Answer key: Heron’s Formula Worksheet
For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: 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.
Completed results table
| Setting | Perimeter | Semiperimeter s | Area |
|---|---|---|---|
| 3, 4, 5 | 12 units | 6 | 6 units² |
| 5, 5, 6 | 16 units | 8 | 12 units² |
| 5, 5, 8 | 18 units | 9 | 12 units² |
| 13, 14, 15 | 42 units | 21 | 84 units² |
| 6, 6, 6 | 18 units | 9 | 15.588 units² |
Answers
Use Heron’s formula A = √[s(s − a)(s − b)(s − c)] to find the area of the 13–14–15 triangle by hand.
s = (13 + 14 + 15) ÷ 2 = 21. A = √(21 × 8 × 7 × 6) = √7,056 = 84 square units, matching the solver.
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?
s = 6, so A = √(6 × 3 × 2 × 1) = √36 = 6 square units. The triangle has a right angle (3² + 4² = 5²), so the legs 3 and 4 are a base and its height: ½ × 3 × 4 = 6.
The 5–5–6 and 5–5–8 triangles have different shapes. Compare their areas. How can that be?
Both have an area of 12 square units. For 5–5–6, s = 8 and A = √(8 × 3 × 3 × 2) = √144 = 12. For 5–5–8, s = 9 and A = √(9 × 4 × 4 × 1) = √144 = 12. Cutting each isosceles triangle along its line of symmetry shows why: base 6 with height 4, and base 8 with height 3, both give ½ × 24 = 12.
Find the area of the equilateral triangle with sides of 6. Compare your answer with the formula A = (√3 ÷ 4) × side².
s = 9, so A = √(9 × 3 × 3 × 3) = √243 ≈ 15.588 square units. (√3 ÷ 4) × 36 ≈ 15.588 as well.
What happened with the sides 2, 3 and 6? Explain using the triangle inequality and using Heron’s formula.
2 + 3 = 5 is less than 6, so the two short sides cannot meet: no triangle exists, and the solver says so. In Heron’s formula, s = 5.5 and s − c = 5.5 − 6 = −0.5 is negative, so you would need the square root of a negative number.
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”.
s = 12, so A = √(12 × 5 × 4 × 3) = √720 ≈ 26.833 square units. Height = 2A ÷ 9 ≈ 5.963 units.