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Area of Triangles

9.3 Area of a Triangle
ADDITIONAL MATHEMATICS • Form 4 • Chapter 9: Solution of Triangles

9.3 Area of a Triangle

The standard formula for the area of a triangle, \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\), requires the perpendicular height to be known. In many geometric and real-life problems, the perpendicular height is not directly given. By applying trigonometric ratios, we can determine the area of any triangle when given two side lengths and an included angle, or by using Heron's formula when all three side lengths are known.

Learning Objectives

  • Derive the formula for the area of a triangle: \(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\).
  • Determine the area of a triangle when two side lengths and an included angle are given.
  • Determine the area of a triangle using Heron's formula when the lengths of all three sides are given.
  • Solve real-world and geometric problems involving the areas of triangles.

Summary of Triangle Area Formulae

Choose the correct formula based on the information given in the problem.

Given 2 sides and an included angle

Use the sine area formula:

\(\text{Area} = \dfrac{1}{2}ab \sin C\)

The angle \(C\) must be the included angle between sides \(a\) and \(b\).

Given all 3 sides

Use Heron's formula in two steps:

Step 1: \(s = \dfrac{a+b+c}{2}\)

Step 2: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

How to Choose the Correct Formula

1

Identify the given information in the triangle.

2

If two sides and the included angle are known, use \(\text{Area} = \frac{1}{2}ab \sin C\).

3

If all three sides are known, calculate \(s = \frac{a+b+c}{2}\).

4

Apply Heron's formula: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\).

Short Explanation

Use the sine area formula or Heron's formula depending on the given information.

1. Deriving the Sine Area Formula

From basic geometry, \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). When the perpendicular height \(h\) is not given, use trigonometric ratios:

\(h = a \sin B \quad \text{or} \quad h = b \sin A\)

Substituting \(h\) into the basic formula gives:

\(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\)

Condition: The angle used must be the included angle between the two given side lengths.

2. Finding Unknown Values from the Area

Given the area and two sides, the included angle \(\theta\) can be found by rearranging:

\(\sin \theta = \dfrac{2 \times \text{Area}}{a \times b}\)

Given the area, one side, and an included angle, the other side length can be calculated algebraically.

3. Heron's Formula

When only the lengths of all three sides \((a, b, c)\) are given, Heron's formula is used without needing to calculate an angle or the perpendicular height.

Historical background: Named after Hero of Alexandria (Heron), a Greek mathematician who recorded it in his work Metrica.

Step 1: Calculate the semi-perimeter, \(s = \frac{a + b + c}{2}\).

Step 2: Substitute into Heron's formula: \(\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}\).

4. Solving Problems Involving Areas of Triangles

Combine Heron's formula, the sine area formula, the Cosine Rule, and the Sine Rule to solve composite geometric shapes, land plots, and 3D solids.

For multi-part figures, find the total area of the larger triangle or composite region, then subtract unshaded or non-target regions to find the required area.

Sine Area Formula

\(\text{Area} = \dfrac{1}{2}ab \sin C\)

Requires 2 sides and 1 included angle.

Heron's Formula

\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

Requires all 3 side lengths.

Try to Answer First

Answer in your mind, then press "Check Answer".

1

Find the area of \(\triangle JKL\) where \(JK = 6\text{ cm}\), \(KL = 4.4\text{ cm}\), and \(\angle JKL = 69^\circ\).

2

A triangle has side lengths of \(5.5\text{ cm}\), \(9.6\text{ cm}\), and \(12.3\text{ cm}\). Calculate its semi-perimeter \(s\) and its area.

3

The area of \(\triangle DEF\) is \(50\text{ cm}^2\). Given \(DE = 8.6\text{ cm}\), \(DF = 14.2\text{ cm}\), and \(\angle EDF = \theta\), find the value of \(\theta\).

Common Mistakes

  • !Using a non-included angle in the formula \(\text{Area} = \frac{1}{2}ab \sin \theta\). The angle must strictly lie between the two chosen sides.
  • !Forgetting to take the square root at the final step of Heron's formula.
  • !Confusing the semi-perimeter \(s = \frac{a+b+c}{2}\) with the full perimeter \(P = a + b + c\).
  • !Rounding off intermediate values too early, which causes inaccuracies in the final area.

Concept Misunderstandings

Misunderstanding

Assuming the formula \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) can use any slant side as height.

Correct Concept

The height must be perpendicular to the base. If the perpendicular height is unknown, use \(\frac{1}{2}ab \sin C\) or Heron's formula.

Misunderstanding

Assuming Heron's formula only works for isosceles or equilateral triangles.

Correct Concept

Heron's formula works for any triangle (scalene, isosceles, equilateral, acute, or obtuse) as long as all three side lengths are known.

