Grade 10 · Core Mathematics · Lesson 4 of 12

Area of a Triangle Using Heron's Formula

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of Polygons. In it, learners state and apply Heron's formula to find the area of triangles with known side lengths.

Strand Measurements and GeometrySub-strand Area of PolygonsLesson code 10CM-25-L440 minutes · 2 slides

KICD learning outcome (c)

Determine the area of a triangle using Heron's formula.

By the end of this lesson, learners should be able to

State and apply Heron's formula to find the area of triangles with known side lengths.
Calculate the semi-perimeter of a triangle and substitute values correctly into the area expression.

Key vocabulary

Heron's formula: A mathematical rule for finding a triangle's area using its three side lengths.; We applied Heron's formula to find the area of a triangular plot of land.
Semi-perimeter: Half the perimeter of a triangle, calculated by taking half of all summed sides.; The semi-perimeter $s$ is calculated before determining the terms under the square root.

Lesson preview

Slide 1 · Heron's Formula for Area of a TriangleModel · 18 min

Figure 1: A triangle with side lengths of 7 metres, 9 metres, and 12 metres alongside a step-by-step calculation of its area using Heron's formula.

When the perpendicular height of a triangle is unknown, we can find its area directly from its side lengths using Heron's formula. Surveyors use this reliable method to confidently establish real-life land boundaries. We first calculate the semi-perimeter, denoted as $s = \frac{a + b + c}{2}$ , with side lengths denoted by $a$ , $b$ , and $c$ . The area is then given by $A = \sqrt{s \times (s - a) \times (s - b) \times (s - c)}$ . This method provides an accurate geometric solution for any valid triangle. Study Figure 1 to see a step-by-step example of this method.

Heron's formula calculates triangle area directly from three given side lengths $a$ , $b$ , and $c$ .
The semi-perimeter $s$ equals half the perimeter, computed as $s = \frac{a + b + c}{2}$ .
Subtract each side length from $s$ to obtain three difference terms: $(s - a)$ , $(s - b)$ , and $(s - c)$ .
Multiply $s$ by all three difference factors before evaluating the final square root value.

Worked example. Calculate semi-perimeter:

s = \frac{7 + 9 + 12}{2} = \frac{28}{2} = 14 m

. Calculate differences:

s - a = 14 - 7 = 7

s - b = 14 - 9 = 5

, and

s - c = 14 - 12 = 2

. Substitute into formula:

Area = \sqrt{14 \times 7 \times 5 \times 2} = \sqrt{980}

. Evaluate root:

Area \approx 31.30 m^{2}

. Answer: $31.30 m^{2}$ .

Check question: A triangular flower bed has sides measuring $6 m$ , $8 m$ , and $12 m$ . Find its area correct to two decimal places using Heron's formula.

$5.92 m^{2}$
$21.33 m^{2}$
$361.99 m^{2}$
$455.00 m^{2}$

Show answer

$21.33 m^{2}$ . The semi-perimeter is $s = \frac{6 + 8 + 12}{2} = 13 m$ . Substituting into Heron's formula gives $Area = \sqrt{13 (13 - 6) (13 - 8) (13 - 12)} = \sqrt{13 \times 7 \times 5 \times 1} = \sqrt{455} \approx 21.33 m^{2}$ .

Slide 2 · Practising Heron's Formula in Real ContextsPractice · 22 min

Figure 2: Two triangles with their side lengths and the complete step-by-step calculations required to find their areas using Heron's formula.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 11 lessons in this sub-strand

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