Areas scale with the square of lengths, volumes with the cube. A 10% longer side gives 21% more area and 33.1% more volume.
How much paint for a wall, how much water in a tank, how much wire around a field? Mensuration is measuring. Every question asks one of three things: the length around (perimeter), the flat space inside (area), or the space a solid fills (volume).
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.
| Measure | Units | Think of it as |
|---|---|---|
| Perimeter | cm, m | fence around a field |
| Area | cm2, m2 | tiles on a floor |
| Volume | cm3, m3, litres | water in a tank |
| Surface area | cm2, m2 | paint on a box |
A rectangular field is 50 m long and 30 m wide. How much fencing goes around it, and how much grass is inside?
Answer: Fencing 160 m, grass 1500 m2
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.
Two angles of a triangle are 48° and 67°. Find the third angle and say what type the triangle is by its angles.
Answer: 65
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.
| Pythagorean triple (a, b, hypotenuse) | Multiples also work |
|---|---|
| 3, 4, 5 | 6, 8, 10 · 9, 12, 15 |
| 5, 12, 13 | 10, 24, 26 · 15, 36, 39 |
| 8, 15, 17 | 16, 30, 34 · 24, 45, 51 |
| 7, 24, 25 | 14, 48, 50 · 21, 72, 75 |
| 20, 21, 29 | 40, 42, 58 · 60, 63, 87 |
| 12, 35, 37 | 24, 70, 74 · 36, 105, 111 |
| 9, 40, 41 | 18, 80, 82 · 27, 120, 123 |
Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.
Answer: 84
The hypotenuse of a right triangle is 25 cm and one side is 7 cm. Find its area.
Answer: 84
An equilateral triangle has side 8 cm. Find its area.
Answer: 16 √(3)
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.
The diagonal of a square is 12 cm. Find its area.
Answer: 72
A garden 40 m × 30 m has a path 2 m wide all around it on the outside. Find the area of the path.
Answer: 296
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 parallelogram has base 15 cm and the perpendicular height to it is 8 cm. Its slanted side is 10 cm. Find its area and perimeter.
Answer: Area 120 cm2, perimeter 50 cm
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.
The diagonals of a rhombus are 16 cm and 12 cm. Find its area, side and perimeter.
Answer: Area 96 cm2, side 10 cm, perimeter 40 cm
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.
The parallel sides of a trapezium are 12 cm and 20 cm, and the distance between them is 7 cm. Find its area.
Answer: 112
An isosceles trapezium has parallel sides 10 cm and 22 cm, and each slanted side is 10 cm. Find its area.
Answer: 128
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).
The circumference of a circle is 44 cm. Find its area (π = 22/7).
Answer: 154
Find the area of a sector of angle 90° in a circle of radius 14 cm (π = 22/7).
Answer: 154
A circular park of radius 20 m has a path 7 m wide around it on the outside. Find the area of the path (π = 22/7).
Answer: 1034
A wheel of radius 35 cm makes how many turns to cover 1.1 km (π = 22/7)?
Answer: 500
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.
| Sides n | Angle sum | Each interior angle (regular) | Each exterior angle | Diagonals |
|---|---|---|---|---|
| 3 | 180 | 60 | 120 | 0 |
| 4 | 360 | 90 | 90 | 2 |
| 5 | 540 | 108 | 72 | 5 |
| 6 | 720 | 120 | 60 | 9 |
| 8 | 1080 | 135 | 45 | 20 |
| 10 | 1440 | 144 | 36 | 35 |
| 12 | 1800 | 150 | 30 | 54 |
Each interior angle of a regular polygon is 140°. How many sides does it have?
Answer: 9
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 room is 10 m long, 8 m wide and 4 m high. Find the cost of painting its four walls at Rs 20 per m2.
Answer: 2880
How many 2 cm cubes can be cut from a cuboid 10 cm × 8 cm × 6 cm?
Answer: 60
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 cylindrical tank has radius 7 m and height 10 m. How many litres does it hold (π = 22/7)?
Answer: 1540000
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 cone has radius 6 cm and height 8 cm. Find its curved surface area, in terms of π.
Answer: 60 π
A conical tent has radius 7 m and height 24 m. Find the canvas needed (π = 22/7).
Answer: 550
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.
Find the volume of a sphere of radius 21 cm (π = 22/7).
Answer: 38808
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.
A bucket is a frustum with top radius 14 cm, bottom radius 7 cm and height 15 cm. Find its capacity in cm3 (π = 22/7).
Answer: 5390
A pyramid has a square base of side 6 m and height 10 m. Find its volume.
Answer: 120
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.
A metal sphere of radius 6 cm is melted and drawn into a wire of radius 0.2 cm. Find the length of the wire in metres.
Answer: 72
Each side of a rectangle is increased by 20%. By what percent does its area increase?
Answer: 44%
Use it when: One solid melted into one or many others.
A hemispherical bowl of radius 9 cm is full of liquid, poured into cylinders of radius 1.5 cm and height 4 cm. How many cylinders?
Answer: 54
Use it when: Any right triangle, cone slant, rhombus side or ladder question.
A ladder 13 m long reaches a window 12 m high. How far is its foot from the wall?
Answer: 5
Use it when: Sides change by x% and y%.
Length +20%, breadth -20%. Change in area?
Answer: -4%
A parallelogram has sides 12 cm and 8 cm, and the height on the 12 cm side is 5 cm. Find its area.
Answer: 60 sq cm
The diagonals of a rhombus are 30 cm and 16 cm. Find its perimeter.
Answer: 68 cm
Each interior angle of a regular polygon is 150°. How many sides does it have?
Answer: 12
How many diagonals does a regular octagon have?
Answer: 20
27 questions with full solutions.