Hard GCSE Perimeter and area Questions

Write down the rule for the area of a triangle.

A triangle is half of a rectangle with the same base and the same perpendicular height, so $A = \frac{1}{2}bh$.

Write down the new base.

The base is doubled.

Write down the new perpendicular height.

The perpendicular height is halved.

Write down the new area.

Substitute the new base and the new height into the rule for the area of a triangle.

Simplify the new area.

The factor of $2$ and the factor of $\frac{1}{2}$ cancel each other out exactly.

Compare the new area with the old one.

The two areas are identical, so the area has not changed.

Test the result on a particular triangle.

Take a triangle with base $6$ cm and perpendicular height $8$ cm, of area $24$ square centimetres.

Change the triangle as the question says.

The new triangle has base $12$ cm and perpendicular height $4$ cm.

Work out the area of the new triangle.

The new area is $24$ square centimetres — exactly the same as before.

Work out the scale factor for the area.

The area is multiplied by $1$, which is another way of saying it is unchanged.

Rule out doubling and halving.

Neither $48$ nor $12$ is the new area, so the area is neither doubled nor halved.

Rule out the factor of four.

Neither $96$ nor $6$ is the new area either. A factor of $4$ would need both lengths to be doubled.

Explain why the two changes cancel.

The area depends on the product of the base and the height, and that product is unchanged when one is doubled and the other is halved.

Note what would change.

The triangle is now long and low rather than short and tall, so its perimeter has changed even though its area has not.

Select the correct statement.

The area is unchanged.

Write down the area of the original rectangle.

Call the length $l$ and the width $w$.

Write down the new length.

Every length is multiplied by $4$.

Write down the new width.

The width is a length too, so it is multiplied by the same number.

Multiply the new length by the new width.

The new area is still length times width.

Simplify the new area.

The two factors of $4$ multiply together to give $16$.

Compare the new area with the old one.

The area has been multiplied by $16$, whatever the original rectangle was.

Test the result on a particular rectangle.

Take a rectangle $3$ cm by $2$ cm, of area $6$ square centimetres.

Scale that rectangle.

The scaled rectangle is $12$ cm by $8$ cm.

Work out the new area.

The new area is $96$ square centimetres.

Divide the new area by the old area.

The example agrees: the area is multiplied by $16$.

Rule out multiplying the area by the length scale factor.

Multiplying the area by $4$ gives $24$, not $96$. This is the most common error.

Rule out the other two multipliers.

Neither gives $96$, so neither can be right.

Rule out the statement that the area is unchanged.

The rectangle gets bigger in both directions, so its area certainly changes.

State the general rule.

Multiplying every length by $k$ multiplies every area by $k^2$, because an area is made from two lengths.

Select the correct statement.

The area is multiplied by $16$.

Call the width x and write the length in terms of x.

The length is $3$ times the width.

Write down the rule for the area of a rectangle.

The area of a rectangle is its length multiplied by its width.

Write the area in terms of x.

Length times width gives an expression with $x$ squared in it, because an area is built from two lengths.

Form an equation from the area.

The area is $108$ square centimetres.

Divide both sides by the coefficient.

This leaves $x^2 = 36$.

Take the positive square root.

A width must be positive, so the negative root is rejected: the width is $6$ cm.

Work out the length.

The length is $18$ cm.

Check the area of the rectangle you have found.

The area comes back to $108$ square centimetres, as it should.

Write down the rule for the perimeter of a rectangle.

The perimeter is the total distance round the outside. A rectangle has two lengths and two widths, so $P = 2(l + w)$.

Add the length and the width.

One length and one width together are $24$ cm.

Double that total.

A rectangle has two lengths and two widths.

Check the perimeter a second way.

The perimeter is $2(3 + 1)$ times the width, which gives the same $48$ cm.

Check the ratio of the sides.

The length really is $3$ times the width.

Check the units of the answer.

A perimeter is a length, so it is measured in centimetres, not in square centimetres.

Work out the perimeter of the rectangle.

The perimeter is $48$ cm.

Call the width x and write the length in terms of x.

The length is $4$ times the width.

Write down the rule for the area of a rectangle.

The area of a rectangle is its length multiplied by its width.

Write the area in terms of x.

Length times width gives an expression with $x$ squared in it, because an area is built from two lengths.

Form an equation from the area.

The area is $144$ square centimetres.

Divide both sides by the coefficient.

This leaves $x^2 = 36$.

Take the positive square root.

A width must be positive, so the negative root is rejected: the width is $6$ cm.

Work out the length.

The length is $24$ cm.

Check the area of the rectangle you have found.

The area comes back to $144$ square centimetres, as it should.

Write down the rule for the perimeter of a rectangle.

The perimeter is the total distance round the outside. A rectangle has two lengths and two widths, so $P = 2(l + w)$.

Add the length and the width.

One length and one width together are $30$ cm.

Double that total.

A rectangle has two lengths and two widths.

Check the perimeter a second way.

The perimeter is $2(4 + 1)$ times the width, which gives the same $60$ cm.

Check the ratio of the sides.

The length really is $4$ times the width.

Check the units of the answer.

A perimeter is a length, so it is measured in centimetres, not in square centimetres.

Work out the perimeter of the rectangle.

The perimeter is $60$ cm.

Split the shape into the rectangle and the triangle.

The two pieces do not overlap, so their areas simply add.

Write down the rule for the area of a rectangle.

The area of a rectangle is its length multiplied by its width.

Work out the area of the rectangle.

The rectangle has an area of $98$ square centimetres.

Subtract the rectangle from the whole shape to get the area of the triangle.

The triangle must have an area of $35$ square centimetres.

Write down the base of the triangle.

The triangle stands on the top side of the rectangle, so its base is the same as the width of the rectangle, $14$ cm.

Write down the rule for the area of a triangle.

A triangle is half of a rectangle with the same base and the same perpendicular height, so $A = \frac{1}{2}bh$.

Form an equation for the perpendicular height.

The only unknown left is the perpendicular height of the triangle.

Multiply both sides by two.

Doubling clears the fraction.

Divide both sides by the base.

Dividing by the base leaves the perpendicular height on its own.

Check the area of the triangle.

The triangle really does have an area of $35$ square centimetres.

Check the area of the whole shape.

The rectangle and the triangle together give $133$ square centimetres, as the question says.

Check the answer with a single expression for the whole shape.

The whole shape is its width times the sum of the height of the rectangle and half the height of the triangle. That gives $\frac{h}{2} = 2.5$, so $h = 5$ again.

Check the overall height of the shape.

The shape stands $12$ cm tall in total, which fits the picture.

Check the units of the answer.

A perimeter is a length, so it is measured in centimetres, not in square centimetres.

Work out the perpendicular height of the triangle.

The perpendicular height of the triangle is $5$ cm.