Length, perimeter, area & circles

Year 7 core

By the end of this topic you should be able to:

convert between metric length units (mm, cm, m, km),
calculate the perimeter of any polygon,
find the area of a rectangle, a triangle, and a parallelogram,
name the parts of a circle (centre, radius, diameter, circumference),
use $\pi$ to relate the circumference of a circle to its radius and diameter.

1. Metric length units

Length conversions

Metric ladder (each step is x10 or /10 between adjacent units)

1 \text{ cm} = 10 \text{ mm}, \quad 1 \text{ m} = 100 \text{ cm}, \quad 1 \text{ km} = 1000 \text{ m}.

Converting

Bigger -> smaller unit: multiply. Smaller -> bigger unit: divide.

$3.5$ m $= 3.5 \times 100 = 350$ cm
$8400$ mm $= 8400 \div 10 = 840$ cm

2. Perimeter

The perimeter is the total distance around the outside of a shape. Add up every side length.

Worked example 1 Perimeter of an irregular polygon

A pentagon has sides $4$ cm, $6$ cm, $5$ cm, $7$ cm, $4$ cm. Find the perimeter.

P = 4 + 6 + 5 + 7 + 4 = 26 \text{ cm}.

3. Area of standard shapes

Area measures the amount of surface a shape covers. Units are squared: mm^2, cm^2, m^2, km^2.

Area formulas (Year 7 core)

Rectangle

A = L \times W.

Triangle

A = \tfrac{1}{2} \times b \times h,

where $b$ is the base and $h$ is the perpendicular height from the base to the opposite vertex.

Parallelogram

A = b \times h,

where $h$ is the perpendicular height - the same as for a triangle on the same base.

Square (special case of rectangle)

A = s^{2}.

Triangle and parallelogram with base b and perpendicular height h.

Worked example 2 Triangle area

A triangle has base $8$ cm and perpendicular height $5$ cm. Find its area.

A = \tfrac{1}{2} \times 8 \times 5 = 20 \text{ cm}^{2}.

Worked example 3 Parallelogram area

A parallelogram has base $12$ m and perpendicular height $7$ m. Find its area.

A = 12 \times 7 = 84 \text{ m}^2.

4. Circles: parts and the role of pi

Parts of a circle: centre, radius r, diameter d, circumference C.

The centre is the point in the middle.
The radius ( $r$ ) is the distance from the centre to any point on the circle.
The diameter ( $d$ ) is the distance straight across, through the centre; it equals $2r$ .
The circumference ( $C$ ) is the total distance around the circle.

Circle relationships

Diameter and radius

d = 2r.

Circumference in terms of pi

C = \pi \times d \;=\; 2 \pi r.

The symbol $\pi$ (pi) is the constant ratio of circumference to diameter. $\pi \approx 3.14159 \approx \tfrac{22}{7}$ .

Worked example 4 Circumference from a radius

A circle has radius $5$ cm. Find its circumference, using $\pi \approx 3.14$ .

C = 2 \pi r = 2 \times 3.14 \times 5 = 31.4 \text{ cm}.

Worked example 5 Diameter from a circumference

A circle has circumference $44$ m. Find its diameter, using $\pi \approx \tfrac{22}{7}$ .

d = \frac{C}{\pi} = 44 \div \frac{22}{7} = 44 \times \frac{7}{22} = 14 \text{ m}.

Practice: Year 7 core

Fluency

Length and perimeter

Convert $250$ cm to metres.
Convert $3.4$ km to metres.
Convert $85$ mm to cm.
Convert $2500$ m to km.
Find the perimeter of a rectangle $12$ cm by $7$ cm.
Find the perimeter of a square with side $9$ cm.
A triangle has sides $5$ cm, $7$ cm, $9$ cm. Find its perimeter.
A regular hexagon has side $4$ cm. Find its perimeter.

Fluency

Area of rectangles, triangles, parallelograms

Find the area of a $15$ cm by $4$ cm rectangle.
Find the area of a square with side $6$ cm.
Find the area of a triangle with base $10$ cm and height $8$ cm.
Find the area of a triangle with base $12$ m and height $5$ m.
Find the area of a parallelogram with base $9$ cm and height $6$ cm.
A rectangle has area $48$ cm^2 and length $8$ cm. Find its width.
A triangle has area $36$ cm^2 and base $9$ cm. Find its height.
A parallelogram has area $60$ m^2 and height $5$ m. Find its base.

Fluency

Circles

Use $\pi \approx 3.14$ unless another value is given.

A circle has radius $4$ cm. Find its diameter.
A circle has diameter $18$ m. Find its radius.
Find the circumference of a circle with radius $10$ cm.
Find the circumference of a circle with diameter $14$ cm, using $\pi \approx \tfrac{22}{7}$ .
A circle has circumference $31.4$ cm. Find its diameter.
True or false: for every circle, circumference $\div$ diameter gives about the same number.

