Geometry is a part of mathematics that deals with forms, shapes and the sizes of objects. There are many different kinds of shapes . Some are two-dimensional like circles, squares or triangles, others are three-dimensional like cubes, balls or cones . Geometry shows us how we can construct or draw such forms and how to measure them.

We need geometry in everyday life . Houses are made up of geometrical shapes. Pilots use geometry when they plan their routes and surveyors need geometry to make maps and measure areas of land.

The world is full of geometric shapes. Egyptian pyramids got their name from geometry and the biggest building in Washington is the Pentagon, a structure that has five sides.


Plane Figures and Shapes

A plane is a flat surface like the top of a table. Plane geometry is also called two-dimensional geometry.


Lines and Angles

The simplest objects in geometry are lines and angles. A line is straight and has two arrowheads which show that it continues in both directions. A ray is a line that has a starting point and then goes on in a certain direction. Two lines that come together and have the same starting point form an angle . It can be measured in degrees.

Angles that make square corners measure 90° and are called right angles. Angles with less than 90° are acute, those with more than 90° are called obtuse. Two lines that never meet are parallel.


lines, rays and angles




When several segments of a line join together they create a closed figure called a polygon.



Rule of PythagorasA polygon with three sides is a triangle. If all three sides have the same length we have an equilateral triangle, two sides of the same length make up an isosceles triangle. A triangle with one right angle is called a right triangle. The side opposite the right angle is the hypotenuse , the other two sides are called legs.

The Greek mathematician Pythagoras , who lived at about 500 B.C. ,found out that there is a connection between the length of the legs and the hypotenuse of a right triangle. If you add up the squares of both legs it is equal to the square of the hypotenuse.




Quadrilaterals are polygons with four sides. They have many more shapes than triangles do.

A quadrilateral with four right angles is a rectangle. If all sides of this rectangle have the same length we call it a square. A parallelogram has opposite sides that are parallel and equal in length. If all sides are the same in length we call it a rhombus.



Other polygons

The term polygon comes from the Greek language and means "many angles". A five-sided polygon is a pentagon. Hexagons (six sides) are shapes we use to tile floors or bees use for their honeycombs. The stop sign we see at a road crossing is an eight-sided polygon , an octagon.



Shapes are not always made up of straight lines. A circle is a curved line. All points on this line have the same distance from the centre. Circles are found everywhere in our daily life—wheels, coins and cups have circular shapes.

The line that cuts the circle into two equal halves is called the diameter. The radius is the line from the centre of the circle to every point on its border.



Solids are three-dimensional objects. They have a length, width and a height.

A polyhedron is a figure with flat faces and straight edges. A cube, for example, is built from a square. It has six faces that are all the same. However, not all polyhedrons are regular. A prism has two parallel bases and its other faces are parallelograms . A pyramid has a base that has four sides , the other faces are triangles.

Some solids are curved. A sphere is a ball-shaped object with a completely round surface. The distance from the middle to every point on the surface of the sphere is always the same.

Cylinders are shapes that we find in cans or pipes. They have two circles as bases and a curved side. The cone is probably the most familiar too us because it is shaped like an ice cream.


Measuring geometrical objects

The area is the surface that a figure covers. It can be measured by putting squares on a shape and then counting them . A rectangle with a length of 5 cm and a width of 4 cm has an area of 20 square centimeters. You can put 20 square centimeter units into it.

Measuring the area of other objects may be more difficult. You can get a triangle's area by multiplying the base line with its height and then dividing it by two.

The area of a circle is the square of the radius multiplied by 3.14159 , a constant called Pi.

Measuring the area of objects

The amount of space that a solid figure fills is called its volume. Basically, you can get the volume when you multiplying the area of the base by the height of the object. The unit for volume is the cube . A box, for example that has a length, width and height of 1 m would have a volume of 1 cubic metre.

The volume of curved or irregular objects are much more difficult to calculate. Mathematicians have complex formulas for finding out the volume of a sphere or a cone.


Downloadable PDF Text- and Worksheets



  • acute = an angle of less than 90°
  • angle = the space between two lines that join each other
  • area = the space that a shape covers
  • arrowhead =the sharp, pointed end of an arrow
  • base =bottom
  • base line =here: the line at the bottom
  • basically =on the whole, normally
  • border =end, margin
  • calculate = to work out the size, length, volume etc... of something
  • certain =special
  • circular = the form and shape of a circle
  • closed =not open
  • coin = a round flat piece of metal that is used for money
  • cone = an object that is round at one end and pointed at the other
  • connection =link, relationship
  • continue = go on
  • cover =spread over
  • crossing =where two or more roads meet
  • cube = an object with six equal sides
  • deal with = is about
  • degree = unit for measuring the size of an angle
  • diameter = a straight line from one side of a circle to the other; it passes through the centre of the circle
  • distance =space between two things
  • equal = the same
  • equilateral = all sides are the same
  • face = part of the outside surface
  • familiar = well known
  • formula =numbers that form an equation or a mathematical rule
  • height =how high something is
  • honeycomb = a structure made by bees. It is made up of six-sides cells in which bees put their honey
  • irregular = a shape that is not smooth or even
  • isosceles = two of the three sides have the same length
  • join = get together, meet
  • leg =side
  • length =how longs something is
  • measure =to find out how big, long or heavy something is
  • obtuse = an angle of more than 90°
  • opposite = on the other side of
  • pipe = a tube through which water or other things can flow
  • plane =a completely flat surface
  • polyhedron = a solid shape with many sides
  • prism =an object that has the same ends and sides that go up
  • quadrilateral = a flat shape with four straight sides
  • ray =a line that goes in a certain direction
  • rectangle = a shape that has four straight sides and four 90° angles
  • segment = part of a line
  • shape = the form that something has
  • size = how big or how small something is
  • solid = hard, with a fixed shape
  • space =room, area
  • sphere = a ball shape
  • square =a shape with four equal sides and right angles
  • structure = building
  • surface =the top part of something
  • surveyor = a person who measures land
  • tile = to cover a floor with small plates
  • triangle = a flat shape with three sides and three angles
  • unit =an amount of something that is used to measure
  • volume =the amount that an object or container fills
  • width =how wide something is