Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 7, lesson 1 (perimeter and area of rectangles): a rectangle is 13 cm by 4 cm. What is its area?
Answer
52 cm². 13 × 4 = 52. - From Unit 2, lesson 3 (percentage of an amount): what is 50% of 76?
Answer
38. 50% is a half, and 76 ÷ 2 = 38. - From Unit 4, lesson 4 (angles in a triangle): two angles of a triangle are 50° and 60°. What is the third?
Answer
70°. The three angles add to 180°, and 50 + 60 = 110, so 180 − 110 = 70.
Words used in this lesson
- base
- the side you measure the height from. For example: The bottom edge of a triangle.
- perpendicular height
- the distance from the base to the top, measured at a right angle to the base. For example: The mast of a sail.
- parallelogram
- a four-sided shape whose opposite sides are parallel. For example: A leaning rectangle.
- slanted side
- a side that leans and is not at a right angle to the base. For example: The edge of a parallelogram that leans.
- right angle
- a square corner, 90°. For example: The corner of a page.
- triangle
- a shape with three straight sides. For example: A sail.
Which length is the height?
Idris is a sailmaker in Zanzibar. A new sail is a triangle, and the cloth is sold by the square metre. He measures the bottom edge of the sail, and the height of the mast. He also measures the long leaning edge. Three lengths, and one of them is not needed for the area. Which one?
First a simpler shape: the parallelogram. Predict: if you lean its top edge further to the right and keep the base and the height the same, does the area go up, go down or stay the same? Then use the lean slider and press Cut and slide.
The parallelogram has base 7 cm and height 4 cm, so its area is 7 × 4 = 28 cm². Its slanted side is 5 cm, which is longer than the height and is not used.
Show me the steps
- The parallelogram has base 7 cm and height 4 cm, leaning 3 cm. Its area is 7 × 4 = 28 cm². The slanted side is 5 cm.
- Press Cut and slide. The triangle at the left end moves to the right end and fills the gap. The shape is now a rectangle 7 cm by 4 cm, still 28 cm².
- Set the lean to 0. The parallelogram is already a rectangle, with the same 28 cm² and a slanted side equal to the height, 4 cm.
- Set the lean to −2, so the top edge leans the other way. The area is still 28 cm². The slanted side is now about 4.5 cm, longer than the height of 4 cm.
- Make the height 6 cm and the lean 3 cm again. The area is 7 × 6 = 42 cm². The height matters; the lean does not.
A parallelogram has area base × height. A triangle has area ½ × base × height. The height is measured at a right angle to the base.
- Parallelogram: cut a triangle off one end and slide it to the other: you have a rectangle with the same base and height, so the area is base × height.
- Triangle: a second copy, turned half a turn, makes a parallelogram with the same base and height. The triangle is half of it: ½ × base × height.
- The height is the straight distance from the top edge or corner down to the base, at a right angle. The slanted side is longer than the height (unless there is no lean) and is not used for area. Any side can be chosen as the base, as long as the height is measured at a right angle to that side.
Back to Idris: he uses the bottom edge and the mast height, and ignores the leaning edge.
A triangle is half of a parallelogram
Add a copy and turn it. Then slide the top corner along the line parallel to the base: the triangle changes shape, but what stays the same?
The triangle has base 6 cm and height 4 cm, so its area is ½ × 6 × 4 = 12 cm². A turned copy would make a parallelogram of area 6 × 4 = 24 cm².
Show me the steps
- The triangle has base 6 cm and height 4 cm. Its area is ½ × 6 × 4 = 12 cm².
- Press Add a copy. The copy, turned half a turn, joins the triangle to make a parallelogram 6 cm by 4 cm, area 24 cm². The triangle is half of that.
- Slide the top corner to 5. The triangle leans the other way but the base and height are the same, so the area is still 12 cm².
- Slide the top corner to −2. The height now meets the extended base line outside the triangle. The area is still ½ × 6 × 4 = 12 cm².
- Make the height 6 cm. The area is ½ × 6 × 6 = 18 cm², and the parallelogram is 36 cm².
A common mistake: “The parallelogram has a base of 15 cm and a slanted side of 13 cm, so the area is 15 × 13 = 195 cm².” The slanted side is not at a right angle to the base. The height is 12 cm (the dashed line), so the area is 15 × 12 = 180 cm². If a question gives you a slanted side, it is there to check that you choose the length at a right angle: find the length that makes a square corner with the base.
Worked examples
A parallelogram has a base of 14 cm, a slanted side of 10 cm and a height of 8 cm. Find its area.
- The base is 14 cm. The height at a right angle to it is 8 cm.
- The slanted side, 10 cm, is not at a right angle to the base, so it is not used.
- Area = base × height = 14 × 8 = 112 cm².
A triangular sail has a base of 11 m and a mast height of 8 m. Find the area of the cloth.
- Base 11 m, height 8 m at a right angle to it.
- A copy would make a parallelogram of 11 × 8 = 88 m².
- The triangle is half of that: 44 m².
Your go. The first step is given. A triangle has a base of 15 cm and a height of 6 cm. Find its area.
- Base 15 cm, height 6 cm at a right angle to it.
- Now find the area of the parallelogram made from two copies, take half of it, then check.
Check the full answer
Two copies make a parallelogram of 15 × 6 = 90 cm². The triangle is half of that: 45 cm².
Check before you go on
1. Which calculation gives the area of this parallelogram in square centimetres?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A parallelogram has a base of 9 cm and a height of 6 cm, measured at a right angle to the base. What is its area in square centimetres?
3. A triangular sail has a base of 8 m and a height of 9 m at a right angle to the base. What is its area in square metres?
Level 2 · Building
4. Find the area of this triangle in square centimetres.
5. A triangle has an area of 36 cm² and a base of 9 cm. How high is the triangle, at a right angle to the base, in centimetres?
6. A parallelogram has an area of 90 m² and a height of 6 m. What is its base, in metres?
Level 3 · Stretch
7. A triangle has a base of 10 cm. Its top corner is 4 cm beyond the right-hand end of the base and 6 cm above the line of the base. What is its area in square centimetres?
8. A parallelogram and a triangle both have a base of 12 cm and a height of 5 cm. How many square centimetres bigger is the area of the parallelogram?
Something to think about
A triangle has a base of 14 cm and a height of 5 cm. Its top corner slides along the line parallel to the base, 5 cm above it, all the way off to the right. What happens to the area as it slides, and to the length of its longest side? Is there a triangle with that base and height that has a perimeter of exactly 34 cm?
One way to think about it
The area stays 35 cm² the whole time, because the base and the height do not change. The longest side keeps growing without limit, so the perimeter grows too. A perimeter of 34 cm is possible: the least perimeter is for the corner above the middle, 14 + 2 × 8.6 = 31.2 cm (each sloping side is about 8.6 cm), and sliding sideways raises it, so 34 cm happens a little way off centre. Same area, many perimeters.
Exit ticket
9. Triangle ABC has a right angle at A, with AB = 9 cm, AC = 12 cm and BC = 15 cm. Which is its area?