSA question:

Using the first three shapes and values as a guide, find the value that corresponds to the last shape.

ID 7278

ID 7388

The measure of the two acute angles in a right triangle are in the ratio 2:7.

What is the measure of the smallest angle of the triangle?