Find areas of trapezoids, rhombuses, kites, and composite polygons

Ontario Grade 6 Mathematics

Spatial Sense

Trapezoids, rhombuses, kites, and composite polygons

Area tells how much flat space a shape covers. We measure area in square units, such as square centimetres. A rectangle with side lengths 5 cm and 3 cm covers 15 square centimetres because it holds 5 rows of 3 one-square-centimetre tiles. We write this as

5\times3=15\text{ cm}^2

. This lesson uses that rectangle idea to find areas of other polygons. A polygon is a closed, flat shape made from straight sides.

What you will learn

Find the area of a trapezoid using its parallel sides and height.
Find the area of a rhombus or kite using its diagonals.
Find the area of a composite polygon by splitting it into familiar shapes.
Choose measurements that match the area rule for each shape.

Use parallel sides and height for a trapezoid

A trapezoid has one pair of parallel sides. Parallel sides stay the same distance apart. These sides are called the bases. The height is the straight, shortest distance between the bases. It meets the bases at a right angle. A slanted side is not the height unless it meets the bases at a right angle.

Imagine two equal trapezoids placed together to make a parallelogram. The parallelogram has a base equal to the two trapezoid bases joined, and its height is unchanged. One trapezoid has half the parallelogram's area. This gives a useful rule: add the lengths of the two bases, multiply by the height, then take half.

For example, if a trapezoid has bases of 8 cm and 4 cm and a height of 3 cm, its area is half of 12 groups of 3 square centimetres. The units are squared because area counts square units.

A=\frac{(b_1+b_2)h}{2}

Identify both parallel bases.
Use the perpendicular height, not a sloping side.
The two base lengths can be added in either order.

Use diagonals for a rhombus or kite

A rhombus has four equal sides. A kite has two pairs of equal sides that meet next to each other. Both shapes can be measured using their diagonals. A diagonal is a straight line joining two opposite corners.

For these shapes, multiply the lengths of the two diagonals and take half. You can picture the diagonals dividing the shape into four triangles. Their combined area is half the product of the diagonal lengths. Measure each diagonal from corner to opposite corner, not along a side.

The diagonals do not need to have the same length. Use both full diagonal lengths in the rule. Give the answer in square units.

A=\frac{d_1d_2}{2}

A diagonal joins opposite corners.
Measure both full diagonals.
Do not substitute the side lengths for the diagonals.

Split a composite polygon into familiar shapes

A composite polygon is a shape made by joining two or more simpler shapes. To find its area, split it into parts whose areas you know how to find. Find each part's area, then add them if the parts make up the whole shape without overlap.

A useful split might make a rectangle and a triangle, or two trapezoids. Draw the split line inside the shape. Mark any lengths needed for each part. The same length can sometimes serve as a height for one part, but check that it meets the correct base at a right angle.

You can also find a missing corner by imagining a larger rectangle around the shape and subtracting the area outside the polygon. For this lesson, use a split or subtraction only when the needed lengths are given or can be found from the labelled sides. Do not count any part twice.

A_{\text{whole}}=A_1+A_2+\cdots

Split the shape into known parts.
Find each part's area with its matching rule.
Add included parts or subtract an excluded part.

Check labels, units, and reasonableness

Before calculating, name the shape and choose its area rule. Then match each measurement to the rule: bases and height for a trapezoid, diagonals for a rhombus or kite, and the needed side lengths for each part of a composite polygon.

Write the unit as a square unit, such as

\text{cm}^2

. A length is measured in centimetres, but an area covers square centimetres. If you split a shape, keep the same area unit for every part.

A quick estimate can help you check. For a trapezoid, the area should be between the areas of rectangles with the shorter and longer base and the same height. For a composite shape, the total should be larger than each included part and smaller than the area of a rectangle that fully contains it.

Use measurements that match the chosen rule.
Label area in square units.
Check that the answer fits the size of the shape.

Worked example

Find the area of a trapezoid

A trapezoid has parallel bases of 8 cm and 4 cm. Its perpendicular height is 4 cm. Find its area.

Trapezoid with labelled bases and height

The parallel bases are 8 cm and 4 cm. The perpendicular height is 4 cm.

Choose the measurements
The two parallel sides are the bases. The 4 cm height is the perpendicular distance between them.
Use the trapezoid rule
Add the base lengths, multiply by the height, and take half.
$A=\frac{(8+4)\times4}{2}$
Calculate
The area is 24 square centimetres.
$A=24\text{ cm}^2$

Answer: The trapezoid's area is

24\text{ cm}^2

.

