Solids

Exercise 1

What you see here is a truncated cubes with edges of $6$ cm. The points $P$ and $R$ are the centers of the edges they are located on. $B Q = 1$ cm.

a

Calculate the lengths of the edges of $Δ P Q R$ .

b

Make a life sized drawing of $Δ P Q R$ and calculate the angles of this triangle.

c

The diagonal plane $D B Q S H$ is a pentagon. make a life sized drawing of this diagonal plane of the truncated cube and calculate its angles.

Exercise 2

Given is a regular pyramid $T . A B C D$ with the square $A B C D$ as the base. The edges of $A B C D$ are $6$ cm and the height $T S$ of the pyramid is $10$ cm. $S$ is the intersection point of $A C$ and $B D$ .

a

Calculate height $T S$ of this pyramid.

b

Explain why quadrangle $B C Q P$ is an isosceles trapezium and calculate the lengths of the edges of this quadrangle.

c

Draw $B C Q P$ life sized and calculate all angles of this trapezium.

Exercise 3

Below you see two figures.

Calculate the length of the line segments with a question mark.

Exercise 4

Here you see a so called hipped roof, a roof with a rectangular base $A B C D$ with the roof-ridge $E F$ situated exactly over the middle of the base. The roof itself consists of twee equilateral triangles and two symmetric trapeziums.

a

Calculate the length of the edges $A E$ , $D E$ , $B F$ and $C F$ .

b

Calculate the size of $∠ A B F$ and $∠ B F C$ .

c

At $3$ m above the attic floor $A B C D$ a rectangular loft is made. Calculate the area of the floor of this loft.

Exercise 5

Beside you see a photo of the building "Willemswerf" in Rotterdam and below you see a view from above of a very simplified version of the building. This strongly simplified version has a height of $80$ m. The bend in the building starts $10$ m above the ground.

a

Calculate the lengths of the edges of that trapezium.

b

Calculate the size of the angles of that trapezium.

Exercise 6

Marianne is in New York for a couple of days. She takes a picture of a tree. She stands at a distance of $10$ m of the tree. In the picture the tree is $2$ cm high. The distance from the lens to the negative in the camera is $6$ cm.

a

Calculate the height of the tree precisely.

b

After development the picture also contains the Statue of Liberty. Coincidentally the Statue of Liberty happens to be exactly the same size as the tree. The statue is $93$ m high. How far was Marianne from the Statue of Liberty?

Exercise 7

What you see here is a cube that has been truncated at the front $A B C D . E F G H$ . $∠ B A D = ∠ A D C = 90 °$ . The dimensions are in the figure.

Calculate the length of $A Q$ .

Exercises