Triangle Formulas (Area, Perimeter, Heron’s Formula)

A triangle is a three-sided polygon. Its area is A = ½·b·h (half the base times the height), or Heron’s formula A = √(s(s−a)(s−b)(s−c)) from the three sides, where s = (a+b+c)/2 is the semiperimeter. Its perimeter is P = a + b + c.

Triangle formulas

Area (base & height): $A = \tfrac{1}{2} b h$
Area (Heron’s): $A = \sqrt{s(s-a)(s-b)(s-c)}$
Semiperimeter: $s = \dfrac{a+b+c}{2}$
Perimeter: $P = a + b + c$

where a, b, c = the three side lengths, b = base, h = height (perpendicular to base), s = semiperimeter (a+b+c)/2, A = area

Area (base & height)

A = \tfrac{1}{2} b h

Area (Heron’s)

A = \sqrt{s(s-a)(s-b)(s-c)}

Worked example

The 3-4-5 triangle (a right triangle) has semiperimeter s = 6 and area 6 square units.

A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{6\cdot 3\cdot 2\cdot 1} = \sqrt{36} = 6

Semiperimeter

s = \dfrac{a+b+c}{2}

Perimeter

P = a + b + c

Worked example

Its perimeter is 12 units.

P = a + b + c = 3 + 4 + 5 = 12

Interactive triangle explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Side a a

Base / side b b

Side c c

Area A 6units²

Perimeter P 12units

Semiperimeter s 6units

Step-by-step substitution

P = a + b + c = 3 + 4 + 5 = 12
s = P / 2 = 12 / 2 = 6
A = √(s(s−a)(s−b)(s−c)) = √(6·3·2·1) = 6

Classroom use: triangle formulas

Who it's for: Grades 6–10 · Middle-school students learning the base-height area rule, and high-school geometry students revisiting it alongside Heron's formula.

Learning objectives

After using this page, students will be able to:

Apply A = ½bh, identifying the height as perpendicular to the chosen base.
Recognise that any of the three sides may serve as the base.
Compute a triangle's perimeter and area from side lengths.

Suggested classroom activity

Use the explorer to hold the base fixed while changing the height, then the reverse. Ask why the ½ appears at all; the composing/decomposing argument (two congruent triangles form a parallelogram) lands well right after students have seen the area stay put under those changes.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`6.G.A.1`	Find the area of triangles, special quadrilaterals, and polygons by composing or decomposing shapes.
`7.G.B.6`	Solve real-world problems involving area, volume, and surface area of 2-D and 3-D objects.
`HSG.MG.A.1`	Use geometric shapes, their measures, and their properties to describe objects.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Common mistakes

Using a slant side instead of the perpendicular height in A = ½bh.
Forgetting to take the square root at the end of Heron’s formula.

Related & next steps

Add triangle to your formula sheet →

Frequently asked questions

What is the area formula for a triangle?

The area of a triangle is A = ½·b·h, half the base times the perpendicular height. A triangle with base 10 and height 6 has area ½ × 10 × 6 = 30 square units.

What is Heron’s formula?

Heron’s formula finds a triangle’s area from its three sides: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2. It needs no height. For sides 3, 4, 5, the area is 6.

What are the 30-60-90 and 45-45-90 triangle ratios?

In a 30-60-90 triangle the sides are in ratio 1 : √3 : 2. In a 45-45-90 triangle they are 1 : 1 : √2. These special right triangles let you find every side from one.

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