Heron’s Formula Worksheet
Goal: Find the area of a triangle from its three sides with Heron’s formula, and use the triangle inequality to decide when three lengths make a triangle.
What to do
- Open the triangle solver and choose “Three sides (SSS)”.
- For each triangle in the table, enter the sides in “Side a”, “Side b” and “Side c”, press “Solve triangle” and record the perimeter and area.
- Work out the semiperimeter s = perimeter ÷ 2 yourself and add it to the table.
- Finally, try the sides 2, 3 and 6, and read the message the solver gives.
Record your results
| Setting | Perimeter | Semiperimeter s | Area |
|---|---|---|---|
| 3, 4, 5 | |||
| 5, 5, 6 | |||
| 5, 5, 8 | |||
| 13, 14, 15 | |||
| 6, 6, 6 |
Questions
Use Heron’s formula A = √[s(s − a)(s − b)(s − c)] to find the area of the 13–14–15 triangle by hand.
Find the area of the 3–4–5 triangle two ways: with Heron’s formula and with A = ½ × base × height. Why does the second way work here?
The 5–5–6 and 5–5–8 triangles have different shapes. Compare their areas. How can that be?
Find the area of the equilateral triangle with sides of 6. Compare your answer with the formula A = (√3 ÷ 4) × side².
What happened with the sides 2, 3 and 6? Explain using the triangle inequality and using Heron’s formula.
Challenge: a triangle has sides 7, 8 and 9. Find its area with Heron’s formula, then use A = ½ × base × height to find the height to the side of length 9. Check with the solver’s “Height to c”.
Free from Geometry Lab · geometrylab.net
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