Heron's formula
When three side lengths are known but no perpendicular height is given, use Heron's formula. Find the perimeter, halve it to get s, then use the square root
\[s=\frac{a+b+c}{2},\quad A=\sqrt{s(s-a)(s-b)(s-c)}\]
- Find the perimeter Add the three side lengths before finding the semi-perimeter \(\displaystyle P=5+6+7=18\text{ cm}\)
- Find the semi-perimeter Divide the perimeter by 2 to find s \(\displaystyle s=\frac{P}{2}=\frac{18}{2}=9\text{ cm}\)
- Use the Heron product Substitute s and the three side lengths into the product under the root \(\displaystyle A=\sqrt{s(s-a)(s-b)(s-c)}\)
- Round the final area Keep the full product until the final square-root calculation \(\displaystyle A=\sqrt{216}=6\sqrt{6}\approx14.70\text{ cm}^{2}\)
Worked example
A triangle has side lengths 5 cm, 6 cm and 7 cm. Find its area to 2 decimal places
\[a=5\text{ cm},\quad b=6\text{ cm},\quad c=7\text{ cm}\]
- \(\displaystyle P=5+6+7=18\text{ cm}\)Add the three side lengths to find the perimeter
- \(\displaystyle s=\frac{18}{2}=9\text{ cm}\)The semi-perimeter is half the perimeter
- \(\displaystyle A=\sqrt{9(9-5)(9-6)(9-7)}=\sqrt{216}\)Substitute the semi-perimeter and side lengths into Heron's formula
- \(\displaystyle A=6\sqrt{6}\approx14.70\text{ cm}^{2}\)The triangle area is approximately 14.70 square centimetres
Common mistakes
Using the full perimeter
✕\(\displaystyle s=5+5+8=18,\quad A=\sqrt{18(13)(13)(10)}\)✓\(\displaystyle s=\frac{5+5+8}{2}=9,\quad A=\sqrt{9(4)(4)(1)}=12\text{ cm}^{2}\)
Heron's s is half the perimeter, so using the full perimeter makes every factor incorrectSkipping the square root
✕\(\displaystyle A=9(9-5)(9-6)(9-7)=216\text{ cm}^{2}\)✓\(\displaystyle A=\sqrt{9(9-5)(9-6)(9-7)}=\sqrt{216}\approx14.70\text{ cm}^{2}\)
The product gives the number under the root, so the square root must be taken to find the areaRounding the semi-perimeter early (sides 4 cm, 5 cm, 6 cm)
✕\(\displaystyle s\approx8,\quad A=\sqrt{8(4)(3)(2)}=\sqrt{192}\approx13.86\text{ cm}^{2}\)✓\(\displaystyle s=7.5,\quad A=\sqrt{7.5(3.5)(2.5)(1.5)}=\sqrt{98.4375}\approx9.92\text{ cm}^{2}\)
Rounding an intermediate value changes the product, so retain the exact semi-perimeter until the final answerDifficulty levels
introUse Heron's formula with three side lengths that give an exact whole-number area and find the perimeter first
standardFind an area to a stated number of decimal places and identify the role of a Heron quantity
challengeUse a half-integer semi-perimeter and retain precision while finding an irrational area
examChoose the correct Heron setup or area from several possible calculations
Questions about this topic
When should I use Heron's formula?+
Use it when the three side lengths are known and a perpendicular height is not supplied
Why is the semi-perimeter used?+
Heron's formula is defined with s equal to half the perimeter, so the side lengths are combined with that value
Why is the square root taken at the end?+
The Heron product is the quantity under the root, and taking its square root gives the area
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity
In this chapter
Round to decimal places and significant figuresScientific notationMultiply and divide in scientific notationFind a side of a right-angled trianglePythagoras in three dimensions10C Perimeter and area: Convert measurement unitsPerimeter and area of standard shapesArea of a triangle from three sidesCircumference and area of a circlePerimeter and area of composite shapesArc length and sector area10E Volume and capacity: Convert measurement unitsVolume of prisms and cylinders10E Volume and capacity: Volume of cones, pyramids and spheresCapacity of a container10F Volume of pyramids: Volume of cones, pyramids and spheresSurface area of solidsSimilar figures and scale factorsTests for similar trianglesSimilar solids and scaling volume