A square is divided into triangles and quadrilaterals as shown. What fraction of the square area is covered by the shaded triangle?

## The shark fin

An isosceles triangle is inscribed in a semicircle with one side along the diagonal and the top vertex somewhere on the semicircle. What’s the maximum fraction shaded?

## The diagonal dilemma

On two faces of a cube a diagonal is drawn that meet in the same vertex. What’s the angle between them?

## The blockhat

Four congruent rectangles are placed in a hat-shaped configuration. What’s the angle between the lines connecting the opposite corners?

## The corner pocket

A snooker player wants to corner a ball starting from a point on one side and bouncing two times from the opposite sides. Given the dimensions of the table in the figure, what’s the length of the track the snooker ball travels?

## The snowman mansion

A square and a half square are stacked in order to form a house-shaped quadrilateral. Inside two circles are closely packed. What’s the angle between the tangency points?

## Balancing balls

Two touching circles are placed on top of a right triangle. What’s the angle between the chords connecting the tangency points?

## The marble box

Two marbles of sizes π and 4π are enclosed in a rectangular box. What is the total area of the box?

## The folded sheet

A rectangular sheet of paper is folded in such a way that two equal angles are formed as shown. What fraction of the resulting quadrilateral is shaded?

## Moon over mountains

A rectangular frame encloses two congruent equilateral triangles and a unit circle. What is difference between width and height of the rectangle?