Formulas · Mathematics

Perimeter and area of flat shapes

6 formulas, each with worked examples. Revision sheet · Practise these formulas

Foundation

Perimeter of a rectangle

P = 2 (l + b)

What each letter means
$P$	the perimeter: the distance all the way round (length)
$l$	the length (length)
$b$	the breadth (width) (length)

Going round a rectangle you walk each length once and each breadth once, twice over.

Why it works

A rectangle has two sides of length l and two of length b. Adding all four: l + b + l + b = 2(l + b).

When to use it

Fencing, borders, frames, lace round a tablecloth: any time you need the distance round a rectangular shape.

How to use it

Add the length and the breadth.
Double the sum.
Give the answer in units of length (m, cm).

To remember: Round the rectangle: length plus breadth, twice.

Other forms

Special case $P = 4 l$ When a square (b = l).
Rearranged $b = \frac{P}{2} - l$ The breadth, from the perimeter and the length.

Worked examples

Story

A garden is 20 m long and 15 m wide. How much fencing goes all the way round it?

$P = 2 (20 + 15) = 2 \times 35 = 70$
It needs 70 m of fencing.

Answer: 70

Picture

Every rectangle below has a perimeter of 24 cm. Which one has the largest area?

l + b = 12, so the rectangles are 11 × 1, 10 × 2, 9 × 3, 8 × 4, 7 × 5 and 6 × 6.
Their areas are 11, 20, 27, 32, 35 and 36 cm².
The square, 6 cm × 6 cm, has the largest area.

Answer: the 6 cm by 6 cm square, with area 36 cm²

Direct

Find the perimeter of a rectangle 12 cm long and 5 cm wide.

$P = 2 (12 + 5) = 2 \times 17 = 34$

Answer: 34

Reverse

A rectangle has a perimeter of 50 cm and a length of 15 cm. How wide is it?

Try it first, then show the working

$b = \frac{50}{2} - 15 = 25 - 15 = 10$

Answer: 10

Exam

The length of a rectangle is 3 cm more than its breadth and its perimeter is 46 cm. Find its length and breadth.

Try it first, then show the working

Let the breadth be w cm; then the length is w + 3 cm.
$2 (w + 3 + w) = 46$
$2 w + 3 = 23$
$w = 10$
So the breadth is 10 cm and the length is 13 cm. Check: 2(13 + 10) = 46.

Answer: length 13 cm, breadth 10 cm

Common mistake: Multiplying length by breadth (that is the area), or adding only one length and one breadth.

Foundation

Area of a rectangle

A = l b

What each letter means
$A$	the area (area)
$l$	the length (length)
$b$	the breadth (width) (length)

A rectangle l units long and b units wide holds l × b unit squares.

Why it works

Rule the rectangle into unit squares: there are b rows, each with l squares, so l × b squares in all. This also works for fractional sides, by using smaller squares.

When to use it

Floors, walls, fields, screens, pages: the space covered by any rectangular surface; and, by splitting, shapes made of rectangles.

How to use it

Make sure both sides are in the same unit.
Multiply length by breadth.
Give the answer in square units (m², cm²).

To remember: Rows times columns.

Other forms

Special case $A = l^{2}$ When a square (b = l).
Rearranged $b = \frac{A}{l}$ The breadth, from the area and the length.

Worked examples

Story

A room is 6 m long and 4 m wide. How many square metres of tiles cover the floor?

$A = 6 \times 4 = 24$
It needs 24 m² of tiles.

Answer: 24

Picture

An L-shaped room is made of an 8 m by 3 m part and a 4 m by 5 m part joined together. Find its floor area.

Split the L into its two rectangles.
$8 \times 3 + 4 \times 5 = 24 + 20 = 44$

Answer: 44 m²

Direct

Find the area of a rectangle 12 cm long and 5 cm wide.

$A = 12 \times 5 = 60$

Answer: 60

Reverse

A rectangle has an area of 72 cm² and a length of 9 cm. How wide is it?

Try it first, then show the working

$b = \frac{72}{9} = 8$

Answer: 8

Exam

The length of a rectangle is twice its breadth and its area is 50 cm². Find its length and breadth.

Try it first, then show the working

Let the breadth be w cm; then the length is 2w cm.
$2 w \times w = 50$
$w^{2} = 25$
$w = 5$
So the breadth is 5 cm and the length is 10 cm.

Answer: length 10 cm, breadth 5 cm

Common mistake: Adding the sides (that is part of the perimeter), or mixing units, such as metres with centimetres.

Foundation

Area of a triangle (half base times height)

A = \frac{1}{2} bh

What each letter means
$A$	the area (area)
$b$	the base (any side) (length)
$h$	the height: the perpendicular distance from the base to the opposite corner (length)

A triangle is exactly half of a parallelogram with the same base and height.

Why it works

Put a copy of the triangle upside down against it: the two make a parallelogram of base b and height h, with area bh. One triangle is half of that.

When to use it

Any triangle whose base and height you know or can find: sails, roofs, ramps, plots, and the triangles you get by cutting up other shapes.