Reasoning

Explain and reason

Two rectangles have the same perimeter. Does it follow that they have the same area? Justify with a numerical example.
Pete writes the area of a triangle with base $6$ and slant side $5$ as $\tfrac{1}{2} \times 6 \times 5 = 15$ . Explain what is wrong.
Is it possible for one shape to have a larger perimeter but a smaller area than another? Give an example.
A pizza of diameter $30$ cm is cut exactly in half. What is the perimeter of each half-pizza (the crust plus the straight cut)?
Without calculating, decide which has the larger area: a square with side $10$ cm or a rectangle $12$ cm by $8$ cm. Explain.

Problem solving

Real-world problems

A rectangular paddock is $80$ m by $45$ m. What is the cost of fencing it at $18 per metre?
A triangular sail has base $3.5$ m and height $4$ m. What is its area in square metres?
A circular garden pond has diameter $3.5$ m. How long is a rope needed to go once around the edge? (Use $\pi \approx \tfrac{22}{7}$ .)
A rectangular garden is $8$ m by $12$ m. Convert the area to square centimetres.
A bicycle wheel has diameter $70$ cm. How far (to the nearest metre) does the bike travel in $20$ turns of the wheel? (Use $\pi \approx \tfrac{22}{7}$ .)

Extension

More area formulas (extension)

Trapezium

A = \tfrac{1}{2}(a + b) \times h,

where $a$ and $b$ are the parallel sides and $h$ is the perpendicular distance between them.

Composite shapes

Break the shape into rectangles and triangles (or start with a big rectangle and subtract the missing piece).

Practice: Extension

Reasoning

Trapezium and composite areas

Find the area of a trapezium with parallel sides $4$ cm and $10$ cm and height $3$ cm.
Find the area of a trapezium with parallel sides $6$ m and $10$ m and height $4$ m.
An L-shape is made of an $8$ m by $5$ m rectangle with a $3$ m by $2$ m rectangle removed from one corner. Find its area.
A path $1$ m wide runs around a $10$ m by $6$ m garden, on the outside. Find the area of the path.
A rectangular piece of cardboard is $40$ cm by $30$ cm. A $5$ cm square is cut from each corner. What is the remaining area?

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Year 7 core - answers

Fluency

Length and perimeter

$2.5$ m
$3400$ m
$8.5$ cm
$2.5$ km
$38$ cm
$36$ cm
$21$ cm
$24$ cm

Fluency

Area of rectangles, triangles, parallelograms

$60$ cm^2
$36$ cm^2
$40$ cm^2
$30$ m^2
$54$ cm^2
$6$ cm
$8$ cm. Method: $\tfrac{1}{2} \times 9 \times h = 36$ .
$12$ m. Method: $b \times 5 = 60$ .

Fluency

Circles

$8$ cm
$9$ m
$62.8$ cm. Method: $2 \times 3.14 \times 10$ .
$44$ cm. Method: $\tfrac{22}{7} \times 14$ .
$10$ cm. Method: $d = C / \pi = 31.4 / 3.14$ .
True - that constant ratio is $\pi$ .

Reasoning

Explain and reason

Not necessarily. Example: a $4 \times 6$ rectangle has perimeter $20$ and area $24$ ; a $2 \times 8$ rectangle also has perimeter $20$ but area $16$ . Same perimeter, different area.
Pete used the slant side, not the perpendicular height. Without the perpendicular height, the area formula cannot be applied directly; more information is needed.
Yes. Example: a thin $1 \times 20$ rectangle has perimeter $42$ and area $20$ , while a $5 \times 5$ square has perimeter $20$ and area $25$ . The first has a larger perimeter but smaller area.
$30 + \tfrac{1}{2}(\pi \times 30) \approx 30 + 47.1 = 77.1$ cm. Method: cut across the pizza is $30$ cm; half the circumference is $\tfrac{1}{2}\pi d$ .
Square wins. Square area $= 100$ ; rectangle area $= 96$ . Among rectangles with the same perimeter ( $40$ here), the square has the greatest area.

Problem solving

Real-world problems

$4500. Method: perimeter $= 2(80 + 45) = 250$ ; cost $= 250 \times 18$ .
$7$ m^2. Method: $\tfrac{1}{2} \times 3.5 \times 4$ .
$11$ m. Method: $C = \pi d = \tfrac{22}{7} \times 3.5 = 11$ .
$960\,000$ cm^2. Method: $96$ m^2; $1$ m^2 $= 10\,000$ cm^2.
$44$ m (to nearest metre). Method: $C = \tfrac{22}{7} \times 70 = 220$ cm per turn $= 2.2$ m; $\times 20 = 44$ m.

Extension - answers

Reasoning

Trapezium and composite areas

$21$ cm^2. Method: $\tfrac{1}{2}(4 + 10) \times 3 = 21$ .
$32$ m^2. Method: $\tfrac{1}{2}(6 + 10) \times 4 = 32$ .
$34$ m^2. Method: $40 - 6$ .
$36$ m^2. Method: outer $12 \times 8 = 96$ ; garden $10 \times 6 = 60$ ; path $= 96 - 60$ .
$1100$ cm^2. Method: $40 \times 30 - 4 \times 5^2 = 1200 - 100$ .

Prefer paper? Print the answer key as a separate booklet: open print view ->