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4.045 Dimensions from area


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.


$A$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.


$A$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.


$A$A $=$= $\frac{1}{2}bh$12bh

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.

A triangle is has a base positioned vertically. The altitude of a triangle is drawn perpendicularly from the vertex of the triangle to the base, creating two right angled triangles. The altitude is the height of the triangle labeled h cm, and the base is labeled 6 cm.

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.

A right-angled triangle is drawn with a dashed line indicating its height 'h cm' dropping from the vertex opposite the right angle to the base. The base is labeled as 10cm. There are arrows on both ends of the base line and the height line indicating that they are measurements. The hypotenuse extends from the endpoint of the base opposite the right angle to the peak of the triangle.



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