Perimeter and area

Geometry, grade 7 Lesson 11 of 18Learning

Perimeter and area

Perimeter

The perimeter of a shape is the length of the line around it: the sum of the lengths of all its sides. The lengths in the drawing are in cm:

The perimeter of the shape: $\displaystyle {8+3+4+5}={20}$ cm.
A perimeter is a length, so it is measured in units of length: cm, m, km.
A fence around a garden, a picture frame, the lines marking the edge of a football pitch – they are all perimeters.

Area

The area of a shape is the number of unit squares that cover it exactly. Each small square in the rectangle below has a side of 1 cm – it is one cm²:

The rectangle has 3 rows, with 5 squares in each row: $\displaystyle {3\cdot5}={15}$ squares.
So the area of the rectangle is 15 cm².
Area is measured in units of area: cm², m².

The rectangle’s area and perimeter

A rectangle with sides $\displaystyle a$ and $\displaystyle b$ has $\displaystyle b$ rows, with $\displaystyle a$ squares in each row: The area $\displaystyle S$ is the product of two sides that meet at a corner:

\displaystyle S

\displaystyle =

\displaystyle a\cdot b

The formula holds also when the sides are not whole numbers: a rectangle 2.5 cm by 4 cm has an area of $\displaystyle {2.5\cdot4}={10}$ cm².
A rectangle has two sides of length $\displaystyle a$ and two of length $\displaystyle b$ , so its perimeter $\displaystyle P$ is:

\displaystyle P

\displaystyle =

\displaystyle 2\cdot(a+b)

The square’s area and perimeter

A square is a rectangle whose sides are all equal. If its side is $\displaystyle a$ , its area is $\displaystyle a\cdot a$ , written $\displaystyle a^{2}$ for short (“a squared”):

\displaystyle S

\displaystyle =

\displaystyle a^2

\displaystyle P

\displaystyle =

\displaystyle 4\cdot a

A square with a side of 6 cm: the area is $\displaystyle {6\cdot6}={36}$ cm², and the perimeter $\displaystyle {4\cdot6}={24}$ cm.

Example: the floor of a room

The floor of Noa’s room is a rectangle 4 m long and 3.5 m wide.
We find the area of the floor and its perimeter.

The area: $\displaystyle {4\cdot3.5}={14}$ m². You need it to buy a carpet or tiles.
The perimeter: $\displaystyle {2\cdot(4+3.5)}={2\cdot7.5}={15}$ m. You need it to buy skirting boards along the walls.

Exercise 1

A basketball court is a rectangle 28 m long and 15 m wide.
Calculate the area of the court and its perimeter.

the area, in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

the perimeter, in m

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Same area, different perimeters

The three rectangles below have the same area, 12 squares, but different perimeters:

The rectangle	Area	Perimeter
12 by 1	12	26
6 by 2	12	16
4 by 3	12	14

The closer the rectangle is to a square, the smaller its perimeter.
The other way round too: rectangles with the same perimeter have different areas. A 9 by 1 rectangle, a 7 by 3 rectangle and a 5 by 5 square all have a perimeter of 20, but their areas are 9, 21 and 25.

Exercise 2

Uri has 24 m of fence.
He wants to fence a vegetable bed in the shape of a rectangle, its sides a whole number of metres long.
Write three rectangles Uri can fence, and the area of each.
Which of all these rectangles has the largest area?

A missing side

When you know the area or the perimeter of a rectangle and one of its sides, you can calculate the other side. Here are two examples.

From the area.
The area of a rectangle is 36 cm².
One of its sides is 4 cm.

The area is the product of the sides: 4 times the other side is 36.
So the other side is $\displaystyle {36:4}={9}$ cm.

From the perimeter.
The perimeter of a rectangle is 30 cm.
One of its sides is 6 cm.

The two sides of 6 cm are together $\displaystyle {2\cdot6}={12}$ cm.
That leaves $\displaystyle {30-12}={18}$ cm for the other two sides.
They are equal, so each of them is $\displaystyle {18:2}={9}$ cm.

Exercise 3

Calculate.

The area of a rectangle is 48 m², and one of its sides is 6 m.
Find the other side.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The perimeter of a rectangle is 26 cm, and one of its sides is 8 cm.
Find the other side.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The perimeter of a square is 36 cm.
Find its area.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A shape made of rectangles

Ido’s room has the shape of the letter L. The lengths in the drawing are in metres.
The example below finds the area of the room in two ways, and then its perimeter.

Cutting (the first drawing): a dashed line cuts the room into two rectangles, $\displaystyle {5\cdot6}={30}$ and $\displaystyle {3\cdot4}={12}$ . The area: $\displaystyle {30+12}={42}$ m².
Completing (the second drawing): we complete the room to a rectangle 8 by 6, whose area is $\displaystyle {8\cdot6}={48}$ . The part we added is a rectangle 3 by 2, of area 6. The area: $\displaystyle {48-6}={42}$ m².
Both ways give the same area. Choose the one with the shorter calculation.
The perimeter: $\displaystyle {8+4+3+2+5+6}={28}$ m.

Exercise 4

The lengths in the drawing are in cm.
Calculate the area of the shape and its perimeter.

the area, in cm²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

the perimeter, in cm

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: geometry, grade 7

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Angles

The sum of angles – the ray $\displaystyle BD$ runs inside $\displaystyle \angle ABC$
$\displaystyle \angle ABC$ $\displaystyle =$ $\displaystyle \angle ABD+\angle DBC$
The difference of angles
$\displaystyle \angle DBC$ $\displaystyle =$ $\displaystyle \angle ABC-\angle ABD$
The bisector $\displaystyle BD$ cuts $\displaystyle \angle ABC$ into two equal angles
$\displaystyle \angle ABD=\angle DBC=\frac{\angle ABC}{2}$
Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
The angles around a point
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$

The angles of a triangle and a quadrilateral

The angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
The two acute angles of a right triangle
$\displaystyle \alpha+\beta=90^{\circ}$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$
The angles of a quadrilateral
$\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ}$

The perimeter and area of a rectangle and a square

The area of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The perimeter of a rectangle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\cdot(a+b)$
The area of a square – $\displaystyle a$ the side
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^2$
The perimeter of a square
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4\cdot a$

The areas of polygons

The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot b}{2}$ $\displaystyle \angle C=90^{\circ}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$

The box and the cube

The surface area of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle 2(ab+bc+ac)$
The surface area of a cube – $\displaystyle a$ the edge
$\displaystyle S$ $\displaystyle =$ $\displaystyle 6a^2$
The volume of a box
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\cdot b\cdot c$
The volume of a cube
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$