Grade 10 · Core Mathematics · Lesson 6 of 12

Determining Area of Quadrilaterals in Different Situations: Kites and Complex Figures

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of Polygons. In it, learners calculate the perpendicular height and total area of a trapezium using sine trigonometric ratios.

Strand Measurements and GeometrySub-strand Area of PolygonsLesson code 10CM-25-L640 minutes · 3 slides

KICD learning outcome (d)

Determine the area of quadrilaterals in different situations.

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

Calculate the perpendicular height and total area of a trapezium using sine trigonometric ratios.
Determine the surface area of a kite using the diagonal product formula Area = 1/2 * d1 * d2 or by summing component triangle areas.
Solve inverse problems to find missing diagonal lengths, side lengths, or interior angles of kites and parallelograms given their areas.

Key vocabulary

Trapezium: A quadrilateral with exactly one pair of parallel sides.; Trapezium $P Q R S$ has parallel sides $P S = 10$ m and $Q R = 18$ m.
Diagonal: A straight line segment joining two non-adjacent vertices of a polygon.; The kite-shaped patch has diagonals of 5 m and 8 m crossing at right angles.
Included angle: The angle formed directly between two specified adjacent side lengths of a polygon.; The angle included between kite sides $B A$ and $B C$ is $∠ A B C = 62^{\circ}$ .
Parallelogram: A quadrilateral with two pairs of parallel and equal opposite sides.; The parallelogram garden has adjacent sides $12 m$ and $8 m$ with area $48 m^{2}$ .
Inverse calculation: Working backwards from a known output, such as area, to calculate an unknown input.; Using an inverse calculation, we found the unknown diagonal of the kite to be $10 m$ .

Lesson preview

Slide 1 · Area of Trapeziums and Kites Using Diagonal and Trigonometric MethodsOpening · 12 min

Figure 1: A diagram illustrating how to calculate the area of a trapezium using trigonometric height and the area of a kite using perpendicular diagonals.

To calculate the area of a trapezium when the perpendicular height is unknown, we use trigonometric ratios on a right-angled triangle formed by a non-parallel side. For a kite, perpendicular diagonals allow us to find area using the formula $Area = \frac{1}{2} d_{1} d_{2}$ . Alternatively, a kite can be split along a diagonal into two triangles, calculating their areas individually and adding them together to obtain the total surface region. Refer to Figure 1 to visualise how these area formulas are derived from the shapes' dimensions.

The height of a trapezium can be determined using sine ratios from a non-parallel inclined side.
Formula for a trapezium area is $Area = \frac{1}{2} (a + b) h$ , substituting the trigonometric height.
The area of a kite with perpendicular diagonals $d_{1}$ and $d_{2}$ equals $\frac{1}{2} d_{1} d_{2}$ .
A kite can also be partitioned into two triangles to sum their separate surface areas.

Worked example. Identify parallel sides

a = 10 m

and

b = 18 m

. Calculate height

h

using

h = R S \sin 40^{\circ} = 6 \sin 40^{\circ} \approx 3.8567 m

. Apply trapezium area formula:

Area = \frac{1}{2} (a + b) h = \frac{1}{2} (10 + 18) \times 3.8567

. Evaluate:

Area = 14 \times 3.8567 = 53.9938 \approx 54.0 m^{2}

. Answer: $54.0 m^{2}$ .

Check question: A kite-shaped vegetable patch has perpendicular diagonals measuring $5 m$ and $8 m$ . What is the total area of the patch?

$10 m^{2}$
$13 m^{2}$
$20 m^{2}$
$40 m^{2}$

Show answer

$20 m^{2}$ . Using the formula $Area = \frac{1}{2} d_{1} d_{2}$ , substitute $d_{1} = 5 m$ and $d_{2} = 8 m$ to get $Area = \frac{1}{2} \times 5 \times 8 = 20 m^{2}$ .

Slide 2 · Area of Kites and Parallelograms via Trigonometric PartitioningModel · 14 min

Figure 2: Diagrams demonstrating how to find the area of a kite by splitting it into triangles and how to calculate a parallelogram's area using trigonometry.

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All 3 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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Determining Area of Quadrilaterals in Different Situations: Kites and Complex Figures

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