A triangular farm plot has two sides measuring 12 m and 7 m with an included angle of 30°. What is the height perpendicular to the 12 m base, and what is the area of the plot?

Height = 3.5 m, Area = 21 m^2. The perpendicular height to the 12 m base is 7 30° = 7 × 0.5 = 3.5 m. Using the formula Area = (1)/(2) × base × height, the area is (1)/(2) × 12 × 3.5 = 21 m^2.

Grade 10 · Core Mathematics · Lesson 1 of 12

Deriving and Applying the Formula for Area of a Triangle

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of Polygons. In it, learners derive the trigonometric area formula for a triangle given two side lengths and an included angle.

Strand Measurements and GeometrySub-strand Area of PolygonsLesson code 10CM-25-L140 minutes · 2 slides

KICD learning outcome (a)

Derive the formula for the area of a triangle given two sides and an included angle.

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

Derive the trigonometric area formula for a triangle given two side lengths and an included angle.
Calculate the area of triangles given two sides and the included angle in practical contexts.

Key vocabulary

Included angle: The angle formed directly between two specified adjacent side lengths of a polygon.; In triangle $P Q R$ , the angle $θ$ between sides $P Q$ and $Q R$ is the included angle.
Altitude: A perpendicular line segment drawn from a vertex of a triangle to its opposite base.; The altitude $h$ drops perpendicularly from vertex $B$ onto the base line segment $A C$ .
Sine ratio: The trigonometric ratio of the length of the opposite side to the hypotenuse.; Using the sine ratio, $\sin 45^{\circ} \approx 0.7071$ gives the altitude height directly.

Lesson preview

Slide 1 · Deriving Triangular Area using TrigonometryOpening · 18 min

Figure 1: A triangle labelled ABC showing an altitude dropped from vertex B to the base, with an angle marked at vertex A.

Recall that the area of any triangle is calculated using one-half multiplied by the base and the perpendicular height. When the vertical height is not directly given, we can construct an altitude from a vertex to the base line. In the resulting right-angled triangle, the sine ratio relates the height to an adjacent side length and an included angle. Expressing height in terms of sine allows us to calculate total area directly without measuring perpendicular height. Use Figure 1 to visualise how the altitude forms a right-angled triangle for the sine calculation.

The standard area formula for a triangle requires knowing the base length and perpendicular height.
Dropping a perpendicular height from a vertex creates a right-angled triangle containing that altitude.
In this right-angled triangle, the sine of the angle equals opposite height divided by hypotenuse side.
Substituting height into the base formula produces a general area equation using two sides and angle.

Worked example. 1. Drop a perpendicular altitude

h

from vertex

B

to side

A C

. 2. In the right-angled triangle formed,

\sin 30^{\circ} = \frac{h}{8}

, so

h = 8 \sin 30^{\circ} = 8 \times 0.5 = 4 cm

. 3. Substitute

h = 4 cm

into

Area = \frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times 4 = 20 {cm}^{2}

. Answer: $Area = 20 {cm}^{2}$ .

Check question: A triangular farm plot has two sides measuring $12 m$ and $7 m$ with an included angle of $30^{\circ}$ . What is the height perpendicular to the $12 m$ base, and what is the area of the plot?

Height $= 3.5 m$ , Area $= 21 m^{2}$
Height $= 7 m$ , Area $= 42 m^{2}$
Height $= 6 m$ , Area $= 36 m^{2}$
Height $= 3.5 m$ , Area $= 42 m^{2}$

Show answer

Height $= 3.5 m$ , Area $= 21 m^{2}$ . The perpendicular height to the $12 m$ base is $7 \sin 30^{\circ} = 7 \times 0.5 = 3.5 m$ . Using the formula $Area = \frac{1}{2} \times base \times height$ , the area is $\frac{1}{2} \times 12 \times 3.5 = 21 m^{2}$ .

Slide 2 · Applying the Sine Area Formula for Any TrianglePractice · 22 min

Figure 2: A triangle with two labelled side lengths and an included angle, accompanied by the corresponding sine area 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

See the school projector WhatsApp us

Deriving and Applying the Formula for Area of a Triangle

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 12 lessons in Area of Polygons