Lesson 5 of 8 · Grades 3 to 6
Area of triangles and parallelograms
A triangle is half of a rectangle, and a parallelogram can be cut and slid into one. These two facts give the area rules for both shapes.
A triangle is half a rectangle
Draw a rectangle and a diagonal line from one corner to the opposite corner. The line cuts the rectangle into two triangles that fit exactly on top of each other. Each triangle is half of the rectangle.
The triangle below has a base of 6 cm and a height of 4 cm, the same as the dashed rectangle around it. The rectangle covers 6 × 4 = 24 cm², and the triangle covers half of that: 24 ÷ 2 = 12 cm². This is where the triangle rule comes from.
The rule for triangles
Because a triangle is half of the matching rectangle, its area is half of base times height: area = (base × height) ÷ 2. The base is any side you choose, and the height is the perpendicular distance from that base to the opposite corner. Perpendicular means the height meets the base at a right angle.
A triangle with a base of 9 m and a height of 6 m has an area of 9 × 6 = 54, then 54 ÷ 2 = 27 m². A triangle with a base of 10 cm and a height of 7 cm has an area of 35 cm², because 10 × 7 = 70 and 70 ÷ 2 = 35.
The parallelogram trick
A parallelogram looks like a rectangle that has been pushed sideways. Imagine cutting a triangle off one slanted end and sliding it to the other end. The shape becomes a rectangle with the same base and the same height. Nothing was added or removed, so the areas are equal.
That gives the parallelogram rule: area = base × height, with no dividing by 2. The height is still the perpendicular distance between the two bases, which is shorter than a slanted side.
Choosing the base and the height
A triangle has three sides, so you can treat any of them as the base. Once you choose a base, the height is the perpendicular distance to the opposite corner. The pair you choose does not change the area, but some pairs are easier to read from a diagram than others.
Keep the height and the slanted side apart. On the parallelogram above, the slanted side is longer than the height, so using it in the rule would give an area that is too big. The dashed line marked as the height is the number you need.
Worked example
A triangle has a base of 12 cm and a height of 9 cm. Find its area.
Step 1: multiply the base by the height. 12 × 9 = 108.
Step 2: a triangle is half of that rectangle, so divide by 2. 108 ÷ 2 = 54.
Step 3: check that the height is perpendicular to the base. It is marked as the height, so it is.
Step 4: label the unit. The answer is 54 cm².
Answer: 54 cm².
Watch out for this
The height must meet the base at a right angle. A slanted side is not a height. Also remember to divide by 2 for triangles only: a parallelogram uses base × height with no halving, because the cut and slide shows it is already a rectangle.
Remember
- Area of a triangle = (base × height) ÷ 2.
- Area of a parallelogram = base × height.
- The height is perpendicular to the base, never slanted.
- Any side of a triangle can be the base, as long as you use its matching height.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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A triangle has a base of 8 cm and a height of 5 cm. What is its area?
Show answer
20 cm². 8 × 5 = 40, and half of 40 is 20.
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A parallelogram has a base of 9 m and a height of 4 m. What is its area?
Show answer
36 m². 9 × 4 = 36.
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A triangle has a base of 12 cm and a height of 3 cm. What is its area?
Show answer
18 cm². 12 × 3 = 36, and half of 36 is 18.
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A parallelogram has a base of 7 cm and a height of 6 cm. What is its area?
Show answer
42 cm². 7 × 6 = 42.
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A triangle has a base of 10 m and a height of 8 m. A rectangle is 10 m by 4 m. Which has the bigger area?
Show answer
They are equal. The triangle is half of 80 = 40 m², and the rectangle is 10 × 4 = 40 m².
Puzzle · Parallelogram area
Puzzle time
This parallelogram has a base of 8 cm and a perpendicular height of 5 cm. What is its area?
Show a hint
A parallelogram is base × height. The slanted side is longer than the height, so it is not used in the rule.
Show the answer
Answer: 40 cm². Area = base × height = 8 × 5 = 40 cm². The slanted side is not the height.