Geometry and measures

55

提示The three together fill half a turn; take the two you know away from that total.

为什么Angles on a straight line total 180°, so x = 180 − 35 − 90 = 55. Angles meeting around a point fill a whole turn of 360° instead.

A reflex angle

提示Measure the smaller gap round the other side, then take that reading away from a full turn.

为什么A half-circle protractor only reads up to 180°, so for a reflex angle measure the acute or obtuse gap that is left over and subtract it from 360°. Check the answer looks bigger than a straight line before you trust it.

80

提示Count how many equal shares the parts make between them, then work out what one share is worth.

为什么The nine parts share 180°, so one part is 20° and the angles are 40°, 60° and 80°. Ratio and angle facts combine often at Year 9, and the check is always that the three angles add back to 180°.

20

提示Let the top corner be x, write the two equal corners in terms of it, and use the total for all three.

为什么The three angles are x, 4x and 4x, so 9x = 180 and x = 20. The base angles are then 80° each, and 20 + 80 + 80 = 180 checks out. Writing an equation beats guessing whenever a triangle is described by a relationship.

30 cm²

提示A triangle fills exactly half of the rectangle that boxes it in.

为什么Area of a triangle is base × perpendicular height ÷ 2, so (10 × 6) ÷ 2 gives the answer in square centimetres. The height must meet the base at a right angle.

24 cm³

提示Count how many unit cubes cover the base, then how many layers stack above it.

为什么Volume is length × width × height. Each layer here holds 12 unit cubes and there are two layers, and the units are cubic because three lengths were multiplied.

They are equal

提示Picture the Z shape — one angle inside each parallel line, on opposite sides of the crossing line.

为什么Alternate angles sit in a Z pattern and always match. Corresponding angles make an F shape and match too, while co-interior angles make a C and total 180°.

360

提示Walk once round the outline and you finish facing the way you set off.

为什么Each exterior angle is the turn you make at a corner, so walking the whole outline turns you through one complete rotation of 360°. For a regular polygon, divide that total by the number of sides.

C = πd

提示The distance round the outside is a little over three times the width across the middle.

为什么π is the fixed ratio of the distance round a circle to the distance straight across it, roughly 3.14 for every circle ever drawn. Since the diameter is twice the radius, C = 2πr says the same thing.

radius

提示Not the full width across the circle — only the length from the centre out to the edge.

为什么The formula is A = πr². Using the width across by mistake makes the answer four times too big, because doubling a length quadruples its square.

25 and 5 cm

提示Pythagoras: the square on the longest side equals the two smaller squares put together.

为什么Undoing the squaring at the end is what turns an area-style number back into a length. The 3, 4, 5 triangle is the best known right-angled triple.

V = πr²h

提示Work out the area of the circular end, then stack that end along its length.

为什么A cylinder is a prism with a circular cross-section, so its volume is cross-section area × height. Every prism works this way, whatever the shape of its end face.