Definition of an isosceles triangle
An isosceles triangle has two equal sides and one unequal side.
The definition of an isosceles triangle has changed slightly over the years. Originally, Euclid, who was sometimes referred to as Euclid of Alexandria and is regarded as the Father of Geometry, defined an isosceles triangle like this: a triangle with two equal sides exactly. However, the terminology has changed slightly over time, while the sentiment remains the same. The more modern definition of an isosceles is a triangle with at least two equal sides. This change is seemingly minor, but it means that, by current standards, equilateral triangles, which have three equal sides, are a particular case of isosceles triangles.
In the case of an isosceles triangle with two equal sides, the equal sides are referred to as the legs of the triangle, and the third side as the base.
Name Origin: Isosceles comes from the Greek roots ‘isos’, meaning equal, and ‘skelos,’ meaning leg.
As well to the isosceles triangle, there are three other types of triangles:
- scalene triangles;
- right-angled triangles;
- equilateral triangles.
All triangles have three sides, but each type has special properties that make it unique. So why not quiz your class to test their memory of the different properties of each kind of triangle?
Properties of an isosceles triangle
- A triangle with two equal sides. These are known as the legs.
- One unequal side is known as the base.
- Two equal angles are opposite sides of equal length.
- All angles are acute (less than 90º).
- The sum of the angles is 180º.
Angles in an isosceles triangle
All angles in an isosceles triangle are smaller than 90º. It means that they are acute angles.
The sum of the three angles in an isosceles triangle is always 180º. So it means we can find out the third angle of the triangle if the other two angles are known.
Let’s look at how you can find the size of the angles in an isosceles triangle.
Finding the size of angles in isosceles triangles
Some maths problems might require children to find the size of missing angles in triangles. For example, it is relatively straightforward for an isosceles triangle since we all know that two angles are the same size.
Here are some tips for finding the size of angles in the isosceles triangle:
- The two angles opposite the two matching sides are the same size. Therefore, if you know one of the angles’ sizes, you see the other’s. For example, if one of them is 46º, you know the matching angle is 46º.
- All of the angles add up to 180º. If you know the size of two of the angles, you can add them together and subtract the sum from 180º to find the other angle. For example, if one angle is 88º and the other is 46º, they add up to 134º. 180º – 134º = 46º.
- If you know the size of one of the equal angles you can subtract the sum of both from 180º to find the size of the unequal angle. The sum for this would be 180º – (46º x 2) = 88º.
Isosceles triangle lines of symmetry
An isosceles triangle has one line of symmetry.
By definition, an isosceles triangle can only have one line of symmetry. So it is because a triangle can only be an isosceles triangle if it has two equal sides.
The line of symmetry on an isosceles triangle can be drawn by connecting the vertex between equal sides and the center of the opposite side.
Other isosceles triangle properties:
- It has rotational symmetry of order one.
- It has two equal angles
There are three other types of triangles that have different properties:
| Triangle | Line(s) of symmetry | Other properties |
| Scalene | A scalene triangle has no lines of symmetry. | It has rotational symmetry of order one. It also has no equal angles or sides. |
| Equilateral | An equilateral triangle has three lines of symmetry. | It has rotational symmetry of order three. It has three equal angles, all at 60°. It also has three equal sides. |
| Right-angled | A right-angled triangle has no lines of symmetry. | It has a rotational symmetry of order one and one angle of 90°. |
Different types of isosceles triangles
There are three vital types of isosceles triangles, i.e., triangles with two equal sides, that we will be discussing. They are as follows:
- Isosceles acute triangle: In this type of isosceles triangle, all three angles are less than 90°, and a minimum of two of its angles have an equal measurement.
- Isosceles right triangle: In this type of isosceles triangle, two of the legs, and their corresponding angles, are of equal measure.
- Isosceles obtuse triangle: In this type of isosceles triangle, one of the three angles is obtuse, meaning it measures between 90° and 180°. The other two angles in this triangle are acute and equal in measurement.
The area of an isosceles triangle
There is a specific formula that you can use to find the area of an isosceles triangle. This formula is as follows:
A = ½ × b × h
Key:
- A = area
- B = base
- H = height
The perimeter of an isosceles triangle
There is a specific formula that you can use to find the perimeter of an isosceles triangle. This formula is as follows:
P = 2a + b
Key:
- P = perimeter.
- A = the length of the equal sides of the isosceles triangle (also known as the legs).
- B = the length of the third, unequal side of the triangle.
