122. Area of Special Quadrilaterals

Learning Intentions

• To understand that the formulas for the area of special quadrilaterals can be developed from the formulas for the area of rectangles and triangles
• find the area of rhombuses, kites and trapeziums

Pre-requisite Summary

• Know that area measures the amount of surface inside a two-dimensional shape
• Know that area is measured in square units such as ${\text{cm}}^2$, ${\text{m}}^2$ and ${\text{mm}}^2$
• Be able to find the area of rectangles using length $\times$ width
• Be able to find the area of triangles using ${\displaystyle \frac{1}{2}}\times \text{base}\times \text{height}$
• Understand that some quadrilaterals can be split into triangles or rearranged into rectangles
• Know the names and basic properties of rhombuses, kites and trapeziums
• Understand that the perpendicular height is used in area formulas, not a sloping side

Worked Examples

Worked Example 1

A trapezium can be split and rearranged to connect its area formula to rectangles and triangles.
a) A trapezium has parallel sides $8\text{ cm}$ and $12\text{ cm}$, and height $5\text{ cm}$. Find its area.
b) A trapezium has parallel sides $7\text{ m}$ and $9\text{ m}$, and height $4\text{ m}$. Find its area.

Worked Example 2

A rhombus can be split by its diagonals into triangles.
a) A rhombus has diagonals $10\text{ cm}$ and $8\text{ cm}$. Find its area.
b) A rhombus has diagonals $14\text{ m}$ and $6\text{ m}$. Find its area.

Worked Example 3

A kite can be split by its diagonals into triangles.
a) A kite has diagonals $12\text{ cm}$ and $9\text{ cm}$. Find its area.
b) A kite has diagonals $16\text{ mm}$ and $5\text{ mm}$. Find its area.

Worked Example 4

Use the area formula for a trapezium:
a) parallel sides $15\text{ cm}$ and $11\text{ cm}$, height $6\text{ cm}$
b) parallel sides $20\text{ m}$ and $14\text{ m}$, height $7\text{ m}$

Worked Example 5

Use the area formula for a rhombus or kite:
a) a rhombus with diagonals $18\text{ cm}$ and $10\text{ cm}$
b) a kite with diagonals $13\text{ cm}$ and $8\text{ cm}$

Problems

Problem 1

A trapezium can be split and rearranged to connect its area formula to rectangles and triangles.
a) A trapezium has parallel sides $6\text{ cm}$ and $10\text{ cm}$, and height $4\text{ cm}$. Find its area.
b) A trapezium has parallel sides $9\text{ m}$ and $13\text{ m}$, and height $3\text{ m}$. Find its area.

Problem 2

A rhombus can be split by its diagonals into triangles.
a) A rhombus has diagonals $12\text{ cm}$ and $7\text{ cm}$. Find its area.
b) A rhombus has diagonals $20\text{ m}$ and $9\text{ m}$. Find its area.

Problem 3

A kite can be split by its diagonals into triangles.
a) A kite has diagonals $14\text{ cm}$ and $6\text{ cm}$. Find its area.
b) A kite has diagonals $18\text{ mm}$ and $4\text{ mm}$. Find its area.

Problem 4

Use the area formula for a trapezium:
a) parallel sides $12\text{ cm}$ and $8\text{ cm}$, height $5\text{ cm}$
b) parallel sides $17\text{ m}$ and $9\text{ m}$, height $4\text{ m}$

Problem 5

Use the area formula for a rhombus or kite:
a) a rhombus with diagonals $24\text{ cm}$ and $5\text{ cm}$
b) a kite with diagonals $15\text{ cm}$ and $12\text{ cm}$

Exercises

Understanding and Fluency

1. Complete each statement:
   a) The area of a trapezium depends on the two ______ sides and the ______
   b) The area of a rhombus can be found using its ______
   c) The area of a kite can be found using its ______

