E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Area of Rhombuses, Kites and Trapezoids

E2.4About 45 minutes

Learning goals

I can find the area of a rhombus or a kite from its diagonals.
I can find the area of a trapezoid by turning two copies into a parallelogram.
I can explain where each area formula comes from by decomposing shapes.

Words to know

area
The amount of surface a shape covers, measured in square units like cm² or m².

decompose
To break a shape into smaller shapes whose areas you know.

height
The distance between a base and the opposite side or vertex, measured at a right angle to the base.

parallel sides of a trapezoid
The two sides of a trapezoid that never meet. Its height is the distance between them.

diagonal
A line segment joining two vertices that are not next to each other.

Remember from Grade 5

In Grade 5 you found the area of a parallelogram (base × height) by cutting off a triangle and moving it to make a rectangle. You found the area of a triangle (base × height ÷ 2) because it is half of a parallelogram. Today you'll use these ideas for three more shapes.

Let's start with a kite

You're building a kite for a spring kite festival. You want to know how much fabric covers it. The kite's crossed sticks are its diagonals: one is 6 cm across and the other is 8 cm long on your plan.

Draw a rectangle around the kite, with sides the same as the diagonals: 6 cm by 8 cm. The diagonals cut the rectangle into 4 smaller rectangles. The kite covers exactly half of each one! So the kite covers half of the big rectangle.

A kite with diagonals of 6 cm and 8 cm, and a rhombus with diagonals of 8 cm and 5 cm, each inside a dashed rectangle. The lengths are drawn in proportion. Each shape covers exactly half of its rectangle.

Area of a rhombus or a kite

Area = diagonal × other diagonal ÷ 2

This works for every kite and every rhombus, because their diagonals are perpendicular. (A rhombus is also a parallelogram, so you can use base × height too, if you know those instead.)

Worked example

Example 1: The kite

Find the area of the kite with diagonals 6 cm and 8 cm.

Rectangle around it: 6 × 8 = 48 cm².
The kite is half: 48 ÷ 2 = 24 cm².

24 cm²

Worked example

Example 2: The rhombus

Find the area of the rhombus with diagonals 8 cm and 5 cm.

8 × 5 = 40.
40 ÷ 2 = 20.

20 cm²

Area of a trapezoid

Take two copies of the same trapezoid. Turn one copy upside down and slide it next to the first. Together they make a parallelogram!

The parallelogram's base is the two parallel sides added together.
Its height is the trapezoid's height.
One trapezoid is half of the parallelogram.

Area of a trapezoid = (one parallel side + the other parallel side) × height ÷ 2

A trapezoid with parallel sides of 9 cm and 5 cm and a height of 4 cm, drawn in proportion. A second copy, turned upside down, fits beside it to make a parallelogram with a base of 14 cm and a height of 4 cm.

Worked example

Example 3: Area of a trapezoid

A trapezoid has parallel sides of 9 cm and 5 cm, and a height of 4 cm. Find its area.

Add the parallel sides: 9 + 5 = 14 cm.
Multiply by the height: 14 × 4 = 56 cm². That's the parallelogram.
Halve it: 56 ÷ 2 = 28 cm².

28 cm²

Worked example

Example 4: Decompose instead

A trapezoid has parallel sides of 4 m and 7 m, a height of 3 m, and two right angles on the left. Find its area by splitting it into a rectangle and a triangle. Check with the formula.

The rectangle on the left is 4 m by 3 m: 4 × 3 = 12 m².
The triangle on the right has a base of 7 − 4 = 3 m and a height of 3 m: 3 × 3 ÷ 2 = 4.5 m².
Total: 12 + 4.5 = 16.5 m².
Formula check: (4 + 7) × 3 ÷ 2 = 11 × 3 ÷ 2 = 16.5 m² ✓.

16.5 m²

Just for fun

What did the trapezoid say to the parallelogram?

"Copy me, flip me over, and together we make you!"

Height, not slant

The height is always measured straight across, at a right angle to the base. The slanted sides of a trapezoid are not the height. If a drawing shows a slanted side length, you usually don't need it for area.

