Mensuration

Placement papers ask 1 or 2 mensuration questions: areas of triangles, rectangles, circles and trapeziums, volumes of cylinders and cones, and 'melt one solid into another'. Once you can picture the shape, each question is one formula. So this lesson draws every shape first, and says what each letter in the formula stands for.

Perimeter is the distance around a flat shape. Walk around a field and count your steps. It is a length, measured in cm or m.

Area is how much flat surface a shape covers. Count how many 1 cm × 1 cm squares fit inside. It is measured in square units: cm², m².

Volume is how much space a solid takes up. Count how many 1 cm cubes fit inside. It is measured in cubic units: cm³, m³. Capacity (how much liquid it holds) is volume too: 1000 cm³ = 1 litre.

Surface area is the total area of all the faces of a solid: the paint you would need to cover a box. 'Curved surface area' (CSA) counts only the curved part; 'total surface area' (TSA) adds the flat ends as well.

Triangles are named in two ways. By their sides: equilateral (all three sides equal), isosceles (two sides equal), scalene (all different). By their biggest angle: acute (every angle under 90 degrees), right-angled (one angle exactly 90), obtuse (one angle over 90).

The three angles of every triangle add up to 180 degrees. So an equilateral triangle has three 60-degree angles, and an isosceles triangle has its two base angles equal.

A triangle can be both kinds at once: a right-angled isosceles triangle has angles 90, 45, 45.

Area of any triangle = 1/2 × base × height. The height is the perpendicular (straight-down) distance from the top corner to the base, not a slanting side. A triangle is exactly half of a rectangle with the same base and height; that is where the 1/2 comes from.

If you only know the three sides, use Heron's formula: find s = half the perimeter, then area = √(s(s - a)(s - b)(s - c)).

In a right-angled triangle, Pythagoras connects the sides: hypotenuse² = base² + height². Some whole-number triples come up again and again. Learn them and many questions need no square roots at all.

Equilateral triangle with side a: height = (√(3)/2) × a, area = (√(3)/4) × a².

A rectangle has opposite sides equal and all angles 90 degrees. Area = length × breadth. Perimeter = 2(length + breadth). The diagonal cuts it into two right triangles, so diagonal = √(l² + b²).

A square is a rectangle with all four sides equal. Area = a². Perimeter = 4a. Diagonal = a × √(2). If you know only the diagonal d, the area is d² / 2.

A parallelogram is a 'pushed-over' rectangle: opposite sides are parallel and equal, but the corners are not 90 degrees.

Area = base × height, where the height is the perpendicular distance between the two parallel sides. Cut the triangle off one end and slide it to the other and you get a rectangle with the same base and height. That is why.

The slanted side is NOT the height. Using it is the most common mistake here.

A rhombus is a parallelogram with all four sides equal: a diamond. Its two diagonals cut each other at right angles and in half.

Area = 1/2 × d1 × d2 (half the product of the diagonals). The diagonals split the rhombus into 4 right triangles, and together those 4 triangles make half of the rectangle d1 × d2.

Because the diagonals meet at 90 degrees, side² = (d1/2)² + (d2/2)². That lets you get the side from the diagonals.

A trapezium has exactly one pair of parallel sides (the top a and bottom b), with a perpendicular height h between them.

Area = 1/2 × (a + b) × h: the average of the two parallel sides times the height. Put two copies of the trapezium together, one upside down, and they make a parallelogram with base (a + b) and height h. Half of that is one trapezium.

If the non-parallel sides are equal, it is an isosceles trapezium; drop two heights and the bottom splits into a, and two equal pieces of (b - a)/2.

Every point on a circle is the same distance r (radius) from the centre. The diameter d = 2r goes all the way across.

Circumference (perimeter) = 2 π r. Area = π r². Use π = 22/7 when the radius is a multiple of 7, otherwise 3.14.

A sector is a slice, like a pizza slice, with angle t at the centre. It is t/360 of the whole circle, so its area is (t/360) × π r² and its curved edge (arc) is (t/360) × 2 π r.

A semicircle is a sector of 180 degrees. Its perimeter is the curved half (π r) PLUS the diameter (2r). Forgetting the diameter is a classic trap.

A ring (a circular path) is a big circle minus a small one: π (R² - r²).

A polygon is any closed shape with straight sides: pentagon (5), hexagon (6), octagon (8). 'Regular' means all sides and all angles are equal.

From one corner you can cut an n-sided polygon into (n - 2) triangles. Each triangle has 180 degrees, so the interior angles add up to (n - 2) × 180. In a regular polygon each interior angle is that divided by n.

The exterior angles (the turn you make at each corner walking around) always add to 360, so each exterior angle of a regular polygon is 360/n.

A regular hexagon is 6 equilateral triangles, so its area is 6 × (√(3)/4) a² = (3 √(3)/2) a².

A cuboid is a box with length l, breadth b and height h. Volume = l × b × h: the number of 1 cm cubes that fill it.

It has 6 rectangular faces in 3 matching pairs (top/bottom, front/back, left/right). Total surface area = 2(lb + bh + hl). The four walls of a room without floor and ceiling (lateral surface) = 2h(l + b).

The longest rod that fits inside goes corner to corner: diagonal = √(l² + b² + h²).

A cube has l = b = h = a: volume a³, surface 6a², diagonal a √(3).

A cylinder is a circle pulled straight up to height h: a pipe, a tin, a drum. Volume = base area × height = π r² h.

Unroll the curved side and it becomes a rectangle: one side is the circumference 2 π r, the other is h. So curved surface area = 2 π r h. Add the two circular ends for the total: 2 π r h + 2 π r² = 2 π r (r + h).

A hollow pipe with outer radius R and inner radius r has volume of material π (R² - r²) h.

A cone is a circle that narrows to a point: an ice-cream cone, a tent. Its height h goes straight up the middle; its slant height l runs down the side.

h, r and l make a right triangle, so l = √(r² + h²).

A cone holds exactly one third of a cylinder with the same base and height: volume = (1/3) π r² h. (Fill a cone with water and pour it into the matching cylinder: it takes three cones to fill it.)

Curved surface area = π r l. Total = π r (l + r).

A sphere is a perfect ball: every point on it is r from the centre. Volume = (4/3) π r³. Surface area = 4 π r² (exactly four circles of the same radius).

A hemisphere is half a ball: a bowl. Volume = (2/3) π r³. Its curved surface is 2 π r²; a solid hemisphere also has a flat top, so total surface = 3 π r².

Cut the top off a cone with a slice parallel to the base and the bottom piece is a frustum: a bucket or a glass. With top radius r, bottom radius R and height h, volume = (1/3) π h (R² + r² + Rr).

A prism has the same shape all the way through (a triangular chocolate box, a pencil). Volume = area of the base shape × length.

A pyramid rises from a base shape to a point. Like a cone, it is one third of the matching prism: volume = (1/3) × base area × height.

If every length of a shape grows by a factor k, its area grows by k² and its volume by k³. Double the radius of a ball and it needs 4 times the paint but holds 8 times the air.

As percentages: if each side rises by x%, the area rises by (2x + x²/100)%. A 10% longer side gives 21% more area and 33.1% more volume.

When one solid is melted and recast into another, nothing is lost: the volume stays the same. Write 'volume before = volume after', cancel π, and solve.