Geometry flashcards
12 cards on Geometry: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 12
Geometry
Angles on a line and round a point
What do angles on a straight line add up to, and angles round a point?
Answer it in your head, then turn the card over.
Angles on a line and round a point
Key points
- Angles on a straight line add up to 180°: half a full turn.
- Angles round a point add up to 360°: one full turn.
- Take the angles you know away from the total to find the one you do not.
Example
On a line with 65°: x = 180° − 65° = 115°
Round a point with 120°, 95° and 80°:
y = 360° − 295° = 65°
Watch out
Never measure an angle on a diagram marked "not drawn accurately": work it out from the facts.
Geometry
Vertically opposite angles
When two straight lines cross, which angles are equal, and how do you write the reason?
Answer it in your head, then turn the card over.
Vertically opposite angles
Key points
- The angles facing each other across the crossing are vertically opposite, and they are equal.
- Each pair of neighbouring angles sits on a straight line, so they add up to 180°.
- Write the fact in words beside each step: the reason earns marks as well as the number.
Two lines cross, making 50°
Next to it: 180° − 50° = 130° (angles on a straight line add to 180°)
Opposite it: 50° (vertically opposite angles are equal)
So the angles are 50°, 130°, 50° and 130°, adding to 360°.
Watch out
Call them vertically opposite, not alternate or corresponding: those two names are for parallel lines.
Geometry
Angles in triangles and quadrilaterals
What do the angles in a triangle add up to, and in a quadrilateral, and why?
Answer it in your head, then turn the card over.
Angles in triangles and quadrilaterals
Key points
- The angles in a triangle add up to 180°.
- The angles in a quadrilateral add up to 360°.
- A diagonal splits any quadrilateral into two triangles, and 2 × 180° = 360°.
Example
Triangle with 48° and 75°: 180° − 123° = 57°
Quadrilateral with 90°, 110° and 85°:
360° − 285° = 75°
Watch out
Use the right total: 180° for three corners, 360° for four. Count the corners before you start.
Geometry
Isosceles and equilateral triangles
What do you know about the angles in an isosceles triangle and an equilateral triangle?
Answer it in your head, then turn the card over.
Isosceles and equilateral triangles
Key points
- An isosceles triangle has two equal sides, and the two base angles opposite them are equal.
- The angle between the two equal sides is the odd one out.
- An equilateral triangle has three equal sides, so each angle is 180° ÷ 3 = 60°.
- Mark equal sides with dashes and equal angles with arcs as you spot them.
Example
40° between the equal sides:
each base angle = (180° − 40°) ÷ 2 = 70°
A base angle of 72°: the third angle = 180° − 2 × 72° = 36°
Watch out
Check which angle you were given: a base angle has a twin, but the angle between the equal sides does not.
Geometry
Angles in parallel lines
When a line crosses two parallel lines, which angles are equal and which add up to 180°?
Answer it in your head, then turn the card over.
Angles in parallel lines
Key points
- Alternate angles are equal: they make a Z shape, on opposite sides of the crossing line.
- Corresponding angles are equal: they sit in matching positions, making an F shape.
- Co-interior angles add up to 180°: they sit between the lines on the same side, making a C shape.
One angle between the parallels is 68°
Its alternate angle is 68° (alternate angles are equal).
Its co-interior angle is 180° − 68° = 112° (co-interior angles add to 180°).
Watch out
Write "alternate angles are equal" as the reason: "Z-angles" helps you spot them but earns no mark.
Geometry
Chains of angle reasons
How do you set out an angle problem that needs more than one fact?
Answer it in your head, then turn the card over.
Chains of angle reasons
Key points
- Find one angle at a time, and write each one on the diagram as you go.
- Give every step its reason, using the full name of the fact.
- Look for parallel-line arrows: they usually unlock the next angle.
Corresponding angle 74°, then find x
The corresponding angle is 74° (corresponding angles are equal).
x sits on a straight line with it:
x = 180° − 74° = 106° (angles on a straight line add to 180°)
Watch out
A right answer with no reasons can miss most of the marks when the question says "give reasons".
Geometry
Interior angle sum of a polygon
How do you find the total of the interior angles of a polygon with n sides?
