The Triangle Circle Calculator uses the three side lengths of a triangle to calculate measurements for an incircle, circumcircle, or excircle. This guide specifically explains this calculator: its exact mode selector, side fields, excircle choice, result labels, formulas, diagram, steps, copy control, and history workflow.
The calculator page is intentionally short and easy to scan. Use the page for the quick computation, then use this guide when you need to identify the correct circle type, understand the geometry behind each output, work through a numerical example, or verify the result before copying it. The calculator has no unit selector, so enter all three side lengths in the same unit and attach that unit to your answer yourself.
What does this Triangle Circle Calculator calculate?
A triangle can define several important circles. The incircle, also called the inscribed circle, lies inside the triangle and touches all three sides. The circumcircle, also called the circumscribed circle, passes through vertices A, B, and C. An excircle, also called an escribed circle, touches one side and the extensions of the other two sides.
Choose one of the calculator's exact mode labels: Incircle / inscribed circle, Circumcircle / circumscribed circle, or Excircle / escribed circle. The selected mode determines the centre label, radius formula, and one mode-specific explanation in the result.
What are Side a, Side b, and Side c?
Side a is the side opposite vertex A, Side b is the side opposite vertex B, and Side c is the side opposite vertex C. The calculator displays these exact field labels with the placeholders Opposite vertex A, Opposite vertex B, and Opposite vertex C. Enter positive numerical side lengths in the fields with IDs represented on the page as tc-a, tc-b, and tc-c.
Every pair of sides must add to more than the third side. In symbols, a + b > c, a + c > b, and b + c > a. If a value is missing, zero, negative, or non-numeric, the calculator displays Enter three positive side lengths. If the triangle inequality fails, it displays These sides do not form a valid triangle. Each pair of sides must add to more than the third side.
What is the semiperimeter?
The semiperimeter is half the triangle's perimeter. The calculator calls it s and displays it as Semiperimeter s:
s = (a + b + c) ÷ 2
The semiperimeter is used in Heron's formula and in the incircle and excircle radius formulas. It is a length, so it has the same unit as the three side inputs.
What is the triangle area?
The calculator finds the area from the three sides with Heron's formula. The displayed result is labeled Triangle area:
Area = √[s(s − a)(s − b)(s − c)]
Area is measured in square units. For example, if all side inputs are in centimetres, the triangle area is in cm2. The calculator does not convert or print a unit, so preserve the input unit when interpreting this output.
What are the circle radius, centre, area, and circumference?
After a successful calculation, the common result lines show Radius, Circle area, and Circumference. The radius is the distance from the selected circle's centre to its boundary. Circle area is the surface enclosed by that circle, and circumference is the complete distance around its boundary.
The result also shows a mode-dependent centre label and coordinates. For an incircle it shows Incenter I = (x, y). For a circumcircle it shows Circumcenter O = (x, y). For an excircle it shows A-excenter IA, B-excenter IB, or C-excenter IC, depending on the selected excircle. The coordinate system used by the diagram places A at (0, 0), B at (c, 0), and C above the horizontal AB axis; these coordinates are a construction aid, not a replacement for the numerical measurements.
Why is π used?
π is the fixed circle constant: the ratio of a circle's circumference to its diameter. Its decimal expansion begins 3.14159265359 and does not terminate. The calculator uses π for the two circular outputs:
Circle area = πr2
Circumference = 2πr
The radius is squared for area because area covers two dimensions. Circumference uses one radius multiplied by 2π because it measures one-dimensional distance around the boundary. The calculator formats displayed numerical values to a maximum of five decimal places, omitting unnecessary trailing zeroes.
Which circle mode should you select?
| Exact mode option | Geometric meaning | Distinct result detail | Useful when |
|---|---|---|---|
| Incircle / inscribed circle | The circle touches all three triangle sides internally. | Shows Incenter I and tangent lengths from A, B, and C. | You need the circle fitted inside the triangle or a radius based on the triangle's area and semiperimeter. |
| Circumcircle / circumscribed circle | The circle passes through all three vertices. | Shows Circumcenter O and confirms that all three vertices are the same distance from it. | You need the circle through A, B, and C. |
| Excircle / escribed circle | The circle touches one side and the extensions of the other two. | Shows the selected excenter and the chosen exradius calculation. | You need the A-, B-, or C-excircle rather than the circle inside the triangle. |
How does the Choose excircle control work?
The extra control appears only after selecting Excircle / escribed circle. Its label is Choose excircle, and its exact options are A-excircle, opposite side a, B-excircle, opposite side b, and C-excircle, opposite side c. Select the option whose excircle you need. The chosen letter determines both the exradius denominator and the centre label.
| Exact excircle option | Radius formula | Centre label |
|---|---|---|
| A-excircle, opposite side a | ra = Area ÷ (s − a) | A-excenter IA |
| B-excircle, opposite side b | rb = Area ÷ (s − b) | B-excenter IB |
| C-excircle, opposite side c | rc = Area ÷ (s − c) | C-excenter IC |
What formulas does the calculator use?
