090. Area of Triangles

Learning Intentions

• To understand that a triangle can be considered as half of a parallelogram
• identify the base and the (perpendicular) height of a triangle
• find the area of a triangle

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 ${\text{cm}}^2$ and ${\text{m}}^2$
• Know that the area of a parallelogram is found using $A=bh$
• Know that a triangle has three sides and three angles
• Know that any side of a triangle can be chosen as the base
• Know that the height is the perpendicular distance from the base to the opposite vertex
• Know that the slanted side is not usually the height unless it is perpendicular to the base

Worked Examples

Worked Example 1

A parallelogram has base $10\text{ cm}$ and height $6\text{ cm}$. A diagonal divides it into two equal triangles. Find the area of one triangle.

Worked Example 2

Identify the base and perpendicular height of each triangle.
a) A triangle with a horizontal base of $8\text{ cm}$ and a vertical height of $5\text{ cm}$
b) A triangle with a sloping side of $7\text{ cm}$, a base of $9\text{ cm}$ and a perpendicular height of $4\text{ cm}$

Worked Example 3

Find the area of a triangle with base $12\text{ cm}$ and perpendicular height $7\text{ cm}$.

Worked Example 4

Find the area of a triangle with base $15\text{ m}$ and perpendicular height $8\text{ m}$.

Worked Example 5

A triangle has base $10\text{ cm}$, perpendicular height $6\text{ cm}$ and another side length of $7\text{ cm}$. Find the area.

Worked Example 6

A triangular garden bed has base $9\text{ m}$ and height $4\text{ m}$. Find its area.

Problems

Problem 1

A parallelogram has base $14\text{ cm}$ and height $5\text{ cm}$. A diagonal divides it into two equal triangles. Find the area of one triangle.

Problem 2

Identify the base and perpendicular height of each triangle.
a) A triangle with a horizontal base of $6\text{ cm}$ and a vertical height of $4\text{ cm}$
b) A triangle with a sloping side of $8\text{ cm}$, a base of $11\text{ cm}$ and a perpendicular height of $5\text{ cm}$

Problem 3

Find the area of a triangle with base $13\text{ cm}$ and perpendicular height $6\text{ cm}$.

Problem 4

Find the area of a triangle with base $18\text{ m}$ and perpendicular height $9\text{ m}$.

Problem 5

A triangle has base $12\text{ cm}$, perpendicular height $5\text{ cm}$ and another side length of $9\text{ cm}$. Find the area.

Problem 6

A triangular banner has base $7\text{ m}$ and height $3\text{ m}$. Find its area.

Exercises

Understanding and Fluency

1. Decide whether each statement is true or false.
   a) A triangle can be seen as half of a parallelogram with the same base and height
   b) The height of a triangle must be perpendicular to the base
   c) The longest side of a triangle is always the base

2. Complete the statements.
   a) The area of a triangle is $A=\mathrm{\_}\mathrm{\_}\mathrm{\_}$
   b) The height is measured $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ to the base
   c) Area is measured in $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ units

3. Identify the base and perpendicular height.
   a) A triangle with base $8\text{ cm}$ and height $3\text{ cm}$
   b) A triangle with base $14\text{ m}$ and height $6\text{ m}$
   c) A triangle with side lengths $5\text{ cm},12\text{ cm},13\text{ cm}$ and a perpendicular height of $4\text{ cm}$ to the $12\text{ cm}$ side

4. Find the area of each triangle.
   a) Base $6\text{ cm}$, height $4\text{ cm}$
   b) Base $9\text{ cm}$, height $7\text{ cm}$
   c) Base $20\text{ m}$, height $5\text{ m}$

5. Find the area of each triangle.
   a) Base $11\text{ cm}$, height $8\text{ cm}$
   b) Base $15\text{ m}$, height $12\text{ m}$
   c) Base $4.5\text{ cm}$, height $2\text{ cm}$

6. A triangle has the following measurements. Find the area.
   a) Base $10\text{ cm}$, height $6\text{ cm}$, slanted side $8\text{ cm}$
   b) Base $14\text{ cm}$, height $9\text{ cm}$, slanted side $11\text{ cm}$
   c) Base $12\text{ m}$, height $7\text{ m}$, slanted side $9\text{ m}$

7. Find the missing height.
   a) Area $24{\text{ cm}}^2$, base $8\text{ cm}$
   b) Area $45{\text{ m}}^2$, base $9\text{ m}$
   c) Area $36{\text{ cm}}^2$, base $12\text{ cm}$

8. Find the missing base.
   a) Area $30{\text{ cm}}^2$, height $5\text{ cm}$
   b) Area $56{\text{ m}}^2$, height $8\text{ m}$
   c) Area $27{\text{ cm}}^2$, height $3\text{ cm}$

Reasoning

9. Explain why the area of a triangle is half the area of a parallelogram with the same base and height.

10. A student uses a slanted side instead of the perpendicular height when finding the area of a triangle. Explain why this is incorrect.

11. Explain why different triangles can have the same base and height and therefore the same area.

12. A triangle has base $12\text{ cm}$, height $5\text{ cm}$ and another side length $7\text{ cm}$. One student says the area is ${\displaystyle \frac{1}{2}}\times 12\times 5$, and another says it is ${\displaystyle \frac{1}{2}}\times 12\times 7$. Determine who is correct and explain why.

Problem-solving

13. A triangular flag has base $18\text{ cm}$ and height $10\text{ cm}$. Find its area.

14. A triangular garden has base $12\text{ m}$ and perpendicular height $9\text{ m}$. How many square metres does it cover?

15. A triangle has area $35{\text{ cm}}^2$ and base $10\text{ cm}$. Find its height.

16. A triangle has area $48{\text{ m}}^2$ and height $6\text{ m}$. Find its base.

17. A rectangular card measures $14\text{ cm}$ by $8\text{ cm}$. It is cut along a diagonal to form two equal triangles. Find the area of one triangle.

18. A triangle-shaped sign has base $2.4\text{ m}$ and height $1.5\text{ m}$. Find its area.

Potential Misunderstandings

• A student may think the slanted side is always the height
• A student may forget that the height must be perpendicular to the base
• A student may think the base must always be the bottom side drawn in the diagram
• A student may confuse area with perimeter
• A student may forget to divide by $2$ after multiplying base and height
• A student may multiply the wrong measurements, especially when an extra side length is given
• A student may not recognise that two congruent triangles can be formed by splitting a parallelogram
• A student may omit square units in the final answer

Next: 091e. Area of Composite Shapes