**Solid geometry**

## Angles in space

**Example 1**

Calculate the angle between the internal diagonal of the rectangular prism and the base.

The bottom diagonal should be calculated first. It can be found by Pythagorean theorem.

Only a positive is valid because it is a length. Now we have a right triangle in which the legs are the diagonal of the prism base and the height of the edge and the hypotenuse is the internal diagonal of the space.

## Cylinders

A cylinder is a solid with similar bases. The height of the cylinder is the perpendicular distance between the bases. When opened, the lateral surface forms a rectangle, one side of which is the height and the other side the perimeter of the base. The volume is obtained by multiplying the area of the base by the height.

A cylinder is a right cylinder if its axis is perpendicular to the bottoms. Otherwise it is an oblique cylinder.

The bottom of a right circular cylinder is a circle. In this case, its volume and lateral area are obtained as follows.

**Example 2**

A right circular cylinder has a base 20 cm in diameter and a height of 38 cm. Calculate the volume of the cylinder in litres and the total area in square centimetres.

Since* 1 dm³ = 1 l*, the lengths are changed to decimetres *20 cm = 2 dm* and *38 cm = 3,8 dm*. The radius of the bottom is *1 dm.*

The volume is *12* litres.

The total area contains the lateral area and the base areas.

## Cones

A cone is a shape whose sides start from the edge of the base and join to the same point on top, the apex.

The volume of the cone is one third of the volume of a cylinder of equivalent height.

A cone is right if the centre of the base is aligned perpendicular to the apex.

The most common cone is a straight circular cone.

Volume of a cone

The lateral surface of a straight circular cone when opened to a plane forms a circular sector with radius* ***s*** *as the side of the cone. The length of the arc of a sector is relative to the arc of the whole circle and is equal to the ratio of areas.

A truncated cone consists of a cone with a part cut off at the top. The removed part is similar with the corresponding whole cone.

**Example 3**

The base of a right circular cone has a diameter of 20 cm and a height of 50 cm. Calculate the volume of the cone and the lateral area.

We use Pythagorean theorem to find *s*

The volume and the lateral surface area

## Spheres

**Example 4**

The location of Helsinki is Latitude 60 ° North. How far is the equator from Helsinki?

The circumference of the earth is about 40 000 km. The angle between Helsinki and the equator is 60 °. We need to find the length of the corresponding arc. We mark the required distance by *E.*

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