For all three modes, the calculator first validates the side lengths, calculates the semiperimeter, and applies Heron's formula:
s = (a + b + c) ÷ 2
Area = √[s(s − a)(s − b)(s − c)]
It then selects the radius formula:
| Selected circle | Radius formula | Additional geometric result |
|---|---|---|
| Incircle | r = Area ÷ s | Tangent lengths from A, B, and C are s − a, s − b, and s − c. |
| Circumcircle | R = abc ÷ (4 × Area) | All three vertices are R units from the circumcenter. |
| A-excircle | ra = Area ÷ (s − a) | The selected centre is IA. |
| B-excircle | rb = Area ÷ (s − b) | The selected centre is IB. |
| C-excircle | rc = Area ÷ (s − c) | The selected centre is IC. |
Finally, it applies Circle area = πr2 and Circumference = 2πr. The displayed centre coordinates are calculated from the same triangle construction used to draw the visualization.
How to use the Triangle Circle Calculator
- Open the Triangle Circle Calculator.
- Select Incircle / inscribed circle, Circumcircle / circumscribed circle, or Excircle / escribed circle.
- If you selected the excircle mode, use Choose excircle to select A-excircle, opposite side a, B-excircle, opposite side b, or C-excircle, opposite side c.
- Enter the length opposite A in the Side a field, the length opposite B in Side b, and the length opposite C in Side c. The placeholders are Opposite vertex A, Opposite vertex B, and Opposite vertex C.
- Keep all three values in the same unit. There is no unit menu on this calculator.
- Check the triangle inequality, then click Click to calculate.
- Review the semiperimeter, triangle area, centre coordinates, radius, circle area, circumference, and mode-specific detail.
Worked example: a 3-4-5 triangle in centimetres
Use Side a = 3 cm, Side b = 4 cm, and Side c = 5 cm. These values form a valid triangle because 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3. The page has no unit field, so “cm” is supplied here as the interpretation of the entered numbers.
| Calculator control | Selection or value |
|---|---|
| Mode | Incircle / inscribed circle |
| Side a | 3 (Opposite vertex A) |
| Side b | 4 (Opposite vertex B) |
| Side c | 5 (Opposite vertex C) |
| Choose excircle | Not shown for the incircle mode |
| Button | Click to calculate |
The common intermediate values are s = (3 + 4 + 5) ÷ 2 = 6 cm and Area = √[6(6 − 3)(6 − 4)(6 − 5)] = 6 cm2. The incircle radius is r = 6 ÷ 6 = 1 cm.
| Output shown by the calculator | 3-4-5 incircle result |
|---|---|
| Semiperimeter s | 6 cm |
| Triangle area | 6 cm2 |
| Incenter I | (3, 1) |
| Radius | 1 cm |
| Circle area | 3.14159 cm2 |
| Circumference | 6.28319 cm |
| Tangent lengths from A, B, and C | 3 cm, 2 cm, and 1 cm |
Compare all three principal modes
Keep the same side inputs and change only the mode. This makes the difference between an internal circle, a circle through the vertices, and an excircle clear.
| Mode selection | Centre shown | Radius | Circle area | Circumference |
|---|---|---|---|---|
| Incircle / inscribed circle | Incenter I = (3, 1) | 1 cm | 3.14159 cm2 | 6.28319 cm |
| Circumcircle / circumscribed circle | Circumcenter O = (2.5, 0) | 2.5 cm | 19.63495 cm2 | 15.70796 cm |
| Excircle / escribed circle, A option | A-excenter IA = (6, 2) | 2 cm | 12.56637 cm2 | 12.56637 cm |
| Excircle / escribed circle, B option | B-excenter IB = (−1, 3) | 3 cm | 28.27433 cm2 | 18.84956 cm |
| Excircle / escribed circle, C option | C-excenter IC = (2, −6) | 6 cm | 113.09734 cm2 | 37.69911 cm |
Try every excircle option
To test the excircle selector, choose Excircle / escribed circle, leave the three side fields at 3, 4, and 5, and run the calculation three times. Select A-excircle, opposite side a to obtain ra = 6 ÷ (6 − 3) = 2 cm. Select B-excircle, opposite side b to obtain rb = 6 ÷ (6 − 4) = 3 cm. Select C-excircle, opposite side c to obtain rc = 6 ÷ (6 − 5) = 6 cm. Each run changes the centre, circle area, circumference, and history entry.
How to use the visualization and steps
After a successful calculation, click Click to visualize it. The page draws triangle ABC, the selected circle, the centre label, a radius line, and a bottom annotation containing r and Circle area. For an incircle, the drawing helps show the circle inside the triangle; for a circumcircle, it shows the circle through the vertices; for an excircle, it shows the selected excenter and circle relative to the triangle. The page explicitly notes that The diagram is a visual aid; use the numerical results for exact measurements.
