Geometry and measures

180°

HintIt is exactly half of a full turn.

The whyA straight line is half a complete rotation, so the angles sitting along it must fill that half turn. Angles meeting around a point fill a whole turn instead.

A protractor

HintIt is the clear plastic semicircle with two scales printed round its curved edge.

The whyPut its centre on the vertex and one arm along the zero line, then read the scale that starts at zero. Two scales exist so you can measure from either side.

70°

HintThe three corners together must fill half a turn.

The whyThe angles of any triangle total 180°, so taking 40 and 70 away from 180 leaves the missing one. Because two angles here match, the triangle is isosceles.

60°

HintShare the triangle total equally between three identical corners.

The whyAll three sides are the same length, so all three angles must match, and 180 ÷ 3 gives each one. An equilateral triangle also has three lines of symmetry.

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. 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.

5 cm

HintSquare each short side, add those results together, then undo the squaring.

The why3² + 4² = 9 + 16 = 25, and the square root of 25 gives the 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.