The climbing frame has the shape of an upright pyramid whose base is an isosceles right triangle whose legs are 266 cm long. Height of the pyramid is 185 cm. There is an additional fence at the top of the pyramid, so the total height of the climbing frame is 269 cm. The climbing frame has a climbing rock on it. Furthermore, the climbing square has a coloured cargo net. The net is square, and each square has a side approximately 10 cm long.
Although this climbing frame resembles climbing frames found in other parks, it relies on a several thousands of years of math and astronomy research. Pythagorean theorem and sundial are some of the oldest scientific achievements that are widely known and can be studied on this element.
The oldest surviving sundial, a clock with which we measure time with the help of the sun, is around 3500 years old and was found in Egypt. It consisted of a gnomon (the pointer that casts the shadow) and a numbered dial on a horizontal plane on which gnomon casts a shadow which moves as the sun’s position in the sky changes throughout the day. A similar sundial can be used on this climbing frame. The fence on the upper side of the pyramid is the gnomon of the large sundial.
In addition to the sundial, we can use this climbing frame to learn about the theorem of Pythagoras, which reads: if a and b are the lengths of the legs of a right triangle and c is the length of its hypotenuse, then c2=a2+b2.
This theorem was named after the Greek mathematician Pythagoras who lived approximately 2600 years ago. Even before Pythagoras, the Egyptians knew that a triangle with side lengths of 3, 4, and 5 was right-angled, while the ancient Indians knew that a triangle with sides of 5, 12, and 13 was right-angled. This knowledge of the right triangle was used in the construction of ancient buildings. It is interesting that the reverse of this theorem also applies: if the sum of the areas of the square over the two shorter sides of the triangle is equal to the area of the square over its longest side, then the triangle is right-angled.
The theorem has been proven in many ways and generalized in many ways. We can see some of these generalizations on this climbing frame. Three sides of the pyramid-shaped climbing frame are right triangles, and the fourth side is an isosceles triangle whose side lengths can be measured or calculated using the Pythagorean theorem.
On this element, you can explore geometric shapes and geometric bodies, measure and calculate the length and area of shapes. Three sides of the pyramid-shaped climbing frame are right triangles, and the fourth side is an isosceles triangle whose side lengths can be measured or calculated using the Pythagorean theorem. With the help of a cargo net, we can approximately measure the lengths of the sides and the area of the right triangles on the sides of the pyramid, as well as the area of the isosceles triangle that makes up the fourth side.
Right-angled triangles belong to mutually perpendicular planes, so we can consider this pyramid as a generalization of a right-angled triangle in three-dimensional space. It can be shown that the sum of the squares of the areas of the right triangles is equal to the square of the area of the triangle that belongs to a plane that is not perpendicular to any of the other three.
The fence that is located above the pyramid is inclined towards the ground at an approximate angle of 45° 19′ 36″, which corresponds to the latitude of the city of Rijeka. Furthermore, when the sun is at its highest point in the sky (12:05 pm in winter and 1:05 pm in summer), the shadow cast by the fence points directly north. This means that the fence is parallel to the meridian that passes through the city of Rijeka at 14° 26′ 32″ east longitude. The fence of this pyramid can serve as a sundial gnomon.
With the help of mathematics and physics, the position and shadows of this climbing frame reveal the exact place on Earth as well as the exact time and part of the year we are currently in.
- Measure the length of the hypothenuse of all right-hand triangles and check your results by applying the Pythagoras’s theorem.
- One side of the climbing frame is made of cargo net. Can this net help you to determine the area of that side (each square of the grid has a side that is 10 cm long)? If you want to cover the triangle made of cargo net with canvas, how much canvas would you need? Measure the area of the right-angled triangle covered by the grid using the lengths of its legs. Try to calculate the same area using the formula for the area of a right triangle. What can you conclude?
- Calculate the areas of all sides of the pyramid. Is the sum of the squares of the areas of right-angle triangles equal to the area of an isosceles triangle that is not right-angled?
- Calculate the lengths of the sides of the triangle on the floor. Can you check if a triangle is right-angled without measuring the angles?
- Try to calculate the volume of this pyramid.
- Show the main sides of the world.
- When is the shadow longest and where does it fall? Are the shadows longer in summer or in winter? Why?
- In which direction does the shadow fall when it is at its longest?
- Do you know what summer and winter times are? Which of those two times is closer to the time shown by your sundial?
- For now, this sundial is not yet finished. What do we have to do to finish it?
