015i - Area of Triangles

Lesson 15 — Area of Triangles

Strand: Measurement | Descriptor: AC9M7M01 | Duration: 45 minutes

Learning Intentions

• To understand why the area of a triangle is half the base times the perpendicular height.
• To calculate the area of a triangle using the established formula and appropriate units.

Success Criteria

I can:

1. Identify the base and the corresponding perpendicular height of any triangle.
2. Apply the formula correctly.
3. Use square units correctly (, ).
4. Find a missing base or height when given the area.

Warmup

(6 minutes — grid paper, individual then pairs)

On centimetre grid paper:

1. Draw a rectangle cm by cm. What is its area?
2. Draw one diagonal. What two shapes have you made?
3. What can you say about the area of each triangle? How do you know?
4. Repeat with a rectangle cm by cm.

Discussion: The diagonal cuts the rectangle into two congruent triangles, so each has half the rectangle’s area — and respectively.

Activities

Activity 1 — Deriving and Stating the Formula (12 min)

Right-angled triangles. From the warmup, a right-angled triangle is exactly half a rectangle.

Non-right-angled triangles — the key move. Draw an acute triangle inside a rectangle whose width is the base and whose height is the triangle’s height. The altitude splits the figure into two smaller rectangles, and in each the triangle takes exactly half. Therefore the halving still holds.

Defining the height precisely — this is where most errors originate.

The height must be perpendicular to the chosen base. It is not a slanted side.

Show three triangles with the same base and same perpendicular height but very different shapes (including an obtuse one where the height falls outside the triangle). All have equal area. Ask why.

I do:

Efficiency tip to model: halve the even factor first. For , : halve the to get .

We do: m, m; cm, cm; cm, cm.

You do: Six triangles from a worksheet, including at least two obtuse ones where the height is drawn outside the triangle, and one where a distractor slant length is given alongside the true height.

Activity 2 — Working backwards (8 min)

Explicit instruction on rearranging.

I do: A triangle has area and base cm. Find its height.

Alternative route to show: double the area first, then divide by the base. , and . Many students find this cleaner.

You do:

1. , cm. Find .
2. , m. Find .
3. , cm. Find .

(Answers: cm; m; cm.)

Activity 3 — Inquiry: Same Area, Different Shape (10 min)

Pairs, dynamic geometry software or grid paper.

Draw a triangle with base cm on a horizontal line, with its apex cm above that line.

1. Calculate its area.
2. Now slide the apex sideways along a line cm above the base, keeping the base fixed. Draw three more triangles this way.
3. What happens to the area? Why?
4. What happens to the perimeter?

Socratic scaffolding:

Prompt	Purpose
Understand: what is changing, and what is staying fixed?	The apex moves; base and perpendicular height stay fixed.
Which quantities does the area formula depend on?	Only and .
So what must happen to the area?	It stays throughout.
Does the shape stay the same?	No — it becomes long and thin as the apex slides far out.
What about the perimeter?	It grows, because the two slanted sides lengthen.
Looking back — what does this tell you?	Area is determined by base and height alone; the side lengths are irrelevant.
Extension	Where must the apex be for the triangle to have the smallest perimeter? (Directly above the midpoint of the base — the isosceles case.)

Answer to 1: .

Checks for Understanding

(5 minutes — exit ticket)

1. Find the area of a triangle with base cm and perpendicular height cm.
2. A triangle has area and height m. Find its base.
3. A triangle has sides cm, cm and cm, with the perpendicular height to the cm base being cm. Find its area.
4. Reasoning. A student calculates the area of a triangle with base cm and slant side cm as . What has gone wrong?
5. Two triangles both have base cm and height cm, but look completely different. Must their areas be equal? Explain.

Answers: 1. ; 2. m; 3. ; 4. The student used the slant side instead of the perpendicular height — the height must meet the base at ; 5. Yes — area depends only on base and perpendicular height.

Common Misconceptions

Misconception	How to pre-empt it
Using a slanted side as the height.	Always require students to mark the right-angle symbol on the height before calculating. Include distractor lengths in every practice set.
Forgetting to halve, giving .	Build the habit of checking against the enclosing rectangle: “is my answer about half of ?”
Believing the height must be inside the triangle.	Deliberately include obtuse triangles where the altitude falls outside. Demonstrate with dynamic software.
Using linear units for area ( instead of ).	Mark units in every model answer. Deduct for missing squared units in exit tickets.
Mixing units within one calculation, e.g. base in m and height in cm.	Insist on converting to a common unit as the first written line.
Assuming a bigger perimeter means a bigger area.	The Activity 3 inquiry addresses this directly.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A triangle has area . If the base is doubled and the height is halved, what is the new area?

Answer

The area is unchanged: .

E2 (AMC Junior style). A right-angled triangle has legs of cm and cm, and hypotenuse cm. Find its area, then find the perpendicular distance from the right angle to the hypotenuse.

Answer

The same area, computed two ways using different base–height pairs.

E3 (Challenge). A triangle has base and height , with area . Both and are whole numbers of centimetres. List all possible pairs .

Answer

We need . The factor pairs of are — 12 ordered pairs, or 6 if order is disregarded.

E4 (Challenge). In a rectangle cm by cm, a point is marked anywhere on one of the longer sides. Triangle is formed by joining to the two opposite corners. Show that its area is always the same, whatever the position of .

Answer

The base is the opposite long side, cm, and the perpendicular height is always the rectangle’s width, cm. So — exactly half the rectangle, regardless of where sits.

Homework

1. Find the area of each triangle: (a) cm, cm (b) m, m (c) cm, cm (d) cm, cm (e) m, m.
2. Find the missing dimension: (a) , cm, find (b) , m, find (c) , cm, find .
3. A triangular sail has a base of m and a height of m. Find its area.
4. A triangular garden bed has an area of and a base of m. Mulch costs 12$ per square metre. Find the height of the bed, and the cost of mulching it.
5. A triangle has base cm and height m. Find its area in . (Convert first.)
6. Reasoning. Explain why two triangles with the same area need not have the same perimeter. Give an example.
7. Challenge. A triangle has an area of . Its height is twice its base. Find both dimensions.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) cm (b) m (c) cm. Q3 — . Q4 — m; cost 2160.5= 50A = 1000\ \text{cm}^2\tfrac12 \times b \times 2b = b^2 = 64b = 8h = 16$ cm.