Click Click to show steps to reveal the working. The button changes to Hide steps while the text is open. The steps include the semiperimeter, Heron's area calculation, the mode-specific radius and centre calculation, circle area, and circumference. If you calculate the 3-4-5 incircle, the steps include s = (a+b+c)/2 = 6, Area = square root of [s(s−a)(s−b)(s−c)] = 6, Incircle radius: r = area/s = 1, the incenter coordinates, tangent lengths, Circle area = πr2 = 3.14159, and Circumference = 2πr = 6.28319.
How to copy, clear, and review history
Click Click to copy result after checking the result. On a successful clipboard copy, the control briefly changes to Copied. The copied text is a compact summary containing the selected circle's radius, centre coordinates, and triangle area; it is not a copy of every line in the result panel. If clipboard access is unavailable, the control can change to Select and copy result.
Click Clear to empty Side a, Side b, and Side c, remove the result and error text, hide the steps and drawing, and disable the visualization and copy controls until a new calculation succeeds. Changing the mode also resets the current calculation, so re-enter the side lengths after switching between incircle, circumcircle, and excircle.
Click Click to view history to open the Recent calculations panel. Each entry identifies the mode and radius and includes the time saved on that device. The panel contains Clear history, which removes the saved entries from the browser's local storage. If there are no entries, the page says Your recent calculations appear here on this device. Selecting a saved history entry restores its mode and side values and runs the calculation again.
Where Do We Use Triangle Circle Outputs in Real Life?
The outputs answer different design and measurement questions. A triangle may represent a roof frame, a surveyed plot, a truss, a triangular panel, or a three-point layout; the relevant circle depends on whether the circle must fit inside, pass through the corners, or relate to one side and the extensions of the others.
| Calculator output | Distinct real-life use |
|---|---|
| Semiperimeter s | Useful in surveying, fabrication, and triangle-based cost estimates because it supports Heron's area calculation and tangent-length checks. |
| Triangle area | Helps estimate the surface of a triangular parcel, panel, sail, sign, or structural face when only the three edge lengths are known. |
| Incenter I and incircle radius | Useful when placing a round feature entirely inside a triangular boundary, such as an inscribed opening, clearance region, or layout mark that must touch all three sides. |
| Tangent lengths | Help compare the three contact distances from vertices to the incircle's touch points, which is useful when laying out equal tangent segments along triangular edges. |
| Circumcenter O and circumradius | Useful for locating the centre of a circle through three surveyed points, positioning a circular path around three anchors, or checking a three-point geometric construction. |
| Excenter and exradius | Useful for external clearance, tangent-layout, and construction problems in which a circle touches one triangle side while relating to the extensions of the other two. |
| Circle area | Estimates the surface of the selected circular region, such as the area of a round cutout, pad, cover, or clearance zone. |
| Circumference | Estimates the boundary length of the selected circle for edging, trim, cable, seal, or other material that follows its perimeter. |
When this calculator is not the right tool
This calculator needs three side lengths and is intended for circles associated with a triangle. If you already know a circle's radius, diameter, circumference, or area and only want the standard circle measurements, use the Basic Circle Calculator. If you specifically need the area of a complete circle from a standard circle measurement, use the Area of a Circle Calculator.
For a circle defined by coordinate data rather than triangle side lengths, use the Circle from 3 Points Calculator. For a theorem or relationship involving angles, chords, or other named circle properties, use the Circle Theorem Calculator. For a problem about a tangent line or an angle bisector rather than an incircle, circumcircle, or excircle, use the Tangent and Angle Bisector Calculator.
Final verification workflow
- Confirm that the problem really gives three side lengths of one triangle and that Side a, Side b, and Side c are matched to the correct opposite vertices.
- Confirm that all three numbers are positive, use the same unit, and satisfy all three triangle inequalities.
- Choose the correct exact mode: Incircle / inscribed circle, Circumcircle / circumscribed circle, or Excircle / escribed circle.
- If using an excircle, confirm the exact Choose excircle selection and the side named as opposite that excircle.
- Click Click to calculate and check Semiperimeter s, Triangle area, the centre label and coordinates, Radius, Circle area, and Circumference.
- Recalculate the radius independently with the appropriate formula: Area ÷ s, abc ÷ (4 × Area), or Area ÷ (s − selected side).
- Use Click to show steps and, when helpful, Click to visualize it. Treat the diagram as a check of the construction, not as the source of exact measurements.
- Only after the values and units are confirmed, use Click to copy result. Use Click to view history for comparisons and Clear before a separate problem.
Open the Triangle Circle Calculator for the calculation, then return to this guide whenever you need to explain the selected triangle circle, reproduce its formulas, or verify the copied result.
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