Geometry Formulas Triangles

The Math Salamanders have a large bank of free printable shape clipart.

Each of the printable shape sheets is available either in color or black and white.

Using this shape clipart will help your child understand to recognize shapes and learn about the different properties that shapes have.

On the sheets with multiple shapes, we have shown the shapes in different sizes and orientations so that your child will recognize variations of the same shape, and start noticing the properties the same shapes all have.

The sheets can be used as part of a Math display, as flashcards, or as printable coloring sheets.

Here you will find our support page on triangles, including properties of triangles and triangle formulas and theorems.

At the bottom of each section you will also find a printable version of parts of this Geometry Formulas Triangles web-page.

Using these sheets will help your child to:

• understand the different properties of triangles;
• learn and use triangle formulas

Triangles have the following properties:

• All triangles have 3 straight sides, 3 corners (vertices), and 3 angles.
• All triangles have angles adding up to 180°.

Equilateral Triangles

Equilateral triangles have the following properties (in addition to the properties above for all triangles):

• Three straight sides of equal length;
• Three angles, all equal to 60°;
• Three lines of symmetry.

Isosceles Triangles

Isosceles triangles have the following properties:

• Two sides of equal length;
• Two equal angles;
• One line of symmetry.

Please note: according to Wikipedia:

This means that there is some dispute as to whether an equilateral triangle is a special case of an isosceles triangle or not!

Most modern textbooks include use the 'at least' definition for isosceles triangles.

Scalene Triangles

Scalene triangles have the following properties:

• No sides of equal length;
• No equal angles;
• No lines of symmetry.

Here is some handy information about the angles in different triangles.

Acute Triangle	Right Triangle	Obtuse Triangle

Acute triangles

Acute triangles have all acute angles (all angles are less than 90°).

It is possible to have an acute triangle which is also an isosceles triangle - these are called acute isosceles triangles.

Right triangles

Right Triangles (right-angled triangles) have one right angle (equal to 90°).

It is possible to have a right isosceles triangle - a triangle with a right angle and two equal sides.

Obtuse triangles

Obtuse triangles have one obtuse angle (angle which is greater than 90°).

It is possible to have a obtuse isosceles triangle - a triangle with an obtuse angle and two equal sides.

If you would like a printable version of the support page above on properties of triangles, use the link below.

This can be shown using the following simple demonstration.

Step 1

Draw a triangle on a piece of paper. Mark the 3 angles a, b and c.

Step 2

Cut out (or tear out) the three angles.

Step 3

Put the angles a, b and c together. They should make a straight line when placed together, which is equivalent to 180°.

The area of a triangle is equal to half of the base multiplied by the perpendicular height of the triangle.

A right triangle showing height and base.	A non-right triangle showing height and base.

Why does this work?

If we look at the rectangle below, we can see that it has twice the area of the blue triangle inside it because the two yellow triangles are exactly the same size as the two sub-divided blue triangles.

The area of this rectangle is base x height, so the area of the triangle is half of this which is \[{1 \over 2} \times base \times height \]

Pythagoras' theorem states that in a right triangle (or right-angled triangle) the sum of the squares of the two smaller sides of the triangle is equal to the square of the hypotenuse.

In other words, \[a^2 + b^2 = c^2 \]

where c is the hypotenuse (the longest side) and a and b are the other sides of the right triangle.

What does this mean?

This means that for any right triangle, the orange square (which is the square made using the longest side) has the same area as the other two blue squares added together.

Other formulae

As a result of the formula \[a^2 + b^2 = c^2 \] we can also deduce that:

• \[b^2 = c^2 - a^2 \; and\]
• \[ a^2 = c^2 - b^2 \]

Square rooting the 3 equations above gives us:

• \[ c = \sqrt {a^2 + b^2} \]
• \[ b = \sqrt {c^2 - a^2} \]
• \[ a = \sqrt {c^2 - b^2} \]

In basic triangle trigonometry, we look at three trigonometric functions: sine, cosine and tangent (sin, cos and tan) with right triangles.

Let us start by naming the parts of this triangle.

θ is one of the angles of the right triangle (but not the right angle!)

h is the hypotenuse - the longest side of the right triangle

o is opposite to the angle

a is adjacent to the angle

The three formulas to rememeber can be summed up in one acronym:

where s stands for sine, c for cosine and t for tangent.

The three formulas are written below:

The basic triangle trigonometry formulas above will only work for right triangles - they will not work for triangles without right angles.

The following formulas below will work for any triangle.

Again, if we start with the naming of the sides and angles of the triangle:

a is the side opposite angle A;

b is the side opposite angle B;

c is the side opposite angle C.

Here you will find a support page packed with a range of geometric formula.

Included in this page are formula for:

• areas and volumes of 2d and 3d shapes
• interior angles of polygons
• angles of 2d shapes
• triangle formulas and theorems

This page will provide a useful reference for anyone needing a geometric formula.

Here you will find a range of geometry cheat sheets to help you answer a range of geometry questions.

The sheets contain information about angles, types and properties of 2d and 3d shapes, and also common formulas associated with 2d and 3d shapes.

Included in this page are:

• images of common 2d and 3d shapes;
• properties of 2d and 3d shapes;
• formulas involving 2d shapes, such as area and perimeter, pythagoras' theorem, trigonometry laws, etc;
• formulas involving 3d shapes about volume and surface area.

Using the sheets in this section will help you understand and answer a range of geometry questions.

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.