Triangle Perimeter using Trigonometry

\[\begin{aligned} P &= a+b+c \\[10pt] \text{where} \\[10pt] a &= \frac{c\sin(A)}{\sin(C)} \\[10pt] b &= \frac{c\sin(B)}{\sin(C)} \end{aligned}\]

Variables

Ptriangle perimeter (m)

atriangle side a (m)

btriangle side b (m)

cknown triangle side (m)

Aangle opposite side a (rad or °)

Bangle opposite side b (rad or °)

Cangle opposite side c (rad or °)

Description

What is this formula?

This trigonometric formula calculates the perimeter of a triangle by first determining unknown sides using the Law of Sines.

The perimeter is the sum of all three side lengths. When not all sides are known directly, trigonometric relationships can be used to calculate the missing values.

When to use it

Use this formula when:

- One side and multiple angles are known

- The perimeter of an oblique triangle is required

- Direct side measurements are unavailable

- Solving surveying or navigation problems

- Working with geometric analysis

This method is useful for non-right triangles.

Example

If:

c = 12 m

A = 45°

B = 60°

C = 75°

Then:

a = (12*sin(45°))/sin(75°)

a ≈ 8.78 m

b = (12*sin(60°))/sin(75°)

b ≈ 10.76 m

P = 8.78 + 10.76 + 12

P ≈ 31.54 m

Applications

- Geometry and trigonometry

- Civil engineering

- Surveying and mapping

- Navigation calculations

- Architecture and design

- Educational mathematics