Answer it in your head, then turn the card over.
Interior angle sum of a polygon
Key points
- Diagonals from one corner split the polygon into n − 2 triangles.
- Each triangle holds 180°, so the interior angles add up to (n − 2) × 180°.
- A pentagon holds 540° and a hexagon 720°.
Formula
sum of interior angles = (n − 2) × 180°
Pentagon: 110°, 120°, 95°, 100° and x
Sum = (5 − 2) × 180° = 540°
x = 540° − 425° = 115°
Watch out
Use n − 2 triangles, not n: a pentagon holds 3 × 180°, not 5 × 180°.
Geometry
Angles of a regular polygon
How do you find the exterior and interior angles of a regular polygon?
Answer it in your head, then turn the card over.
Angles of a regular polygon
Key points
- The exterior angles of any polygon add up to 360°: walking round it is one full turn.
- For a regular polygon, each exterior angle = 360° ÷ n.
- An interior and an exterior angle sit on a straight line, so interior = 180° − exterior.
- Working back: the number of sides = 360° ÷ the exterior angle.
Example
Regular octagon: exterior = 360° ÷ 8 = 45°
interior = 180° − 45° = 135°
Exterior angle 24°: 360 ÷ 24 = 15 sides
Watch out
360 ÷ n is the exterior angle, not the interior one: a regular hexagon's interior angle is 120°, not 60°.
Geometry
Areas of triangles and parallelograms
How do you find the area of a rectangle, a triangle and a parallelogram?
Answer it in your head, then turn the card over.
Areas of triangles and parallelograms
Key points
- Rectangle and parallelogram: area = base × height.
- Triangle: area = ½ × base × height, because it is half the rectangle round it.
- The height is the perpendicular height, at right angles to the base, never a slanted side.
Formulae
triangle = ½ × b × h
parallelogram = b × h
Example
Triangle, base 6 cm, height 4 cm: ½ × 6 × 4 = 12 cm²
Parallelogram, base 8 cm, height 5 cm: 8 × 5 = 40 cm²
Watch out
Forgetting the half for a triangle doubles the answer: base 6 cm and height 4 cm gives 12 cm², not 24 cm².
Geometry
Area of a trapezium
How do you find the area of a trapezium?
Answer it in your head, then turn the card over.
Area of a trapezium
Key points
- Add the two parallel sides and halve: that is their average.
- Multiply the average by the perpendicular height.
- Area is always in square units, such as cm² or m².
Formula
area = ½(a + b)h
Sides 5 cm and 9 cm, height 4 cm
Average of the parallel sides: (5 + 9) ÷ 2 = 7 cm
Area = 7 × 4 = 28 cm²
Watch out
Leaving out the half gives 56 cm², twice the real area.
Geometry
Area of a compound shape
How do you find the area of an L-shaped compound shape?
Answer it in your head, then turn the card over.
Area of a compound shape
Key points
- Split it into rectangles (and triangles if needed).
- Find any missing lengths by adding or subtracting the sides you are given.
- Work out each part's area and add them. Or take the missing corner away from the big rectangle.
10 by 8 cm, less a 4 by 3 cm corner
Split: 10 × 5 = 50 and 6 × 3 = 18, so 50 + 18 = 68 cm²
Or subtract: 10 × 8 − 4 × 3 = 80 − 12 = 68 cm²
Watch out
Find each missing length before you multiply: here the short sides are 10 − 4 = 6 cm and 8 − 3 = 5 cm.
Geometry
Perimeter of a compound shape
How do you find the perimeter of a compound shape?
Answer it in your head, then turn the card over.
Perimeter of a compound shape
Key points
- The perimeter is the distance all the way round the outside.
- Work out any missing sides first, by adding and subtracting.
- Trace the outline once and add every edge. Lines inside the shape never count.
The L-shape: 10 by 8 cm, less 4 by 3 cm
Missing sides: 10 − 4 = 6 cm and 8 − 3 = 5 cm
Perimeter = 10 + 8 + 6 + 3 + 4 + 5 = 36 cm
Watch out
Adding each rectangle's own perimeter counts the dividing line inside, which is not part of the outline.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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