Triangles: Types, Properties, and Formulas

Explore different types of triangles in geometry, their properties, and formulas used to calculate angles, perimeter, and area.

Angles of a Triangle

A triangle is a closed two-dimensional figure with three sides and three interior angles: \( \angle A, \angle B, \angle C \).

Note:

The sum of all interior angles of a triangle is always \( 180^\circ \):

\( \angle A + \angle B + \angle C = 180^\circ \)

Perimeter of a Triangle

The perimeter of a triangle is given by:

\( P = a + b + c \)

Area of a Triangle

The area \( A \) of a triangle can be found using different formulas:

• Base and height formula: \( A = \dfrac{1}{2} b h \).
• Heron's formula (when all sides are known): \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \dfrac{a+b+c}{2} \) is the semi-perimeter.
• Using sine: \( A = \dfrac{1}{2} a b \sin C \).

Isosceles Triangle

An isosceles triangle has two equal sides and two equal angles.

• Two equal sides: \( a = b \).
• Two equal angles: \( \angle A = \angle B \).
• The altitude, median, and angle bisector from the third vertex coincide.

Equilateral Triangle

An equilateral triangle has all three sides and angles equal.

• All sides are equal: \( a = b = c \).
• All angles are equal: \( \angle A = \angle B = \angle C = 60^\circ \).

Right Triangle & Pythagorean Theorem

A right triangle has one right angle (\( 90^\circ \)). The longest side opposite the right angle is the hypotenuse.

Pythagorean Theorem:

\( c^2 = a^2 + b^2 \)

Sine Law

The Sine Law helps solve non-right triangles:

\( \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \)

Cosine Law

The Cosine Law is useful for solving non-right triangles:

\( a^2 = b^2 + c^2 - 2bc \cos A \)

More Resources

• Triangle Problems
• Questions on Angles
• Geometry Tutorials