We solve problems

Problems Geometry Plane geometry Area Area (other) Area of a circle, sector, segment

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

Problem #PRU-100600

Problems Geometry Plane geometry Area Area of quadrilateral

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$ .

Problem #PRU-100601

Problems Geometry Plane geometry Area Area of a parallelogram

In a parallelogram $A B C D$ , point $E$ belongs to the side $A B$ , point $F$ belongs to the side $C D$ and point $G$ belongs to the side $A D$ . What is more, the marked red segments $A E$ and $C F$ have equal lengths. Prove that the total grey area is equal to the total black area.

Problem #PRU-100105

Problems Geometry Plane geometry Area Area ratio Triangle area

10–13 3.0

Which triangle has the largest area? The dots form a regular grid.

Problem #PRU-100106

Problems Geometry Plane geometry Area Area ratio Area ratio (other)

11–13 3.0

What is the ratio between the red and blue area? All shapes are semicircles.

Problem #PRU-100107

Problems Geometry Plane geometry Area Triangle area

11–13 4.0

In a parallelogram $A B C D$ , point $E$ belongs to the side $C D$ and point $F$ belongs to the side $B C$ . Show that the total red area is the same as the total blue area:

Problem #PRU-100108

Problems Geometry Plane geometry Area Area of a polygon Finding the area by rearrangment

11–13 4.0

The figure below is a regular pentagram. What is larger, the black area or the blue area?

Problem #PRU-100109

Problems Geometry Plane geometry Area Area ratio Area ratio (other)

11–13 3.0

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

Problem #PRU-100110

Problems Geometry Plane geometry Area Finding the area by rearrangment

11–13 4.0

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$ .

Problem #PRU-100111

Problems Geometry Plane geometry Area Area (other) Finding the area by rearrangment

11–13 4.0

In a parallelogram $A B C D$ , point $E$ belongs to the side $A B$ , point $F$ belongs to the side $C D$ and point $G$ belongs to the side $A D$ . What is more, the marked red segments $A E$ and $C F$ have equal lengths. Prove that the total grey area is equal to the total black area.