E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and AreaArea of a Parallelogram
E2.5About 40 minutes
Learning goals- I can explain why a parallelogram has the same area as a rectangle with the same base and height.
- I can use the formula Area = base × height to find the area of a parallelogram.
- I can tell the height apart from the slanted side.
- I can find a missing base or height when I know the area.
Words to know
base
The side of the parallelogram we measure from. Any side can be the base.
height
The straight-up distance from the base to the opposite side. It makes a right angle with the base.
perpendicular
Meeting at a right angle (90°).
formula
A short math rule written with letters, like A = b × h.
Remember from Grade 4
In Grade 4 you learned that the area of a rectangle is length × width, and you used division to find a missing side. In the lesson on congruent shapes you met the parallelogram: a four-sided shape with 2 pairs of parallel sides. Today you will discover how to find its area.
Let's start with a garden bed
A community garden has a flower bed shaped like a parallelogram. The gardeners want to cover it with mulch. They need to know its area, but it isn't a rectangle. What can they do?
Here's a clever trick: cut and move. Cut a triangle off one end and slide it to the other end. The parallelogram turns into a rectangle with exactly the same area!
Why it works
Cutting and moving a piece doesn't add or remove any area. The new rectangle has:
- a length equal to the parallelogram's base,
- a width equal to the parallelogram's height.
So the area of the parallelogram is:
Area = base × height, or A = b × h
The height must be perpendicular (at a right angle) to the base. It is not the slanted side!
Worked exampleExample 1: On a grid
Find the area of the parallelogram in the picture above.
- The base is 6 units.
- The height (dashed line, straight up from the base) is 3 units.
- A = b × h = 6 × 3 = 18.
18 square units. That matches the rectangle: 6 × 3 = 18.
A parallelogram with a base of 8 cm, a slanted side of 6 cm and a height of 5 cm. The dashed height makes a right angle with the base. (Drawn smaller than real size, but in proportion.)Worked exampleExample 2: Ignore the slanted side
Find the area of the parallelogram above: base 8 cm, height 5 cm, slanted side 6 cm.
- The formula uses the base and the height: A = b × h.
- The slanted side (6 cm) is not the height, because it is not perpendicular to the base. Ignore it for area.
- A = 8 × 5 = 40.
Worked exampleExample 3: Find a missing height
A parallelogram has an area of 36 m² and a base of 9 m. What is its height?
- A = b × h, so 36 = 9 × h.
- What times 9 makes 36? Divide: 36 ÷ 9 = 4.
- Check: 9 × 4 = 36. ✓
Square units for area
Area is always measured in square units: cm², m², or "square units" on a grid. Lengths (base, height) are in cm or m. Writing the right unit shows you understand what you measured.
Worked exampleExample 4: Compare shapes
A rectangle is 7 cm by 4 cm. A parallelogram has a base of 7 cm and a height of 4 cm. Which has the bigger area?
- Rectangle: 7 × 4 = 28 cm².
- Parallelogram: 7 × 4 = 28 cm².
They have the same area. A parallelogram has the same area as a rectangle with the same base and height.
Just for funA parallelogram is just a rectangle leaning on a wall after a long day. Slide its triangle back to the other end, and it stands up straight again, with exactly the same area!
Hands-on activityCut-and-move parallelograms
You need: grid paper (or plain paper and a ruler), scissors (with a grown-up), tape and a pencil.
- On grid paper, draw a parallelogram with a base of 8 squares and a height of 4 squares. (Start the top side 2 squares to the right of the bottom side.)
- Count the squares inside (put the part squares together). Write your count.
- Draw a dashed height line from a top corner straight down to the base.
- Cut out the parallelogram, then cut along the dashed line.
- Slide the triangle to the other end and tape it. What shape do you have? What are its length and width?
- Check: does base × height (8 × 4) match your count?
(Answer: 32 square units. The pieces make an 8 by 4 rectangle.)
Math talk: "Why doesn't the slanted side length matter for area?"
Your turn
1A parallelogram has a base of 7 cm and a height of 4 cm. What is its area?
2A parallelogram has a base of 10 m and a height of 3 m. What is its area?
3A parallelogram has a base of 5 cm, a height of 6 cm and a slanted side of 7 cm. What is its area?
30 cm². A = 5 × 6 = 30. The slanted side isn't used.
4On a grid, a parallelogram has a base of 4 units and a height of 2 units. What is its area?
8 square units. 4 × 2 = 8.
5A parallelogram has an area of 24 cm² and a base of 6 cm. What is its height?
6Which do you use for the area of a parallelogram: the slanted side or the height? Why?
The height. It is perpendicular to the base, just like the width of the rectangle you get by cutting and moving.
More practice: warm-up
1Write the formula for the area of a parallelogram.
A = b × h (area = base × height).
2Base 3 cm, height 2 cm. Area?
3Base 9 m, height 1 m. Area?
4What does perpendicular mean?
Meeting at a right angle (90°).
5Is area measured in cm or cm²?
cm² (square centimetres).
More practice: on your own
1Base 12 cm, height 5 cm. Area?
2A parallelogram has an area of 45 m² and a height of 5 m. What is its base?
3A parallelogram-shaped parking lot has a base of 15 m and a height of 8 m. What is its area?
4Parallelogram P has base 6 cm and height 4 cm. Parallelogram Q has base 8 cm and height 3 cm. Which has the bigger area?
Neither. They're equal. P: 6 × 4 = 24 cm². Q: 8 × 3 = 24 cm².
5A parallelogram on a grid has corners at (1, 1), (6, 1), (8, 4) and (3, 4). What is its area?
15 square units. The base from (1, 1) to (6, 1) is 5 units. The height from y = 1 up to y = 4 is 3 units. 5 × 3 = 15.
Challenge
1A rectangle is 6 cm by 4 cm. A parallelogram has a base of 6 cm and slanted sides of 4 cm, and it leans over. Which has the bigger area? Explain.
The rectangle. Its area is 24 cm². The parallelogram's slanted sides are 4 cm, but since it leans, its height is less than 4 cm. So its area is less than 6 × 4 = 24 cm².
2A parallelogram has a base of 8 cm and a height of 6 cm. What happens to its area if you double the base and keep the same height?
The area doubles, from 8 × 6 = 48 cm² to 16 × 6 = 96 cm².
3Draw two different parallelograms that each have an area of 12 square units. Describe them.
Model answers: base 6 and height 2; base 4 and height 3; base 12 and height 1; or base 3 and height 4. Any pair where base × height = 12 works, and each can lean by any amount.
Quiz
1. What is the formula for the area of a parallelogram?
2. A parallelogram has base 9 cm and height 4 cm. What is its area?
3. A parallelogram has base 10 cm, height 6 cm and slanted side 7 cm. What is its area?
4. Why does a parallelogram have the same area as a rectangle with the same base and height?
5. A parallelogram has area 40 m² and base 8 m. What is its height?
6. The height of a parallelogram is:
Big idea: The Ontario curriculum (E2.5) asks Grade 5 students to use the area relationships among rectangles, parallelograms and triangles to develop the formulas for the area of a parallelogram and a triangle, and solve related problems. The key insight is that a parallelogram can be cut and rearranged into a rectangle with the same base and height.
Common difficulties:
- Using the slanted side instead of the height. This is a very common error. Ask: "Does it make a right angle with the base?"
- Memorizing without understanding. Do the cut-and-move activity so the formula makes sense.
- Forgetting square units.
How to help: Help with scissors during the activity. Ask your child to explain why the formula works, not just use it.