Hands-on activity

Paper trapezoid puzzle

You need: 2 sheets of paper, a ruler, a pencil and scissors (with a grown-up's help).

Fold one sheet in half and draw a trapezoid on the top layer: parallel sides of 6 cm and 12 cm, 5 cm apart. Cut through both layers to get two identical trapezoids.
Turn one trapezoid upside down and fit it beside the other. What shape do you get? Measure its base and height and find its area.
Halve that area. Does it match (6 + 12) × 5 ÷ 2 = 45 cm²?
On the second sheet, draw a kite: cross a 10 cm line and an 8 cm line at a right angle, so that the 10 cm line cuts the 8 cm line exactly in half but crosses it closer to one end of the 10 cm line. Join their ends. Draw a rectangle around it. Cut out the kite and cut the 4 leftover corner triangles from the rectangle. Do the 4 triangles cover the kite exactly?

Math talk: "Why is the kite exactly half of the rectangle?"

Your turn

A kite has diagonals of 10 cm and 6 cm. Find its area.

A rhombus has diagonals of 12 cm and 7 cm. Find its area.

A trapezoid has parallel sides of 6 cm and 10 cm and a height of 5 cm. Find its area.

A rhombus has a base of 5 cm and a height of 4 cm. Find its area.

A trapezoid has parallel sides of 3 m and 8 m and a height of 4 m. Find its area.

A toy kite has diagonals of 60 cm and 90 cm. How much fabric covers it?

More practice: warm-up

A kite has diagonals of 4 cm and 6 cm. What is the area of the rectangle drawn around it? What is the kite's area?

Which three lengths do you need for the area of a trapezoid?

Work out (3 + 5) × 2 ÷ 2.

A rhombus has diagonals of 2 cm and 6 cm. Find its area.

What units do we use for area?

More practice: on your own

A garden bed is a trapezoid with parallel sides of 2.5 m and 4.5 m, and a height of 2 m. Find its area.

A kite has an area of 48 cm². One diagonal is 12 cm. How long is the other diagonal?

A trapezoid has an area of 30 cm² and parallel sides of 4 cm and 8 cm. Find its height.

A trapezoid has parallel sides of 6 cm and 12 cm, slanted sides of 5 cm each, and a height of 4 cm. Ali uses 5 cm as the height. What's wrong? What is the correct area?

A rhombus has diagonals of 9 cm and 6 cm. Find its area.

Challenge

A trapezoid has parallel sides of 6 cm (top) and 10 cm (bottom) and a height of 4 cm. A diagonal splits it into two triangles. Find the area of each triangle and check that they add up to the trapezoid's area.

A rhombus and a kite both have diagonals of 8 cm and 10 cm. Do they have the same area? Are they congruent?

A trapezoid has an area of 45 m², a height of 5 m and one parallel side of 12 m. How long is the other parallel side?

Check your understanding

Quiz

A kite has diagonals of 8 cm and 12 cm. What is its area?

Which is the area formula for a trapezoid?

A trapezoid has parallel sides of 5 m and 7 m and a height of 4 m. What is its area?

Why is the area of a kite half of the rectangle drawn around it?

A rhombus has diagonals of 10 cm and 4 cm. What is its area?

Which length should you not use as the height of a trapezoid?

For parents and teachers

Big idea: The Ontario curriculum (E2.4) asks Grade 6 students to determine the areas of trapezoids, rhombuses, kites and composite polygons by decomposing them into shapes with known area relationships. The formulas come from Grade 5 ideas: rectangles, parallelograms and triangles. This lesson covers the three shapes; the next lesson covers composite polygons.

Common difficulties:

Using a slanted side as the height. The height always meets the base at a right angle.
Forgetting to divide by 2. Ask "What shape is this half of?"
Memorizing without understanding. Ask your child to show you, with paper cut-outs, why each formula works.

How to help: Do the paper trapezoid puzzle together (supervise scissors). Look for trapezoids and kites around you: kites at a spring festival, some road signs, a dustpan or a lampshade outline.