Checkpoint 6 Maths · Geometry and Measure · 6Gg.03

Area of Rectangles and Right-angled Triangles

Find the area of rectangles, see why a right-angled triangle is half a rectangle, estimate areas, and solve Checkpoint problems with compound shapes.

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Use what you know about the area of rectangles to estimate and calculate the area of right-angled triangles. (6Gg.03)
This means: find the area of a rectangle (length × width), see that a right-angled triangle is exactly half of a rectangle, estimate areas by rounding or by counting squares, and solve problems with these shapes, including shapes made of rectangles and triangles.

In this lesson you will learn what area means, how to find the area of any rectangle, and then the big idea of the page: every right-angled triangle is half of a rectangle. Once you understand that one picture, you never need to learn a formula by heart, because you can always work it out again.

1. Key words

Word / phrase	Meaning / use
area	المساحة — the amount of flat surface a 2D shape covers
square centimetre (cm²)	سنتيمتر مربع — the area of a square 1 cm long and 1 cm wide
square metre (m²)	متر مربع — the area of a square 1 m by 1 m (for rooms, gardens, fields)
rectangle	مستطيل — 4 straight sides, 4 right angles, opposite sides equal
right angle	زاوية قائمة — an angle of exactly 90°, shown with a small square
right-angled triangle	مثلث قائم الزاوية — a triangle with one right angle
base	القاعدة — one of the two sides that meet at the right angle
height (perpendicular height)	الارتفاع العمودي — the other side at the right angle; it is at 90° to the base
estimate	تقدير — a sensible answer that is close to the exact answer, found quickly
perimeter	المحيط — the distance all the way round a shape (NOT the same as area)

2. Area is counted in squares

Area tells us how many unit squares cover a shape. If each square is 1 cm long and 1 cm wide, each square has an area of 1 cm² (say "one square centimetre"). The small ² shows that we are measuring in squares, not in lines.

A rectangle 5 cm long and 3 cm wide is covered by 15 squares, so its area is 15 cm².

We do not need to count every square. There are 3 rows and each row has 5 squares, so we multiply: 3 × 5 = 15. That is why:

Area of a rectangle

Area of a rectangle = length × width (some books say base × height). The answer is in square units: cm², m² or mm². Always multiply two lengths in the same unit.

Rectangle	Working	Area
8 cm by 3 cm	8 × 3	24 cm²
12 m by 9 m	12 × 9	108 m²
square with side 7 cm	7 × 7	49 cm²
6.5 cm by 4 cm	6.5 × 4	26 cm²
2.25 m by 3 m	2.25 × 3	6.75 m²

Area or perimeter? Perimeter is the distance around the edge (add the sides). For the 5 cm by 3 cm rectangle, the perimeter is 5 + 3 + 5 + 3 = 16 cm, but the area is 15 cm². Different questions, different units: perimeter in cm, area in cm².

3. A right-angled triangle is half a rectangle

Draw a rectangle and cut it from one corner to the opposite corner (along a diagonal). You get two identical right-angled triangles. Each one has exactly half the area of the rectangle.

The 6 cm by 4 cm rectangle has area 24 cm². The diagonal cuts it into two equal right-angled triangles of 12 cm² each.

Area of a right-angled triangle

Area = base × height ÷ 2. In words: make the rectangle that the triangle is half of, find its area, then halve it. The base and the height are the two sides that meet at the right angle.

Look for the small square that marks the right angle. The two sides that touch that square are the base and the height. The third side is the longest side. It is opposite the right angle and it is slanted. We do not use it to find the area. It does not matter which way round the triangle is drawn: turn the page in your head until the right angle is at the bottom.

Three right-angled triangles in different positions. In each one, the base and the height are the two sides at the right angle.

Find the right angle (the small square).
Write down the two sides that meet there: these are the base and the height. Ignore the longest (slanted) side.
Multiply them: this is the area of the whole rectangle.
Divide by 2: this is the area of the triangle.
Write the unit with a small ²: cm², m² or mm².