2. Find the area of each trapezium:
   a) parallel sides $5\text{ cm}$ and $9\text{ cm}$, height $4\text{ cm}$
   b) parallel sides $11\text{ cm}$ and $15\text{ cm}$, height $3\text{ cm}$
   c) parallel sides $7\text{ m}$ and $13\text{ m}$, height $6\text{ m}$

3. Find the area of each trapezium:
   a) parallel sides $14\text{ cm}$ and $10\text{ cm}$, height $8\text{ cm}$
   b) parallel sides $20\text{ mm}$ and $12\text{ mm}$, height $5\text{ mm}$
   c) parallel sides $18\text{ m}$ and $6\text{ m}$, height $7\text{ m}$

4. Find the area of each rhombus:
   a) diagonals $8\text{ cm}$ and $6\text{ cm}$
   b) diagonals $15\text{ cm}$ and $4\text{ cm}$
   c) diagonals $22\text{ m}$ and $10\text{ m}$

5. Find the area of each kite:
   a) diagonals $10\text{ cm}$ and $7\text{ cm}$
   b) diagonals $18\text{ cm}$ and $9\text{ cm}$
   c) diagonals $30\text{ mm}$ and $12\text{ mm}$

6. Decide which formula is most suitable, then find the area:
   a) a rhombus with diagonals $16\text{ cm}$ and $11\text{ cm}$
   b) a trapezium with parallel sides $9\text{ cm}$ and $13\text{ cm}$, height $5\text{ cm}$
   c) a kite with diagonals $20\text{ cm}$ and $14\text{ cm}$

7. Find the missing measurement:
   a) A rhombus has area $48{\text{ cm}}^2$ and one diagonal $8\text{ cm}$. Find the other diagonal.
   b) A kite has area $60{\text{ cm}}^2$ and one diagonal $12\text{ cm}$. Find the other diagonal.
   c) A trapezium has area $40{\text{ cm}}^2$, parallel sides $6\text{ cm}$ and $10\text{ cm}$. Find the height.

8. A shape is split into simpler shapes to justify its formula.
   a) Explain how a trapezium formula can come from rectangles and triangles.
   b) Explain how a rhombus formula can come from triangles.
   c) Explain how a kite formula can come from triangles.

Reasoning

9. Explain why the area formula for a trapezium uses the average of the two parallel sides.

10. A student says the area of a rhombus is found by multiplying its side lengths. Explain the mistake.

11. Noah says a kite and a rhombus must use different area formulas because they are different shapes. Is he correct? Explain.

12. Explain why the diagonals are used in the area formulas for rhombuses and kites.

13. A student uses a sloping side instead of the perpendicular height in the area formula for a trapezium. Describe the error.

Problem-solving

14. A trapezium has parallel sides $16\text{ m}$ and $10\text{ m}$, and height $9\text{ m}$. Find its area.

15. A rhombus-shaped garden has diagonals $14\text{ m}$ and $12\text{ m}$. Find its area.

16. A kite-shaped window has diagonals $18\text{ cm}$ and $10\text{ cm}$. Find its area.

17. A trapezium has area $72{\text{ cm}}^2$, height $6\text{ cm}$, and one parallel side $8\text{ cm}$. Find the other parallel side.

18. A rhombus has area $90{\text{ cm}}^2$ and one diagonal $15\text{ cm}$. Find the other diagonal.

19. A composite design is made from a trapezium of area $54{\text{ cm}}^2$ and a kite of area $36{\text{ cm}}^2$. Find the total area.

Potential Misunderstandings

• Students may confuse area formulas with perimeter formulas
• Students may use side lengths instead of diagonals for a rhombus or kite
• Students may forget to divide by $2$ when using the diagonals of a rhombus or kite
• Students may forget that only the parallel sides are used in the trapezium formula
• Students may use a sloping side instead of the perpendicular height in a trapezium
• Students may think the formulas for rhombuses, kites and trapeziums are unrelated to rectangles and triangles
• Students may average the wrong lengths in a trapezium
• Students may omit square units in their final answers