Area of a Triangle Using Two Sides and an Included Angle

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

1. Derive the formula Area = 1/2 a b sin C using basic trigonometry for triangle height.
2. Calculate the area of any triangle when given two side lengths and the included angle.
3. Perform inverse calculations to find unknown angles or side lengths when the triangle area is known.

Key vocabulary

Included Angle

The interior angle located directly between two specified adjacent side lengths of a polygon.

In triangle ABC, angle C is the included angle between sides a and b.

Sine Area Formula

A trigonometric formula calculating triangle area using two side lengths and their included angle.

We applied the sine area formula to calculate the flower bed's total surface area.

Inverse Calculation

Working backwards from a known result like area to determine an unknown side or angle.

We performed an inverse calculation to determine the acute included angle from given area.

Ambiguous Case

A situation where a sine value corresponds to both an acute and obtuse triangle angle.

Because sine is positive up to 180 degrees, the ambiguous case offers two angle options.

Lesson preview

Slide 1 · Deriving and Applying the Sine Formula for Triangle AreaOpening · 18 min

In primary school, you calculated triangle area using half of base times height. When perpendicular height is not directly given, trigonometry allows us to express height using an angle. In any triangle with two sides $a$ and $b$ and included angle $C$, the height relative to side $a$ is $h=b\sin C$. Substituting this height into the standard formula gives $\text{Area}=\frac{1}{2}ab\sin C$, which works for any triangle.

• The perpendicular height of a triangle can be expressed as $h=b\sin C$ using basic trigonometry.
• The area of any triangle equals half the product of two sides and the sine of their included angle.
• The included angle must be the angle located directly between the two given side lengths.
• The formula $\text{Area}=\frac{1}{2}ab\sin C$ applies equally to acute and obtuse triangles.

Worked example. Identify known values: side $a=14\text{ m}$, side $b=18\text{ m}$, and included angle $C={62}^{\circ }$. State the sine area formula: $\text{Area}=\frac{1}{2}ab\sin C$. Substitute the values: $\text{Area}=\frac{1}{2}\times 14\times 18\times \sin ({62}^{\circ })$. Calculate $\sin ({62}^{\circ })\approx 0.88295$ and evaluate: $\text{Area}=126\times 0.88295=\mathrm{111.2517...}$ Round the final answer to two decimal places: $111.25{\text{ m}}^2$. Answer: $111.25{\text{ m}}^2$.

Check question: Triangle $XYZ$ has side lengths $XY=10\text{ cm}$ and $YZ=15\text{ cm}$, with an included angle of $\mathrm{\angle }XYZ={40}^{\circ }$. Find its area to two decimal places.

1. $48.21{\text{ cm}}^2$
2. $57.45{\text{ cm}}^2$
3. $75.00{\text{ cm}}^2$
4. $96.42{\text{ cm}}^2$

Show answer

$48.21{\text{ cm}}^2$. Using the sine area formula: $\text{Area}=\frac{1}{2}ab\sin C$ $\text{Area}=\frac{1}{2}\times 10\times 15\times \sin ({40}^{\circ })$ $\text{Area}=75\times \mathrm{0.64278...}$ $\text{Area}=\mathrm{48.209...}\approx 48.21{\text{ cm}}^2$.

Slide 2 · Solving Inverse Problems and Finding Unknown DimensionsPractice · 22 min

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

See the school projectorWhatsApp us

All 12 lessons in Area of Polygons

1. 1Deriving and Applying the Formula for Area of a TriangleDerive the formula for the area of a triangle given two sides and an included angle
2. 2Area of a Triangle Using Two Sides and an Included AngleWork out the area of a triangle given two sides and an included angle
3. 3Determining Area of a Triangle using Heron's FormulaDetermine the area of a triangle using Heron's formula
4. 4Area of a Triangle Using Heron's FormulaDetermine the area of a triangle using Heron's formula
5. 5Determining Area of Quadrilaterals in Different Situations: Parallelograms and TrapeziumsDetermine the area of quadrilaterals in different situations
6. 6Determining Area of Quadrilaterals in Different Situations: Kites and Complex FiguresDetermine the area of quadrilaterals in different situations
7. 7Area of Regular Polygons: Heptagons and OctagonsWork out the area of regular heptagon and octagon
8. 8Area of Regular Polygons: Heptagons and Octagons (Lesson 2)Work out the area of regular heptagon and octagon
9. 9Determining Area of Irregular Polygons by Shape SubdivisionDetermine the area of irregular polygons in different situations
10. 10Determining Area of Irregular Polygons in Real SituationsDetermine the area of irregular polygons in different situations
11. 11Exploring Area of Polygons in Real-Life SituationsExplore the area of polygons as used in real-life situations
12. 12Applying Polygon Area Calculations in Real-Life SituationsExplore the area of polygons as used in real-life situations