Triangle from Three Sides Calculator: Heron's Formula, Angles, Inradius and Circumradius

Area and angles of a triangle from its three sides.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the three side lengths in the same unit.

The calculator checks that they can form a triangle.

It gives the area, the angles, the heights and the radii of the inscribed and circumscribed circles.

Formulas used

  • s = \frac{a + b + c}{2}
  • A = \sqrt{s(s - a)(s - b)(s - c)}
  • r = \frac{A}{s},\ R = \frac{abc}{4A}

Worked example

With the default numbers in the calculator above:

  1. Half the perimeter: 6
  2. Area by Heron's formula: 6
  3. Angle A (opposite side a): 36.87 degrees
  4. Angle B (opposite side b): 53.13 degrees
  5. Angle C (opposite side c): 90 degrees
  6. Inradius (the radius of the inscribed circle): 1
  7. Circumradius (the radius of the circle through the corners): 2.5
  8. Height on side a: 4

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Side a 5; Side b 6; Side c 7.

  1. Half the perimeter: 9
  2. Area by Heron's formula: 14.6969
  3. Angle A (opposite side a): 44.415 degrees
  4. Angle B (opposite side b): 57.122 degrees
  5. Angle C (opposite side c): 78.463 degrees
  6. Inradius (the radius of the inscribed circle): 1.63299
  7. Circumradius (the radius of the circle through the corners): 3.57217
  8. Height on side a: 5.87878

Common questions

What is Heron's formula?

A formula for the area of a triangle when you know only its three sides.

When can three lengths make a triangle?

When each length is shorter than the sum of the other two.

What are the inradius and circumradius?

The inradius is the radius of the circle that fits inside the triangle touching all three sides. The circumradius is the radius of the circle through all three corners.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any geometry or trigonometry text (Heron's formula, the cosine rule).

Need a calculator we do not have?

Tell us what you would like to calculate. We read every request and build the most asked-for ones.

We store your request (and your email if you give it) to decide what to build. We do not store the numbers you type into calculators.

Embed this calculator on your own page

Copy this code into your page. The calculator runs inside the frame and links back to Drafting Central.

Need this drawn, designed or checked?

Drafting Central is a mechanical engineering and drafting business working with customers across Australia and around the world. A calculator gives you the numbers; we can turn them into drawings and designs.

Need help with a real project?

Talk to Drafting Central →

← All engineering calculators