Summary

  • Sine Area Formula: \(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\) (requires 2 sides and 1 included angle).
  • Semi-perimeter: \(s = \frac{a + b + c}{2}\).
  • Heron's Formula: \(\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}\) (requires all 3 side lengths).

Short Activity

Objective: Test your understanding of triangle area formulae, semi-perimeter and Heron's formula.

A. Objective Quiz

1 In \(\triangle PQR\), \(PQ = 7\text{ cm}\), \(PR = 10\text{ cm}\), and the included angle \(\angle QPR = 125^\circ\). What is the area of \(\triangle PQR\)?

2 A triangle has sides of length \(8\text{ m}\), \(8\text{ m}\), and \(6\text{ m}\). What is its semi-perimeter, \(s\)?

B. Fill in the Blanks

3 To find the area of a triangle given two side lengths and an angle, the angle must be the ________ angle between the two given sides.

4 In Heron's formula, the variable \(s\) represents the ________ of the triangle, calculated as \(\frac{a+b+c}{2}\).

C. Matching / Drag and Drop

Drag each formula or term to the correct definition or expression. On a phone, tap a term first, then tap the matching box.

Formula / Term
Area of triangle with included angle \(C\)
Semi-perimeter (\(s\))
Heron's formula
Height \(h\) in terms of angle \(B\)
Basic triangle area formula
Metrica
Definition / Expression
5\(\frac{1}{2}ab \sin C\)
6\(\frac{a + b + c}{2}\)
7\(\sqrt{s(s - a)(s - b)(s - c)}\)
8\(a \sin B\)
9\(\frac{1}{2} \times \text{base} \times \text{height}\)
10Book written by Hero of Alexandria containing Heron's formula
Answer key: 1 A; 2 C; 3 included; 4 semi perimeter; 5 \(\frac{1}{2}ab \sin C\); 6 \(\frac{a+b+c}{2}\); 7 \(\sqrt{s(s-a)(s-b)(s-c)}\); 8 \(a \sin B\); 9 \(\frac{1}{2} \times \text{base} \times \text{height}\); 10 Book written by Hero of Alexandria containing Heron's formula.

Keywords

Included angle Semi-perimeter Heron's formula Perpendicular height Sine area formula
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.

Area of Triangles

9.3 Area of a Triangle
ADDITIONAL MATHEMATICS • Form 4 • Chapter 9: Solution of Triangles

9.3 Area of a Triangle

The standard formula for the area of a triangle, \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\), requires the perpendicular height to be known. In many geometric and real-life problems, the perpendicular height is not directly given. By applying trigonometric ratios, we can determine the area of any triangle when given two side lengths and an included angle, or by using Heron's formula when all three side lengths are known.

Learning Objectives

  • Derive the formula for the area of a triangle: \(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\).
  • Determine the area of a triangle when two side lengths and an included angle are given.
  • Determine the area of a triangle using Heron's formula when the lengths of all three sides are given.
  • Solve real-world and geometric problems involving the areas of triangles.

Summary of Triangle Area Formulae

Choose the correct formula based on the information given in the problem.

Given 2 sides and an included angle

Use the sine area formula:

\(\text{Area} = \dfrac{1}{2}ab \sin C\)

The angle \(C\) must be the included angle between sides \(a\) and \(b\).

Given all 3 sides

Use Heron's formula in two steps:

Step 1: \(s = \dfrac{a+b+c}{2}\)

Step 2: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

How to Choose the Correct Formula

1

Identify the given information in the triangle.

2

If two sides and the included angle are known, use \(\text{Area} = \frac{1}{2}ab \sin C\).

3

If all three sides are known, calculate \(s = \frac{a+b+c}{2}\).

4

Apply Heron's formula: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\).

Short Explanation

Use the sine area formula or Heron's formula depending on the given information.

1. Deriving the Sine Area Formula

From basic geometry, \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). When the perpendicular height \(h\) is not given, use trigonometric ratios:

\(h = a \sin B \quad \text{or} \quad h = b \sin A\)

Substituting \(h\) into the basic formula gives:

\(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\)

Condition: The angle used must be the included angle between the two given side lengths.

2. Finding Unknown Values from the Area

Given the area and two sides, the included angle \(\theta\) can be found by rearranging:

\(\sin \theta = \dfrac{2 \times \text{Area}}{a \times b}\)

Given the area, one side, and an included angle, the other side length can be calculated algebraically.

3. Heron's Formula

When only the lengths of all three sides \((a, b, c)\) are given, Heron's formula is used without needing to calculate an angle or the perpendicular height.

Historical background: Named after Hero of Alexandria (Heron), a Greek mathematician who recorded it in his work Metrica.