You can halve first if it is easier. For a triangle with base 14 cm and height 9 cm: 14 ÷ 2 = 7, then 7 × 9 = 63 cm². This is the same as 14 × 9 = 126 and 126 ÷ 2 = 63 cm². Choose the order that makes the mental maths easy (this uses the laws of arithmetic).

الفكرة الأساسية بالعربي

المثلث القائم الزاوية هو نص مستطيل بالظبط. اضرب الضلعين اللي عند الزاوية القائمة (المربع الصغير) وبعدين اقسم على ٢. الضلع الطويل المايل ماندخلوش في الحساب خالص. وماتنساش تكتب الوحدة مربعة: cm².

4. Estimating area

Checkpoint questions often say estimate. An estimate is a quick, sensible answer. There are two good ways to estimate area.

Method A: round the lengths. Round each length to a number that is easy to multiply (usually the nearest whole number or nearest 10), then multiply (and halve for a triangle). Example: a rectangle 7.9 cm by 5.2 cm has area about 8 × 5 = 40 cm². A right-angled triangle with sides 19.6 cm and 10.3 cm at the right angle has area about 20 × 10 ÷ 2 = 100 cm².

Method B: count squares on a grid. Count the whole squares inside the shape. Then count the part squares and treat each part square as about half a square (or count only the parts that are half or more). This is useful when the shape is drawn on squared paper.

Counting squares: 8 whole squares and 8 part squares give an estimate of about 12 cm². The rule gives 6 × 4 ÷ 2 = 12 cm², so the estimate is a good one.

Use an estimate to check your exact answer. If you calculate 299 cm² but your estimate was 30 cm², you know something has gone wrong (perhaps a decimal point in the wrong place).

5. Shapes made of rectangles and triangles

Many shapes can be split into a rectangle and a right-angled triangle. Find each area separately, then add them. Sometimes it is easier to find a big rectangle and subtract the part that is missing.

Split the shape along the dashed line: rectangle A (10 m by 6 m) and right-angled triangle B (base 4 m, height 6 m).

Draw a dashed line to split the shape into a rectangle and a right-angled triangle.
Work out any missing lengths (opposite sides of a rectangle are equal; lengths along one side add up).
Find the area of each part.
Add the parts (or subtract a cut-out part from a whole rectangle).
Give the unit.

6. Worked examples

Worked example 1 (no calculator) A rectangular poster is 9 cm long and 6 cm wide. (a) Work out the area of the poster. [1] (b) Ahmed has a piece of card 19.8 cm by 4.1 cm. Estimate its area. [1]

(a) Area = length × width = 9 × 6 = 54 cm².
(b) Round: 19.8 ≈ 20 and 4.1 ≈ 4. Estimate = 20 × 4 = 80 cm². (The exact area is 81.18 cm², so 80 is a good estimate.)
Common mistake: writing 54 cm (a length) instead of 54 cm² (an area).

Worked example 2 A right-angled triangle has sides 8 cm, 6 cm and 10 cm. The right angle is between the 8 cm side and the 6 cm side. Calculate the area of the triangle. [1]

The sides at the right angle are 8 cm and 6 cm. The 10 cm side is the longest side, so we do not use it.
Rectangle: 8 × 6 = 48 cm². Triangle: 48 ÷ 2 = 24 cm².
Common mistakes: 8 × 10 ÷ 2 = 40 (used the longest side) or 8 × 6 = 48 (forgot to halve).

Worked example 3 (calculator allowed) A right-angled triangle has base 9.8 cm and height 6.1 cm. (a) Estimate the area. Show your working. [1] (b) Calculate the exact area. [1]

(a) 9.8 ≈ 10 and 6.1 ≈ 6. Estimate = 10 × 6 ÷ 2 = 30 cm².
(b) 9.8 × 6.1 = 59.78, then 59.78 ÷ 2 = 29.89 cm².
Check: 29.89 is close to the estimate of 30, so the answer is sensible.

