Geometry and measures

55

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

The whyAngles 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

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

The whyA 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

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

The whyThe 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

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

The whyThe 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

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

The whyArea 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

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

The whyVolume 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

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

The whyAlternate 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

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

The whyEach 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

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

The whyπ 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

HintNot the full width across the circle — only the length from the centre out to the edge.

The whyThe 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

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

The whyUndoing 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

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

The whyA 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.