Direct answer

What are the main triangle formulas?

The main triangle formulas are area $A = 1/2 bh$ , perimeter $P = a + b + c$ , and the right-triangle formula $a^{2} + b^{2} = c^{2}$ . Use base and perpendicular height for area, side lengths for perimeter, and the Pythagorean theorem only for right triangles.

Short answer

Triangle formulas at a glance

Area: $A = \frac{1}{2} \times b \times h$ Perimeter: $P = a + b + c$ Right triangle: $a^{2} + b^{2} = c^{2}$

Definition

What is a triangle?

A triangle is a 2D polygon with three sides, three vertices, and interior angles that add up to 180 degrees.

Diagram

Triangle diagram

Triangle variables: base b, height h, and side lengths a, b, and c.

Quick formula table

Triangle Formulas quick table

Formula	Equation	Variables
Area from base and height	$A = \frac{1}{2} \times b \times h$	b = base, h = height
Perimeter	$P = a + b + c$	a, b, c = side lengths
Right triangle	$a^{2} + b^{2} = c^{2}$	c = hypotenuse
Equilateral triangle area	$A = (3 /4) s^{2}$	s = side length
Heron's formula	$A = s (s - a) (s - b) (s - c)$	s = semiperimeter

Formula explanation

How to use these formulas

Use base and height when the height is known. Use side lengths for perimeter. Use the Pythagorean theorem only for right triangles.

Calculator

Triangle Calculator

Enter measurements to get the result, unit, formula, and calculation steps.

Examples

Step-by-step examples

Find triangle area

Given b = 10 cm and h = 6 cm.
Use $A = 1/2 bh$ .
$A = \frac{1}{2} \times 10 \times 6 = 30 c m^{2}$ .

Find perimeter

Given sides 4, 5, and 7 m.
Use $P = a + b + c$ .
$P = 4 + 5 + 7 = 16 m$ .

Right triangle hypotenuse

Given legs 3 and 4.
Use $a^{2} + b^{2} = c^{2}$ .
$c = 9 + 16 = 5$ .

Find height from area

Given A = 48 cm² and b = 12 cm.
Use $h = 2 A / b$ .
$h = 2 \times 48/12 = 8 c m$ .

Equilateral triangle area

Given s = 6 m.
Use $A = (3 /4) s^{2}$ .
$A = 93 \approx 15.59 m^{2}$ .

Reverse calculations

Solve backwards from known values

Task	Formula	When to use it
Find height from area	$h = 2 A / b$	Use when area and base are known.
Find base from area	$b = 2 A / h$	Use when area and height are known.
Find hypotenuse	$c = a^{2} + b^{2}$	Use for right triangles only.

Common mistakes

Using a slanted side as height. Height must be perpendicular to the base.
Using the Pythagorean theorem on a triangle that is not right.
Forgetting to check the triangle inequality before using three side lengths.

Practice

Practice problems with answers

1. Find the area when b = 14 cm and h = 5 cm.

$A = \frac{1}{2} \times 14 \times 5 = 35 c m^{2}$ .

2. Find the perimeter of sides 6 m, 8 m, and 9 m.

$P = 6 + 8 + 9 = 23 m$ .

3. A right triangle has legs 5 and 12. Find the hypotenuse.

$c = 25 + 144 = 13$ .

4. A triangle has A = 45 in² and b = 9 in. Find h.

$h = 2 A / b = 90/9 = 10 in$ .

5. An equilateral triangle has side 8 cm. Find its area.

$A = (3 /4) \times 64 = 163 \approx 27.71 c m^{2}$ .

Related formulas

FAQ

Triangle Formulas FAQ

What is the basic triangle area formula?

The basic formula is $A = \frac{1}{2} \times b \times h$ .

What is the perimeter of a triangle?

Add the three side lengths: $P = a + b + c$ .

When do I use the Pythagorean theorem?

Use it only for right triangles, where c is the hypotenuse opposite the right angle.