How to use it

Choose a side as the base.
Find the height to that side (perpendicular, from the opposite corner).
Multiply base by height and halve.

To remember: Half the box it fits in.

Other forms

Rearranged $h = \frac{2 A}{b}$ The height to a side, from the area.
Special caseAn equilateral triangle of side a has height (√3/2)a, so its area is (√3/4)a².When all three sides equal a.

Worked examples

Story

A triangular sail has a base of 4 m and a height of 6 m. How much cloth is in it?

$A = \frac{1}{2} \times 4 \times 6 = 12$
The sail has 12 m² of cloth.

Answer: 12

Picture

A right-angled triangle has shorter sides 6 cm and 8 cm. Find its area and the height to its longest side.

The two shorter sides meet at the right angle, so one is the base and the other the height: area = 1/2 × 6 × 8 = 24 cm².
The longest side is √(6² + 8²) = √100 = 10 cm.
Height to it: 2 × 24 ÷ 10 = 4.8 cm.

Answer: area 24 cm², height 4.8 cm

Direct

A triangle has a base of 10 cm and a height of 6 cm. Find its area.

$A = \frac{1}{2} \times 10 \times 6 = 30$

Answer: 30

Reverse

A triangle has an area of 45 cm² and a base of 9 cm. What is its height?

Try it first, then show the working

$h = \frac{2 \times 45}{9} = 10$

Answer: 10

Exam

Find the area of an equilateral triangle of side 6 cm.

Try it first, then show the working

The height splits the base into 3 and 3, so h² = 6² − 3² = 27, and h = 3√3 cm.
$\frac{1}{2} \times 6 \times 33 = 93$
That is about 15.6 cm², the same as (√3/4) × 6².

Answer: 9√3 cm², about 15.6 cm²

Common mistake: Forgetting the half, or using a slanted side as the height.

Class 9

Area of a parallelogram

A = bh

What each letter means
$A$	the area (area)
$b$	the base (any side) (length)
$h$	the height: the perpendicular distance between the base and the opposite side (length)

A parallelogram has the same area as a rectangle with the same base and the same height.

Why it works

Cut off the right triangle at one slanted end and slide it to the other end. The pieces fit together into a rectangle b long and h high, and nothing was added or lost.

When to use it

Slanted fields and tiles, and comparing shapes on the same base between the same parallel lines (they all have equal area).

How to use it

Choose a side as the base.
Find the height: the perpendicular distance to the opposite side, not the slanted side.
Multiply base by height.

To remember: Base times the straight-up height.

Other forms

Rearranged $h = \frac{A}{b}$ The height, from the area and the base.
Same formula, another formParallelograms on the same base and between the same parallel lines have equal areas, because they share b and h.

Worked examples

Story

A field shaped like a parallelogram has a side of 50 m and is 30 m across from that side to the opposite one. Find its area.

$A = 50 \times 30 = 1500$
The field is 1500 m².

Answer: 1500

Picture

Two different parallelograms stand on the same 6 cm base, between the same pair of parallel lines 4 cm apart. Compare their areas.

Both have base 6 cm and height 4 cm (the gap between the parallel lines).
$6 \times 4 = 24$
Both are 24 cm²: equal, however slanted they are.

Answer: both 24 cm²

Direct

A parallelogram has a base of 8 cm and a height of 5 cm. Find its area.

$A = 8 \times 5 = 40$

Answer: 40

Reverse

A parallelogram has an area of 96 cm² and a base of 12 cm. What is its height?

Try it first, then show the working

$h = \frac{96}{12} = 8$

Answer: 8

Exam

A parallelogram has sides 10 cm and 6 cm, and the distance between the longer sides is 4 cm. Find the distance between the shorter sides.

Try it first, then show the working

Using the longer side as base: area = 10 × 4 = 40 cm².
Using the shorter side as base: 6 × h = 40.
$h = \frac{40}{6} = \frac{20}{3}$

Answer: 20/3 cm, about 6.7 cm

Common mistake: Multiplying the two sides together. The slanted side is longer than the height, so that gives too big an answer.

Class 9

Heron's formula (area of a triangle from its sides)

A = s (s - a) (s - b) (s - c)

What each letter means
$A$	the area of the triangle (area)
$s$	the semi-perimeter: half the perimeter, (a + b + c) ÷ 2 (length)
$a$	the first side (length)
$b$	the second side (length)
$c$	the third side (length)

The area of a triangle can be found from its three sides alone, with no height: work out the semi-perimeter s, then take the square root of s(s − a)(s − b)(s − c).

Why it works

Drop a height and use Pythagoras on the two right triangles it makes to find the height from the sides; putting that into half base times height and simplifying gives exactly this expression. The book checks it on an equilateral triangle, an isosceles triangle and the 3, 4, 5 triangle.

When to use it

You know all three sides but not a height: land plots measured by their edges, triangles given only by side lengths.

How to use it

Add the three sides and halve: that is s.
Work out s − a, s − b and s − c.
Multiply s and those three together.
Take the square root.

To remember: Half the perimeter, then s times each "s minus a side".

