Area Arena

Area Arena

Area is the space *inside* a shape, measured in square units. Perimeter is the distance *around* it, measured in ordinary units. Mixing them up is a classic dropped mark. This module ends on a garden that needs both, and on the question that separates people who have understood the difference from people who have memorised two formulas.

The words for it ▭

Five terms. The third is the one that quietly loses marks, because a diagram will often give you a length you must not use:

Which height?

A triangle has a base of 8 cm. Its sloping side is labelled 6 cm, and a dashed line inside it, at right angles to the base, is labelled 5 cm. What is its area?

• 20 cm², using ½ × 8 × 5, because the dashed perpendicular line is the height the formula asks for
• 24 cm², using ½ × 8 × 6, because 6 cm is the length of the side going up from the base
• 40 cm², using 8 × 5, because area is base times height
• It cannot be found without knowing the third side

The four polygons

Four formulas, and each one is really the rectangle formula in disguise: • Rectangle = length × width. The one everything else is built from. • Triangle = ½ × base × height. Any triangle is exactly half of the rectangle drawn round its base and height. • Parallelogram = base × height. Slice the triangle off one end and slide it to the other, and you have a rectangle of the same base and height. • Trapezium = ½ × (a + b) × height. That is the average of the two parallel sides, multiplied by the height: a rectangle whose width is the mean of the two ends. Every height here is the perpendicular one.

Area of a trapezium

A trapezium has parallel sides of 5 cm and 9 cm, and a perpendicular height of 4 cm. What is its area, in cm²?

Round the edge, or inside?

A circle has two formulas and they are constantly confused with one another. Setting them side by side makes the difference hard to lose:

Area of a circle

A circle has a radius of 5 cm. Its area can be written as k × π cm². What is the value of k?

From area back to perimeter

A square has an area of 36 cm². What is its perimeter?

• 24 cm, because a side must be 6 cm and a square has four equal sides
• 36 cm, the same number as the area
• 9 cm, since 36 ÷ 4 = 9
• 144 cm, since 36 × 4 = 144

Two routes, one answer

An awkward shape is simpler shapes joined together. Take an L-shape: a tall part 2 wide and 5 tall, with a foot 4 wide and 2 tall attached at the bottom right. Route one, split and add. The tall rectangle is 2 × 5 = 10. The foot is 4 × 2 = 8. Together, 18. Route two, surround and subtract. The whole L fits inside a rectangle 6 wide and 5 tall, which is 30. The missing piece at the top right is 4 wide and 3 tall, which is 12. So 30 − 12 = 18. The same answer, which is the point: choose whichever split has fewer awkward pieces. Subtracting is usually quicker when the shape is a rectangle with a bite out of it.

Split the L-shape

This L-shape has been split into two rectangles, with their side lengths marked. Tap the rectangle with the LARGER area.

Area of the L-shape

The L-shape is made of a rectangle 2 cm by 5 cm and a rectangle 4 cm by 2 cm. What is its total area, in cm²?

Turfing a garden

A rectangular lawn measures 12 m by 7 m, with a rectangular pond 3 m by 2 m cut out of the middle. Four decisions.

• You are ordering turf to cover the grass. Which measurement do you need?
• How much turf, allowing for the pond?
• You also want a fence round the outside of the garden. How much do you need?
• The pond is enlarged. What happens to the two quantities?

Which are true?

Select ALL THREE statements that are TRUE.

• The height in a triangle or parallelogram formula is the perpendicular height, not the length of a sloping side
• A composite area can be found by adding the parts or by subtracting a cut-out from a larger shape, and both give the same answer
• Cutting a hole in the middle of a garden changes the area to be turfed but not the length of fence needed round the outside
• The area of a circle is π multiplied by the diameter squared
• Perimeter is measured in square units, like area
• Doubling the radius of a circle doubles its area

The arena rules

The area of a shape is the space inside it, measured in _____ units, while the perimeter is the distance around it. In the triangle and parallelogram formulas, the height must be the _____ height rather than a sloping side. The area of a circle is π multiplied by the _____ squared, and doubling that radius makes the area _____ times as large.