Triangles have three sides and three interior angles. They can be classified based on side length as equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides unequal). Triangles can also be classified based on angle measures as acute (all angles less than 90 degrees), right (one angle of 90 degrees), or obtuse (one angle greater than 90 degrees). The Pythagorean theorem can be used to calculate the length of the hypotenuse (longest side) of a right triangle given the lengths of the other two sides. The area of any triangle can be calculated by multiplying the base by the height.

1. Triangles
What do the sail on a windsurfing board, a set square
and a warning road sign all have in common? They are
triangular in shape. Triangles come in many different
sizes, but they all have three sides.
A triangle is a type of polygon. Polygons are two-
dimensional (flat) shapes with straight sides. There are
different types of polygon, with different numbers of
sides. A triangle is a polygon with three sides.
The three sides of a triangle are all straight lines. These
lines meet one another at the corners of the triangle, and
create an angle at each of the three corners. The internal
angles are the three angles on the inside of the triangle.
SIDES OF A TRIANGLE
Not all triangles look the same. Triangles have different names
depending on the length of their sides.
If all of the sides of a triangle are the same length it is called
an equilateral triangle. If two of the sides of a triangle are the
same length and the third is a different length it is called an
isosceles triangle. If all three sides of a triangle are different
lengths it is called a scalene triangle.

2. In the drawing above, the red triangle is an equilateral triangle,
the blue one is an isosceles triangle and the green one is a
scalene triangle.
ANGLES OF A TRIANGLE
Triangles can also be divided up according to the sizes of their
internal angles. If all three angles are smaller than 90 degrees,
the triangle is called an acute-angled triangle. If one of the
angles is exactly 90 degrees, it is called a right-angled triangle.
If one of the angles is bigger than 90 degrees, it is called an
obtuse-angled triangle.
In the drawing above, the red triangle is an acute-angled
triangle, the blue one is a right-angled triangle and the green
one is an obtuse-angled triangle.
In a right-angled triangle, the side opposite the right angle (the
90-degree angle) is called the hypotenuse. The hypotenuse is
always the longest side of a right-angled triangle.
The size of the three internal angles of a triangle will always
add up to 180 degrees. In an equilateral triangle, where all the
sides are the same length, the internal angles are also all the
same size. Each of the three internal angles of an equilateral
triangle is 60 degrees (3 x 60 degrees = 180 degrees). An
equilateral triangle is a type of acute-angled triangle.
Because the three internal angles of a triangle add up to 180
degrees, it is possible to work out the size of one angle if you
already know the size of the other two. Just add together the

3. two angles that you know, and then take this total away from
180 degrees. This will give you the size of the third angle.
PYTHAGORAS’ THEOREM
In a right-angled triangle, it is possible to work out the length
of one of the sides if you already know the length of the other
two sides. This is done using Pythagoras’ theorem. This
theorem, or rule, states that the square of the hypotenuse is
equal to the sum of the square of the other two sides. So, if the
hypotenuse is A and the other two sides are B and C, then A2
=
B2
+ C2
.
If you have a right-angled triangle, you can work out the length
of the hypotenuse if you know that the other two sides are 3
centimetres and 4 centimetres long:
(length of the hypotenuse)2
= 32
+ 42
(length of the hypotenuse)2
= 9 + 16
(length of the hypotenuse)2
= 25
So, the length of the hypotenuse = 5
AREA OF A TRIANGLE
It is possible to calculate the area of a triangle if you know the
length of its base and height. The base is any one side of the
triangle. The height is the perpendicular distance from the base
of the triangle to the opposite corner. To find the
perpendicular, you need a line that is at right angles to the
base, or an extension of the base, and that touches the
opposite corner. The length of this line is the height of the
triangle.

4. The area of a triangle is:
DRAWING A TRIANGLE
If you know what length you want the three sides to be, it is
possible to draw a triangle using a ruler and compass. You start
by using the ruler to draw a line the length of one of the sides.
If you want a triangle with three sides that are 4 centimetres, 5
centimetres and 6 centimetres long, you could start by using a
ruler to draw a line 6 centimetres long.
Using the ruler to measure, open up the compass to the length
of one of the other sides of the triangle. Place the point of the
compass right at one end (A) of the line you have drawn and
then draw an arc with the compass.

5. Using the ruler again, open the compass to the length of the
final side of the triangle. Place the point of the compass at the
other end (B) of the line, and draw another arc that crosses the
first one (at C).
Finally, draw a line from each end (A and B) of the first line to
the point where the arcs cross (C) to complete the triangle.
.