Other forms

Same formula, another formIn the sides alone: A = (1/4)√((a + b + c)(−a + b + c)(a − b + c)(a + b − c)).
Special caseFor an equilateral triangle of side a, s = 3a/2 and the formula gives (√3/4)a².When a = b = c.

Worked examples

Story

A triangular park has sides of 50 m, 120 m and 130 m. Find its area.

$s = \frac{50 + 120 + 130}{2} = 150$
$A = 150 \times 100 \times 30 \times 20$
$= 9000000 = 3000$
Check: 50² + 120² = 130², so it is right-angled, and 1/2 × 50 × 120 = 3000 too.

Answer: 3000

Picture

Check Heron's formula on the triangle with sides 3, 4 and 5 units (the book's Example 5).

$s = \frac{3 + 4 + 5}{2} = 6$
$A = 6 \times 3 \times 2 \times 1 = 36 = 6$
Half base times height gives 1/2 × 3 × 4 = 6 as well.

Answer: 6

Direct

Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.

$s = \frac{13 + 14 + 15}{2} = 21$
$A = 21 \times 8 \times 7 \times 6$
$= 7056 = 84$

Answer: 84

Reverse

A triangle has sides 13 cm, 14 cm and 15 cm. Find the height to the 14 cm side.

Try it first, then show the working

From Heron's formula its area is 84 cm².
$h = \frac{2 \times 84}{14} = 12$

Answer: 12 cm

Exam

The sides of a triangle are in the ratio 3 to 4 to 5 and its perimeter is 144 cm. Find its area.

Try it first, then show the working

The parts add to 3 + 4 + 5 = 12, and 144 ÷ 12 = 12, so the sides are 36, 48 and 60 cm.
$s = \frac{144}{2} = 72$
$A = 72 \times 36 \times 24 \times 12 = 746496 = 864$

Answer: 864 cm²

Common mistake: Using the full perimeter instead of half of it, or forgetting the square root at the end.

Class 9

Brahmagupta's formula (area of a cyclic quadrilateral)

A = (s - a) (s - b) (s - c) (s - d)

What each letter means
$A$	the area of the quadrilateral (area)
$s$	the semi-perimeter: (a + b + c + d) ÷ 2 (length)
$a$	the first side (length)
$b$	the second side (length)
$c$	the third side (length)
$d$	the fourth side (length)

For a quadrilateral whose four corners lie on one circle (a cyclic quadrilateral), the area comes from the four sides alone: the square root of (s − a)(s − b)(s − c)(s − d).

Why it works

Brahmagupta gave this in 628 CE. A diagonal splits the quadrilateral into two triangles; because the corners are on one circle, opposite angles add to 180°, and combining the two triangle areas simplifies to this product. With one side shrunk to 0 the quadrilateral becomes a triangle and the formula becomes Heron's.

When to use it

Rectangles, squares, isosceles trapeziums and any other quadrilateral whose corners lie on a circle, when you know the four sides.

How to use it

Check that the quadrilateral is cyclic (its corners lie on one circle).
Add the four sides and halve: that is s.
Multiply (s − a), (s − b), (s − c) and (s − d).
Take the square root.

To remember: Four sides, four brackets, one square root.

Other forms

Special case $A = s (s - a) (s - b) (s - c)$ When d = 0: the quadrilateral becomes a triangle, and this is Heron's formula.

Worked examples

Story

A plot is an isosceles trapezium with parallel sides of 10 m and 4 m and slanting sides of 5 m each. Find its area.

$s = \frac{10 + 5 + 4 + 5}{2} = 12$
$A = 2 \times 7 \times 8 \times 7 = 784 = 28$
Check: its height is √(5² − 3²) = 4 m, and 1/2 × (10 + 4) × 4 = 28 m².

Answer: 28

Picture

Verify Brahmagupta's formula for a rectangle 3 units by 4 units (the book's Example 6).

$s = \frac{3 + 4 + 3 + 4}{2} = 7$
$A = 4 \times 3 \times 4 \times 3 = 144 = 12$
The same as 3 × 4.

Answer: 12

Direct

A square has sides of 5 cm (a square is cyclic). Find its area with Brahmagupta's formula.

$s = \frac{5 + 5 + 5 + 5}{2} = 10$
$A = 5 \times 5 \times 5 \times 5 = 25$
The same as 5 × 5.

Answer: 25

Reverse

A square has an area of 49 cm². Using Brahmagupta's formula, find its side.

Try it first, then show the working

For a square of side a, s = 2a, so each bracket is s − a = a.
A = √(a⁴) = a² = 49, so a = 7.

Answer: 7 cm

Exam

Show that when one side becomes 0, Brahmagupta's formula becomes Heron's formula.

Try it first, then show the working

Put d = 0: then s = (a + b + c) ÷ 2 and s − d = s.
So √((s − a)(s − b)(s − c)(s − 0)) = √(s(s − a)(s − b)(s − c)), which is Heron's formula.

Answer: with d = 0 it is exactly Heron's formula

Common mistake: Using it for a quadrilateral that is not cyclic: four sides alone do not fix the area (a square can be pushed over into a thinner rhombus with the same sides).