Geometry and measures

180°

提示It is exactly half of a full turn.

为什么A 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

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

为什么Put 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°

提示The three corners together must fill half a turn.

为什么The 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°

提示Share the triangle total equally between three identical corners.

为什么All 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

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. 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.

5 cm

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

为什么3² + 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

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.