Worked example 4 A right-angled triangle is drawn on a centimetre-square grid (the triangle in section 4). Its two sides at the right angle are 6 squares and 4 squares long. (a) Estimate the area by counting squares. [1] (b) Use the rectangle to find the exact area. [1]

(a) Whole squares: 8. Part squares: 8, each about half, so 8 ÷ 2 = 4. Estimate ≈ 8 + 4 = 12 cm².
(b) The triangle is half of a 6 cm by 4 cm rectangle: 6 × 4 = 24, 24 ÷ 2 = 12 cm².

Worked example 5 (working backwards) A right-angled triangle has an area of 18 cm². One of the sides at the right angle is 4 cm. Find the length of the other side at the right angle. [2]

The triangle is half of a rectangle, so the rectangle has area 18 × 2 = 36 cm². (1 mark for 36 seen)
Rectangle: 4 × ? = 36, so ? = 36 ÷ 4 = 9 cm.
Check: 4 × 9 ÷ 2 = 18 ✓
Common mistake: 18 ÷ 4 = 4.5 (forgot that the triangle is only half the rectangle).

Worked example 6 (compound shape) The diagram in section 5 shows the plan of a garden. Work out its total area. [2]

Rectangle A: 10 × 6 = 60 m².
Triangle B: base 4 m, height 6 m: 4 × 6 ÷ 2 = 12 m².
Total: 60 + 12 = 72 m². (1 mark for a correct method for one part, 1 mark for 72)
Common mistake: using 14 m × 6 m = 84 m², treating the whole shape as a rectangle.

Common mistakes

Using the longest (slanted) side in the calculation. Only the two sides at the right angle are used.
Forgetting to divide by 2 for a triangle.
Dividing by 2 twice: for example, halving the base and then halving the answer again.
Writing cm instead of cm². Area is always in square units.
Mixing up area and perimeter: perimeter is adding the sides; area is the space inside.
Multiplying lengths in different units (for example 2 m × 50 cm). Change them to the same unit first.
In a compound shape, counting a part twice or using a length that belongs to the other part.

How Checkpoint asks this

"Calculate the area of the triangle." A labelled diagram, often with all three sides given: choose the two at the right angle.
"Estimate the area." Show the rounded numbers you used (for example 10 × 6 ÷ 2). An exact answer with no rounding may not get the mark.
Shapes on a centimetre-square grid: count the squares, or find the rectangle around the triangle.
"Draw a right-angled triangle with an area of 10 cm²." Choose two sides that multiply to 20 (for example 5 cm and 4 cm).
Working backwards: the area is given and you find a missing side. Double the area first.
"Tick (✓) to show whether Sara is correct. Explain how you know." Give a number reason, for example "6 × 4 ÷ 2 = 12, not 24".
Calculator questions (Paper 2) can use decimals such as 7.5 cm × 3.2 cm. On Paper 1 the numbers are easier to work out in your head or on paper.
Diagrams may say "not drawn to scale": use the numbers written, not a ruler.

خلي بالك

المساحة وحدتها مربعة (cm² أو m²) — المحيط وحدته عادية (cm أو m).
لو المساحة معطاة وعايز ضلع: اضرب المساحة في ٢ الأول، وبعدين اقسم على الضلع التاني.

7. Practice

Remember

Area is the space inside a shape, measured in square units: cm², m², mm².
Area of a rectangle = length × width.
A right-angled triangle is half of a rectangle: area = base × height ÷ 2, using the two sides at the right angle.
Never use the longest (slanted) side for the area.
Estimate by rounding the lengths or counting squares, and use the estimate to check your answer.
For shapes made of parts: split, find each area, then add (or subtract a cut-out).

All pages in this course

Start here

Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests