# 4.045 Dimensions from area

Lesson

### Finding a side length

Sometimes we might know the area of a rectangle, and either the length or width. Using division, we can work out the missing value. In the case of a square, since it has equal length and width, if we know its area, we can work out its side length by taking the square root of the area.

#### Worked examples

##### Example 1

A rectangle has an area of $42$42 cm2 and a length of $7$7 cm, how wide is the rectangle?

Think: The area is $\text{length}\times\text{width}$length×width. To find the length we need to know what number multiplied by $7$7 is $42$42. We can write this formally in an equation and reverse the multiplication by dividing.

Do:

 $A$A $=$= $L\times W$L×W Write the formula. $42$42 $=$= $7\times W$7×W Substitute in known values. $42\div7$42÷7 $=$= $W$W Divide $42$42 by $7$7. $\therefore W$∴W $=$= $6$6 cm
##### Example 2

A square has an area of $64$64 cm2, what is the side length of the square?

Think: The area is length multiplied by itself, to find the length we need to know what number multiplied by itself is $64$64. We can write this formally in an equation and use the square root to find the length.

Do:

 $A$A $=$= $L^2$L2 Write the formula. $64$64 $=$= $L^2$L2 Substitute in known values. $\sqrt{64}$√64 $=$= $L$L Take the positive square root of $64$64. $\therefore L$∴L $=$= $8$8 cm

#### Practice questions

##### Question 1

Find the width of a rectangle that has an area of $27$27 mm2 and a length of $9$9 mm.

##### Question 2

Find the perimeter of a square whose area is $49$49cm2.

### Finding dimension of a triangle

Just as with rectangles we could be given the area and asked to find the base or height. To do so, we use the area formula and work backwards.

#### Worked example

##### Example 3

A triangle has an area of $24$24 cm2 and a base of $6$6 cm. What is the height of the triangle?

Think: The area is half the base multiplied by the height. To undo this, we can double the area and then divide by the given length. We can write this formally in an equation and rearrange the equation to find the height.

Do:

 $A$A $=$= $\frac{1}{2}bh$12​bh Write the formula. $24$24 $=$= $\frac{1}{2}\times b\times6$12​×b×6 Substitute in known values. $48$48 $=$= $6b$6b Multiply both sides by $2$2. $\frac{48}{6}$486​ $=$= $\frac{6b}{6}$6b6​ Divide $48$48 by $6$6. $\therefore b$∴b $=$= $8$8 cm

#### Practice questions

##### Question 3

Find the value of $h$h in the triangle with base length $6$6 cm if its area is $54$54 cm2.

##### Question 4

A gutter running along the roof of a house has a cross-section in the shape of a triangle. If the area of the cross-section is $50$50 cm2, and the length of the base of the gutter is $10$10 cm, find the perpendicular height $h$h of the gutter.

### Outcomes

#### ACMEM024

calculate areas of rectangles and triangles