Step 1: Calculate the semi-perimeter, \(s = \frac{a + b + c}{2}\).

Step 2: Substitute into Heron's formula: \(\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}\).

4. Solving Problems Involving Areas of Triangles

Combine Heron's formula, the sine area formula, the Cosine Rule, and the Sine Rule to solve composite geometric shapes, land plots, and 3D solids.

For multi-part figures, find the total area of the larger triangle or composite region, then subtract unshaded or non-target regions to find the required area.

Sine Area Formula

\(\text{Area} = \dfrac{1}{2}ab \sin C\)

Requires 2 sides and 1 included angle.

Heron's Formula

\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

Requires all 3 side lengths.

Try to Answer First

Answer in your mind, then press "Check Answer".

1

Find the area of \(\triangle JKL\) where \(JK = 6\text{ cm}\), \(KL = 4.4\text{ cm}\), and \(\angle JKL = 69^\circ\).

2

A triangle has side lengths of \(5.5\text{ cm}\), \(9.6\text{ cm}\), and \(12.3\text{ cm}\). Calculate its semi-perimeter \(s\) and its area.

3

The area of \(\triangle DEF\) is \(50\text{ cm}^2\). Given \(DE = 8.6\text{ cm}\), \(DF = 14.2\text{ cm}\), and \(\angle EDF = \theta\), find the value of \(\theta\).

Common Mistakes

  • !Using a non-included angle in the formula \(\text{Area} = \frac{1}{2}ab \sin \theta\). The angle must strictly lie between the two chosen sides.
  • !Forgetting to take the square root at the final step of Heron's formula.
  • !Confusing the semi-perimeter \(s = \frac{a+b+c}{2}\) with the full perimeter \(P = a + b + c\).
  • !Rounding off intermediate values too early, which causes inaccuracies in the final area.

Concept Misunderstandings

Misunderstanding

Assuming the formula \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) can use any slant side as height.

Correct Concept

The height must be perpendicular to the base. If the perpendicular height is unknown, use \(\frac{1}{2}ab \sin C\) or Heron's formula.

Misunderstanding

Assuming Heron's formula only works for isosceles or equilateral triangles.

Correct Concept

Heron's formula works for any triangle (scalene, isosceles, equilateral, acute, or obtuse) as long as all three side lengths are known.

Summary

  • Sine Area Formula: \(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\) (requires 2 sides and 1 included angle).
  • Semi-perimeter: \(s = \frac{a + b + c}{2}\).
  • Heron's Formula: \(\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}\) (requires all 3 side lengths).

Short Activity

Objective: Test your understanding of triangle area formulae, semi-perimeter and Heron's formula.

A. Objective Quiz

1 In \(\triangle PQR\), \(PQ = 7\text{ cm}\), \(PR = 10\text{ cm}\), and the included angle \(\angle QPR = 125^\circ\). What is the area of \(\triangle PQR\)?

2 A triangle has sides of length \(8\text{ m}\), \(8\text{ m}\), and \(6\text{ m}\). What is its semi-perimeter, \(s\)?

B. Fill in the Blanks

3 To find the area of a triangle given two side lengths and an angle, the angle must be the ________ angle between the two given sides.

4 In Heron's formula, the variable \(s\) represents the ________ of the triangle, calculated as \(\frac{a+b+c}{2}\).

C. Matching / Drag and Drop

Drag each formula or term to the correct definition or expression. On a phone, tap a term first, then tap the matching box.

Formula / Term
Area of triangle with included angle \(C\)
Semi-perimeter (\(s\))
Heron's formula
Height \(h\) in terms of angle \(B\)
Basic triangle area formula
Metrica
Definition / Expression
5\(\frac{1}{2}ab \sin C\)
6\(\frac{a + b + c}{2}\)
7\(\sqrt{s(s - a)(s - b)(s - c)}\)
8\(a \sin B\)
9\(\frac{1}{2} \times \text{base} \times \text{height}\)
10Book written by Hero of Alexandria containing Heron's formula
Answer key: 1 A; 2 C; 3 included; 4 semi perimeter; 5 \(\frac{1}{2}ab \sin C\); 6 \(\frac{a+b+c}{2}\); 7 \(\sqrt{s(s-a)(s-b)(s-c)}\); 8 \(a \sin B\); 9 \(\frac{1}{2} \times \text{base} \times \text{height}\); 10 Book written by Hero of Alexandria containing Heron's formula.

Keywords

Included angle Semi-perimeter Heron's formula Perpendicular height Sine area formula
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.
Chapter : Solution of Triangles
Topic : Solve problems involving areas of triangles
Form 4 Additional Mathematics
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