electronicity.space

USS · Paper II

Geometry & mensuration

Lines, angles, triangles, symmetry, perimeter, area and volume.

6 of the 50 questionsWrite this topic →
What the paper asks

Six questions. Angles, triangles, and mensuration — perimeter, area, surface area and volume. Mensuration takes the most marks off Class 7 students because the formulas are close enough to be swapped under time pressure. Units are the fastest way to catch that.

Angles
  • Acute < 90°, right = 90°, obtuse between 90° and 180°, straight = 180°, reflex > 180°.
  • Complementary angles add to 90°. Supplementary angles add to 180°.
  • Angles of a triangle add to 180°; of a quadrilateral, 360°.
  • The exterior angles of any polygon add to 360°, whatever the number of sides.
  • Interior angle sum of an n-sided polygon = (n − 2) × 180°.
Mensuration — the formulas, with units
ShapeFormulaUnit
Square perimeter4 × sidecm
Square areaside²cm²
Rectangle perimeter2(l + b)cm
Rectangle areal × bcm²
Triangle area½ × base × heightcm²
Parallelogram areabase × heightcm²
Circle circumference2πrcm
Circle areaπr²cm²
Cube surface area6 × side²cm²
Cube volumeside³cm³
Cuboid volumel × b × hcm³
Units are a free check
  • A length answer is in cm, an area in cm², a volume in cm³.
  • If the question asks for area and your working gives a length, you used the wrong formula. This catches the perimeter-for-area swap every time.
  • For a circle with r = 7 and π = 22/7: circumference 44 cm, area 154 cm². Same numbers in, very different answers out.
Circles, and what happens when the radius doubles
  • Use π = 22/7 when the radius is a multiple of 7 — it cancels and the arithmetic stays whole.
  • Diameter = 2 × radius. If a question gives the circumference and asks for the diameter, do not stop at the radius.
  • Doubling the radius multiplies the area by four, not by two, because the radius is squared. r = 7 gives 154 cm²; r = 14 gives 616 cm².
Triangles and quadrilaterals
  • Equilateral — three equal sides, each angle 60°. Isosceles — two equal. Scalene — none equal.
  • Pythagoras: in a right-angled triangle, a² + b² = c². Legs 3 and 4 give a hypotenuse of 5 — square first, then add, then take the root.
  • Square diagonals: equal, bisect each other, meet at 90°. Rectangle diagonals: equal and bisect, but not at 90°.
  • A rectangle has 2 lines of symmetry; a square has 4.
  • A cube has 6 faces, 8 vertices, 12 edges.
Worked through

Find the area of a circle with radius 14 cm. (π = 22/7)

  1. Area = πr² = 22/7 × 14².
  2. 14² = 196.
  3. 22/7 × 196 = 22 × 28 = 616.

The circumference of a circle is 44 cm. Find its diameter. (π = 22/7)

  1. 2πr = 44, so 2 × 22/7 × r = 44.
  2. 44r/7 = 44, so r = 7.
  3. Diameter = 2r = 14. The question asked for the diameter, not the radius.

A right-angled triangle has legs 3 cm and 4 cm. Find the hypotenuse.

  1. a² + b² = 3² + 4² = 9 + 16 = 25.
  2. c = √25 = 5.
Where the marks go

These are the mistakes, not the rules — each one is something a student who knows the topic still does under time pressure.

  1. Giving the perimeter when the area was asked, or 2πr when πr² was wanted. Check the units.
  2. Forgetting the ½ in the triangle area formula — that gives the surrounding rectangle instead.
  3. Confusing surface area with volume for a cube. 6s² is cm²; s³ is cm³.
  4. Doubling the area when the radius doubles. It quadruples.
  5. Adding the legs instead of squaring them in Pythagoras. 3 + 4 = 7 is not the hypotenuse.
← PreviousAlgebra
Write it

Sit 10 questions on this topic

Read the explanation on every one you get wrong — that is where the revision above actually sticks.

 Or the whole paper

Sit Paper II against the clock

Fifty questions in ninety minutes, drawn to the real subject mix. The scorecard will tell you what to revise next.

← All USS revision