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# Area of triangles

Area, volume, circles and Pythagoras · Area

<!-- Speaker note: Students double a triangle with a turned copy to make a parallelogram, see why A = ½bh, find heights that fall outside obtuse triangles, and work backwards from an area (AC9M7M01, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** two copies of any triangle, one turned round, make a parallelogram with the same base and height. So a triangle's area is half of base × height: A = ½bh.

- I can explain why a triangle's area is half of base × height.
- I can find the area of any triangle, including one whose height falls outside it.
- I can find a missing base or height from an area.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the parallelogram area (previous lesson). The one idea: two copies of a triangle make a parallelogram with the same base and height, so the triangle is half. The common mistake is forgetting the half (Omar's mistake). Obtuse triangles have a height outside the shape, onto the base extended. -->

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## Why it matters

> Sails, roof gables and the slices of a pizza cut into wedges are triangles. How much fabric does a triangular sail need?

<!-- Speaker note: A triangle also has a **base** (any side) and a **perpendicular height**: the distance from the opposite corner (the apex) straight down to the base, at a right angle. Two copies of a triangle, one turned round, fit together into a parallelogram with the same base and height. So the triangle is **half** of it: $$A = \tfrac{1}{2} \times b \times h$$ -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A triangle has base 6 cm and perpendicular height 4 cm. What is its area?**

- 20 cm²
- 24 cm²
- 10 cm²
- 12 cm²

<!-- Speaker note: What happens: 12 cm². Two copies make a 6 by 4 parallelogram, 24 cm², so one triangle is half: 12 cm². 24 is the parallelogram. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 24 found the parallelogram. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fmeasurement%2Ftriangles%2Fstep-2&code=widget%3A+area%0Ashape%3A+triangle%0Ab%3A+6%0Ah%3A+4%0Aslider+s%3A+-2%2C+8%2C+2%0Acut%3A+on%0Aunits%3A+cm%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/triangles/?full=step-2#step-2)

<!-- Speaker note: The dashed orange line is the height, from the top straight down to the base. -->

<!-- Speaker note: Teacher: ask what shape the two copies make, and why it has the same height. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 12 cm². Two copies make a 6 by 4 parallelogram, 24 cm², so one triangle is half: 12 cm². 24 is the parallelogram.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Halve it

Here is the reason, then the routine.

Now try one: base 10 cm, height 7 cm.

$$6 \times 4 = 24$$

1) $A = \tfrac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$ — One triangle is half of it.
2) $A = \tfrac{1}{2} \times 10 \times 7 = 35 \text{ cm}^2$ — Multiply the base by the height, then halve (or halve first: 5 × 7).

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 6 × 4 = 24 and say "Two copies of the triangle make a parallelogram with the same base and height." Line 1: write A = 1 ÷ 2 × 6 × 4 = 12 cm^2 and say "One triangle is half of it." Line 2: write A = 1 ÷ 2 × 10 × 7 = 35 cm^2 and say "Multiply the base by the height, then halve (or halve first: 5 × 7)." -->

<!-- Speaker note: Teacher: on the obtuse picture, ask students to trace the height from the top to the dashed extension. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Halve it

**A triangle has base 10 cm and perpendicular height 7 cm. What is its area, in cm²?**

* Answer: 35

<!-- Speaker note: ½ × 10 × 7 = 35 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Omar finds the area of a triangle. Click the line where his working goes wrong.**

```
 1  A triangle has base 8 cm and perpendicular height 5 cm.
 2  Area = base × height
 3  = 8 × 5
 4  = 40 cm²
```

* Answer: line 2

<!-- Speaker note: Line 2: a triangle is half of base × height. ½ × 8 × 5 = 20 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A height outside

**Where is this triangle's perpendicular height?**

- It has no height, because the top is not above the base
- Along the longest side
- Outside the triangle: straight down from the top to the base, extended

* Answer: Outside the triangle: straight down from the top to the base, extended

<!-- Speaker note: Extend the base (dashed) and drop the height from the top at a right angle. The area is still ½ × 4 × 4 = 8 cm². -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **A height outside:** Each task changes one thing.
- **Area 12:** 
- **Another triangle:** 
- **Backwards: the height:** 
- **A right angle:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/triangles/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a height outside the triangle on the board, building an area from a base, a new triangle, a missing height (backwards), then a right-angled triangle where the height is a side. Hints, in order: Move the top past either end of the base. / ½ × 4 × h = 12, so 4 × h = 24. / Set b to 4 first. For students who need more support: Use the board for every triangle, and cut paper triangles in half pairs. For early finishers: Draw four different triangles on squared paper with area 12 cm², including one with its height outside. The hard part, on purpose: Working backwards from an area to a height (doubling before dividing). Ask students to write ½ × 8 × h = 20 and undo one step at a time, rather than giving them the rule. When students get stuck: A student forgets to halve (Omar's mistake). Ask: What shape has area base × height? Fix: Two triangles make that parallelogram; one triangle is half. / A student says an obtuse triangle has no height. Ask: Where would a plumb line from the top land? Fix: Extend the base; the height meets it at a right angle. / A student uses the sloping side of a right-angled triangle. Ask: Which two sides meet at the right angle? Fix: Those two are a base and its height. -->

<!-- Speaker note: Teacher: for the right-angled triangle, ask which two sides are at the right angle. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Halve it

**Base 12 m, perpendicular height 5 m. Area, in m²?**

* Answer: 30

<!-- Speaker note: ½ × 12 × 5 = 30 m². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A decimal answer

**Base 7 cm, perpendicular height 3 cm. Area, in cm²?**

* Answer: 10.5

<!-- Speaker note: ½ × 7 × 3 = 10.5 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Half a rectangle

**A 10 cm by 6 cm rectangle is cut along a diagonal. What is the area of one triangle, in cm²?**

* Answer: 30

<!-- Speaker note: Half of 10 × 6 = 60: 30 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: parallelograms

**Review: a parallelogram has base 7 cm and perpendicular height 6 cm. Area, in cm²?**

* Answer: 42

<!-- Speaker note: 7 × 6 = 42 cm². -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: area 15

Then find a second triangle with the same area and a different base.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/triangles/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building a triangle with a given area means choosing a base and height whose product is double it. Hints, in order: ½ × b × h = 15, so b × h = 30. / Which pairs multiply to 30 and fit the sliders? -->

<!-- Speaker note: Teacher: collect base and height pairs; all multiply to 30. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A triangle has base 9 cm and height 6 cm. Find its area. A triangle with area 24 cm² has base 8 cm: find its height.**

* ½ × 9 × 6 = 27 cm². ½ × 8 × h = 24, so h = 6 cm.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: Area problems (node 2): choosing the base and height, units, and shapes made of rectangles and triangles. Where it leads: area of trapeziums (Year 8), and the cross-section of triangular prisms (Volume). -->