GCSE Maths · Perimeter, Area and Circles · Higher · Summary

GCSE Maths Perimeter, Area and Circles (Higher): summary and key words

Part of the GCSE Maths perimeter, area and circles chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Perimeter and area of a rectangle / square	Perimeter = 2(l + w) or 4s (add the sides, unit cm). Area = l × w or s × s (unit cm²)
Area of a triangle	½ × base × perpendicular height. Use the dotted height, never a slanting side
Area of a parallelogram	base × perpendicular height
Area of a trapezium	½ × (a + b) × h, where a and b are the parallel sides
Compound shapes	Split and add, or complete a rectangle and subtract. Find missing lengths first. Perimeter: add every outside side
Missing side / height	Put the numbers in the formula and undo it: divide by the other length; double the area first for a triangle or trapezium
Units of area	1 m² = 10 000 cm²; 1 cm² = 100 mm²; 1 km² = 1 000 000 m². Square the length conversion
Area problems in context	Same units, find the area, divide by what one item covers, round UP, multiply by the price. Fencing uses perimeter
Circle words	Centre, radius, diameter (d = 2r), chord, tangent, arc, sector, segment, circumference
Circumference of a circle	C = πd = 2πr. Use the π button, or leave π in the answer (“in terms of π”)
Area of a circle	A = πr². Use the radius; square it first, then multiply by π
Semicircle and quarter circle	Area ½πr² or ¼πr². Perimeter = curved part + straight edges (diameter, or two radii)
Composite shapes with circles	Split and add or subtract; keep full calculator values, round once at the end; or keep π for exact answers
Working backwards from C or A	From C: divide by π to get d, halve for r. From A: divide by π, then square root
Arc length and sector area	Fraction = θ/360. Arc = fraction × 2πr. Area = fraction × πr². Sector perimeter = arc + 2r
(Higher) Segments	Area = sector − triangle (½r² sin θ). Exact: sin 30° = ½, sin 60° = √3/2, sin 90° = 1
(Higher) Algebraic and exact forms	Write the area/perimeter with x, simplify, solve. Keep π and √ as symbols. Show-that: finish on the stated expression

Can you…?

I can find the perimeter and area of rectangles, squares, triangles, parallelograms and trapezia, and say where the formulas come from.
I can split a compound shape into simple pieces (or subtract a piece) and find missing lengths.
I can convert between cm², m², mm² and km² and solve area problems about paint, tiles, turf and fencing.
I can name the parts of a circle: centre, radius, diameter, chord, tangent, circumference, arc, sector and segment.
I can find circumferences and areas of circles, semicircles and quarter circles, with a calculator and in terms of π.
I can find the area of composite shapes made with circles and work backwards to find a radius or diameter.
I can find arc lengths and areas of sectors, and the angle or radius of a sector.
(Higher) I can find the area and perimeter of segments, give exact answers with π and surds, and work with algebraic lengths.

Key terms

Perimeter: The distance all the way round the outside of a shape. Measured in length units such as cm or m.
Area: The amount of flat space inside a shape. Measured in square units such as cm² or m².
Perpendicular height: The distance from the base to the opposite corner (or the parallel side), measured at 90° to the base.
Parallelogram: A four-sided shape with two pairs of parallel sides.
Trapezium: A four-sided shape with exactly one pair of parallel sides.
Compound (composite) shape: A shape made by joining simple shapes, or by cutting a piece out of a simple shape.
Circumference: The distance all the way round a circle.
Radius: A straight line from the centre of a circle to its circumference.
Diameter: A straight line across a circle through the centre. It is twice the radius.
Chord: A straight line joining two points on a circle.
Tangent: A straight line that touches a circle at exactly one point.
Arc: A part of the circumference of a circle.
Sector: A slice of a circle between two radii and an arc.
Segment: The region of a circle between a chord and an arc.
π (pi): The number 3.14159… equal to the circumference of any circle divided by its diameter.
In terms of π: An exact answer that keeps the symbol π, for example 12π cm².
Surd: An exact square root that is not a whole number, such as √3.