Check: A rectangle with the shorter base and the same height would have area

4\times4=16\text{ cm}^2

. One with the longer base would have area

8\times4=32\text{ cm}^2

. The answer, 24 square centimetres, is between these values.

Worked example

Find the area of a rhombus

A rhombus has diagonals of 8 cm and 6 cm. Find its area.

Rhombus with diagonals

The horizontal diagonal is 8 cm and the vertical diagonal is 6 cm.

Identify the diagonals
Each diagonal runs from one corner to the opposite corner. The given measurements are the full diagonal lengths.
Use the diagonal rule
Multiply the diagonal lengths, then take half.
$A=\frac{8\times6}{2}$
Calculate
The area is 24 square centimetres.
$A=24\text{ cm}^2$

Answer: The rhombus's area is

24\text{ cm}^2

.

Check: The rule uses the two diagonal lengths, not the lengths of the rhombus's sides.

Worked example

Find the area of a composite polygon

A polygon is shaped like a rectangle with a triangular roof. The rectangle is 10 cm wide and 6 cm high. The triangle's base is 10 cm and its perpendicular height is 3 cm. Find the whole area.

Composite polygon: rectangle and triangular roof

The lower rectangle is 10 cm by 6 cm. The triangle above it has base 10 cm and perpendicular height 3 cm.

Split the shape
The rectangle and triangle meet along an edge. Their areas can be added to get the area of the whole polygon.
Find the rectangle's area
Multiply its width by its height.
$A_{\text{rectangle}}=10\times6=60\text{ cm}^2$
Find the triangle's area
The triangle has half the area of a rectangle with the same base and height.
$A_{\text{triangle}}=\frac{10\times3}{2}=15\text{ cm}^2$
Add the parts
The two parts cover the whole polygon without overlap.
$A_{\text{whole}}=60+15=75\text{ cm}^2$

Answer: The whole polygon's area is

75\text{ cm}^2

.

Check: The rectangle alone covers 60 square centimetres, so the whole shape must be larger. Adding the roof's 15 square centimetres gives 75 square centimetres.

Common mistakes and how to avoid them

Using a sloping side as the trapezoid's height.

Correction: Use the perpendicular distance between the parallel bases.

Using a rhombus's side lengths instead of its diagonals.

Correction: Measure from corner to opposite corner in both directions.

Adding only one part of a composite polygon, or counting a part twice.

Correction: Split the whole shape clearly, find each included part once, and combine the areas.

Writing an area answer in ordinary length units.

Correction: Use square units, such as

\text{cm}^2

.

Lesson summary

A trapezoid's area is half the sum of its parallel bases multiplied by its height.
A rhombus's or kite's area is half the product of its diagonal lengths.
For a composite polygon, find the areas of familiar parts and combine them.
Use the correct measurements and write the answer in square units.

Check your understanding

Question 1

A trapezoid has bases of 9 cm and 5 cm and a perpendicular height of 3 cm. What is its area?

21 cm²
42 cm²
27 cm²
14 cm²

Show answer and explanation

21 cm²

Add the bases, multiply by the height, and take half:

(9+5)\times3\div2=21\text{ cm}^2

.

Question 2

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

40 cm²
20 cm²
28 cm²
14 cm²

Show answer and explanation

20 cm²

Multiply the diagonal lengths and take half:

10\times4\div2=20\text{ cm}^2

.

Question 3

A composite shape is made from a rectangle with area 32 cm² and a trapezoid with area 18 cm². They join without overlap. What is the whole area?

14 cm²
50 cm²
576 cm²
32 cm²

Show answer and explanation

50 cm²

The two parts make up the whole shape, so add their areas:

32+18=50\text{ cm}^2

.

Key terms

Area: The amount of flat space a shape covers, measured in square units.
Base: One of the parallel sides used in the area rule for a trapezoid.
Height: The perpendicular distance between bases, or the perpendicular measurement used for a shape's area.
Diagonal: A straight line joining two opposite corners of a polygon.
Composite polygon: A polygon made by joining simpler shapes.
Perpendicular: Meeting at a square corner, or right angle.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation E2.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E2.4 · Find areas of trapezoids, rhombuses, kites, and composite polygons

Ontario Grade 6 Mathematics

Trapezoids, rhombuses, kites, and composite polygons

What you will learn

Use parallel sides and height for a trapezoid

Use diagonals for a rhombus or kite

Split a composite polygon into familiar shapes

Check labels, units, and reasonableness

Find the area of a trapezoid

Find the area of a rhombus

Find the area of a composite polygon

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review