Sometimes we know the area or perimeter of a given shape and we would like to work out its dimensions (for example width, length, height or radius). The methods required vary depending on the shape and whether it is the area or the perimeter that is known.
Finding dimensions from perimeter
In the case of polygons, if we know the perimeter then we can work out an unknown side length by subtracting known side lengths from the perimeter. For circles if we know the circumference then we will need to re-arrange the formula for the circumference in order to find the radius. For a sector, if we know the arc length then we also need to know the angle in order to work out the radius from the arc length formula. Alternatively, we might know the arc length and the radius, and then we can work out the angle.
Practice questions
Question 1
Find the side length $y$y indicated on the diagram. The perimeter of the shape is $29$29 m.
Question 2
Find the length, in centimetres, of the longest side of the parallelogram shown. Use $s$s as the length of the side.
Question 3
Consider the circle below.
What is the radius $r$r of the circle?
Round your answer to two decimal places.
Question 4
The sector below has a contained angle with a measure of $115^\circ$115° and an arc length of $23$23 cm. Find the radius $r$r of the sector in centimetres.
Round your answer to two decimal places.
Question 5
The sector below has a radius of $4.2$4.2 cm and an arc length of $3.3$3.3 cm. Find the measure of the contained angle $\theta$θ in degrees.
Round your answer to two decimal places.
Finding dimensions from area
Area calculations typically involve a formula expressing the area in terms of a product of dimensions of the shape. Therefore, if we know the area of a shape and we want to work backwards to a length, width, height, radius or an angle, we will need to re-arrange the relevant area formula. You might like to remind yourself of these area formulas.
Practice questions
Question 6
Find the width of the rectangle shown with an area of $24$24 m2 and a length of $8$8 m.
Question 7
Find the value of $h$h in the triangle with base length $6$6 cm if its area is $54$54 cm2.
Question 8
Find the height $\left(h\right)$(h) if the area of the trapezium shown is $24$24 cm2.
Start by substituting the given values into the formula for the area of a trapezium.
$A=\frac{1}{2}\left(a+b\right)h$A=12(a+b)h
Question 9
The sector below has an area of $30$30 m2 and a contained angle measuring $64^\circ$64°.
Find the length of the radius.
Round your answer to the nearest decimal place.