Area of triangles
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.
Sails, roof gables and the slices of a pizza cut into wedges are triangles. How much fabric does a triangular sail need?
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:
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
A 6 by 4 triangle
How would two copies of the triangle fit together?
Add a turned copy
The dashed orange line is the height, from the top straight down to the base.
Press **Add a turned copy**. What do the two triangles make? Then move the top along. Was your prediction right?
Use the board. This ticks when you get there.
Half of base times height
Here is the reason, then the routine.
- Two copies of the triangle make a parallelogram with the same base and height.
- One triangle is half of it.
- Multiply the base by the height, then halve (or halve first: 5 × 7).
Now try one: base 10 cm, height 7 cm.
Read Omar's working. What shape has area base × height?
A triangle has base 10 cm and perpendicular height 7 cm. What is its area, in cm²?
Omar finds the area of a triangle. Click the line where his working goes wrong.
One change at a time
Each task changes one thing.
Need a hint?
Move the top past either end of the base.
Need a hint?
½ × 4 × h = 12, so 4 × h = 24.
Another hint
Set b to 4 first.
Move the top so the height falls outside the triangle. Does the area change?
Use the board. This ticks when you get there.
Make a triangle with base 4 cm and area 12 cm².
Use the board. This ticks when you get there.
Base 9 cm, perpendicular height 4 cm. Area, in cm²?
A triangle has area 20 cm² and base 8 cm. What is its perpendicular height, in cm?
Triangles
Base 12 m, perpendicular height 5 m. Area, in m²?
Base 7 cm, perpendicular height 3 cm. Area, in cm²?
A 10 cm by 6 cm rectangle is cut along a diagonal. What is the area of one triangle, in cm²?
Review: a parallelogram has base 7 cm and perpendicular height 6 cm. Area, in cm²?
Area 15
Then find a second triangle with the same area and a different base.
Need a hint?
½ × b × h = 15, so b × h = 30.
Another hint
Which pairs multiply to 30 and fit the sliders?
Make a triangle with area 15 cm².
Use the board. This ticks when you get there.
For teachers
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.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- 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.
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.
Running the lesson
- Predict: students who chose 24 found the parallelogram.
- Run: ask what shape the two copies make, and why it has the same height.
- Investigate: on the obtuse picture, ask students to trace the height from the top to the dashed extension.
- Modify: for the right-angled triangle, ask which two sides are at the right angle.
- Make: collect base and height pairs; all multiply to 30.
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.
Stretch and support
Stretch: Draw four different triangles on squared paper with area 12 cm², including one with its height outside.
Support: Use the board for every triangle, and cut paper triangles in half pairs.
Where it 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).
What each step is for
- a 6 by 4 triangle: Committing to an area for a triangle with base 6 and height 4 shows whether they halve, or multiply as for a parallelogram.
- add a turned copy: Adding a turned copy shows two triangles make a parallelogram with the same base and height, so the triangle is half; moving the apex keeps the area.
- half of base times height: A = ½bh; Omar forgets the half, giving the parallelogram. The height can fall outside the triangle, onto the base extended.
- one change at a time: 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.
- triangles: Whole and decimal answers and half a rectangle, mixed; forgetting the half is the planted slip; the review keeps parallelograms fresh.
- area 15: Building a triangle with a given area means choosing a base and height whose product is double it.
- Explain why a triangle's area is half of base × height.
- Find the area of any triangle, including one whose height falls outside it.
- Find a missing base or height from an area.