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
• Solve 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 and
• Know that the area of a parallelogram is found Use
• 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 and height . 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 and a vertical height of

b) A triangle with a sloping side of , a base of and a perpendicular height of

Worked Example 3

Find the area of a triangle with base and perpendicular height .

Worked Example 4

Find the area of a triangle with base and perpendicular height .

Worked Example 5

A triangle has base , perpendicular height and another side length of . Find the area.

Worked Example 6

A triangular garden bed has base and height . Find its area.

Problems

Problem 1

A parallelogram has base and height . 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 and a vertical height of

b) A triangle with a sloping side of , a base of and a perpendicular height of

Problem 3

Find the area of a triangle with base and perpendicular height .

Problem 4

Find the area of a triangle with base and perpendicular height .

Problem 5

A triangle has base , perpendicular height and another side length of . Find the area.

Problem 6

A triangular banner has base and height . Find its area.

Exercises

Understanding and Fluency

Exercise 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

Exercise 2.

Complete the statements.

a) The area of a triangle is

b) The height is measured to the base

c) Area is measured in units

Exercise 3.

Identify the base and perpendicular height.

a) A triangle with base and height

b) A triangle with base and height

c) A triangle with side lengths and a perpendicular height of to the side

Exercise 4.

Find the area of each triangle.

a) Base , height

b) Base , height

c) Base , height

Exercise 5.

Find the area of each triangle.

a) Base , height

b) Base , height

c) Base , height

Exercise 6.

A triangle has the following measurements. Find the area.

a) Base , height , slanted side

b) Base , height , slanted side

c) Base , height , slanted side

Exercise 7.

Find the missing height.

a) Area , base

b) Area , base

c) Area , base

Exercise 8.

Find the missing base.

a) Area , height

b) Area , height

c) Area , height

Reasoning

Exercise 9.

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

Exercise 10.

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

Exercise 11.

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

Exercise 12.

A triangle has base , height and another side length . One student says the area is , and another says it is . Determine who is correct and explain why.

Problem-solving

Exercise 13.

A triangular flag has base and height . Find its area.

Exercise 14.

A triangular garden has base and perpendicular height . How many square metres does it cover?

Exercise 15.

A triangle has area and base . Find its height.

Exercise 16.

A triangle has area and height . Find its base.

Exercise 17.

A rectangular card measures by . It is cut along a diagonal to form two equal triangles. Find the area of one triangle.

Exercise 18.

A triangle-shaped sign has base and height . 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 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