Shapes — triangles, quadrilaterals, circles and solids
Triangles by side and by angle with the four rules that decide most questions; the quadrilateral family and the one-way relationships inside it; the parts of a circle; and the faces, edges and vertices of the seven solids the papers ask about, with Euler's check.
What this heading covers
Shapes are asked in two ways, and they reward different work. Either the question names a property and asks which shape has it, or it gives a shape and asks for a count — sides, angles, faces, edges, vertices, diagonals. The first needs the family relationships held straight. The second needs a small table held exactly.
Almost nothing here is calculated. It is recalled, so it is worth recalling correctly.
Triangles
By sides
- Equilateral — three equal sides, and therefore all three angles 60°.
- Isosceles — two equal sides, and the angles opposite them equal.
- Scalene — no two sides equal, no two angles equal.
By angles
- Acute — all three angles under 90°.
- Right — exactly one angle of 90°.
- Obtuse — exactly one angle over 90°.
A triangle can have at most one right angle and at most one obtuse angle, because the three must add to 180°. "A triangle with two right angles" is impossible, and questions do offer it.
The four rules that decide most triangle questions
- Angle sum = 180°.
- Exterior angle = the sum of the two opposite interior angles. It follows from the angle sum and a linear pair, and it shortcuts a great many questions.
- Triangle inequality — the sum of any two sides is greater than the third. A question giving three lengths and asking whether a triangle is possible is applying exactly this. Test the two shortest against the longest; if that passes, all three do.
- Pythagoras, for a right triangle only — hypotenuse² = base² + height². The triples worth recognising on sight are 3-4-5, 5-12-13 and 8-15-17, along with their multiples (6-8-10, 9-12-15).
Area = ½ × base × height. The height is the perpendicular from the vertex to the base, not a slanting side.
Median, altitude, bisector are distinguished in questions: a median joins a vertex to the midpoint of the opposite side; an altitude is the perpendicular from a vertex to the opposite side; an angle bisector cuts the angle in half. In an equilateral triangle all three coincide — which is why that is the shape the question chooses when it wants them to be the same line.
Quadrilaterals
The angle sum of every quadrilateral is 360°.
| Shape | Defining properties |
|---|---|
| Square | 4 equal sides, 4 right angles; diagonals equal, bisect each other at 90° |
| Rectangle | Opposite sides equal, 4 right angles; diagonals equal and bisect each other |
| Parallelogram | Opposite sides parallel and equal, opposite angles equal; diagonals bisect each other |
| Rhombus | 4 equal sides, opposite angles equal; diagonals bisect each other at 90° |
| Trapezium | Exactly one pair of parallel sides |
| Kite | Two pairs of adjacent sides equal; one diagonal bisects the other at 90° |
The relationships run one way
Every square is also a rectangle, a rhombus and a parallelogram. Every rectangle and every rhombus is a parallelogram. The reverse is false in each case.
Questions are built on that asymmetry: "Every rhombus is a square" is false; "Every square is a rhombus" is true. Read which way the sentence points before answering.
The quickest way to tell a rectangle from a rhombus by diagonals alone: a rectangle has diagonals that are equal; a rhombus has diagonals that meet at 90°. A square has both, which is what makes it a square.
Circles
- The radius joins the centre to any point on the circle.
- The diameter passes through the centre and equals 2 × radius. It is the longest chord.
- A chord joins any two points on the circle.
- An arc is a part of the circle itself.
- A sector is bounded by two radii and an arc — the shape of a slice.
- A segment is bounded by a chord and an arc.
- A tangent touches the circle at exactly one point, and is perpendicular to the radius there.
Circumference = 2πr = πd. Area = πr². Use π as 22/7 when the radius is a multiple of 7, and 3.14 otherwise — the question is usually built so one of the two divides cleanly.
Sector and segment are swapped often. A sector has two straight edges; a segment has one.
Solids
Faces, edges and vertices are asked directly, so the table is worth holding exactly.
| Solid | Faces | Edges · vertices |
|---|---|---|
| Cube | 6 | 12 · 8 |
| Cuboid | 6 | 12 · 8 |
| Cylinder | 3 (2 flat, 1 curved) | 2 · 0 |
| Cone | 2 (1 flat, 1 curved) | 1 · 1 (the apex) |
| Sphere | 1 (curved) | 0 · 0 |
| Square pyramid | 5 | 8 · 5 |
| Triangular prism | 5 | 9 · 6 |
Euler's formula checks any polyhedron: F + V − E = 2. For a cube, 6 + 8 − 12 = 2. Use it when a question gives two of the three counts and asks for the third, and use it to check your own recall before committing.
Euler's formula applies to polyhedra — solids with flat faces. The cylinder, cone and sphere have curved surfaces and do not satisfy it, which is itself a question that gets set.
A net is the flat shape that folds into a solid. Questions show a net and ask which solid it makes, or show a solid and ask which net is wrong. Count the faces in the net first: a cube net must have exactly six squares.
A question of the kind the paper sets
The diagonals of a quadrilateral are equal and bisect each other at right angles. What is it?
Take the conditions one at a time.
- Diagonals bisect each other — it is a parallelogram.
- Diagonals are equal — among parallelograms, that means a rectangle.
- Diagonals meet at right angles — among parallelograms, that means a rhombus.
A shape that is both a rectangle and a rhombus is a square.
The question is not testing the square. It is testing whether each diagonal property is attached to the right shape, and it fails anybody who has learned the table as six separate rows rather than as a family.
Mistakes that cost marks here
- Reading the family relationship backwards — answering that every rhombus is a square.
- Using Pythagoras on a triangle that is not right-angled. It holds for right triangles only.
- Taking a slanting side as the height when finding the area of a triangle. The height is perpendicular to the base.
- Swapping sector and segment. Two straight edges means a sector.
- Applying Euler's formula to a cylinder or cone. It is for flat-faced solids.
- Giving a cone two vertices, or a cylinder some. A cone has one, a cylinder none.
- Forgetting the triangle inequality and accepting any three lengths as a triangle.
In one line
Hold the triangle's four rules and the quadrilateral family's one-way relationships, keep the faces-edges-vertices table exact and check it with F + V − E = 2, and remember that Pythagoras needs a right angle and the parallelogram properties need the diagonals to bisect.
For the classroom
Solids are the part children get wrong on paper and right in their hands. A box, a tin and a ball on the table, counted together, fixes the faces-edges-vertices table far more firmly than the table does — and a child who has counted the two flat faces and one curved face of a tin does not later award a cylinder six faces.
For quadrilaterals, cutting a set of shapes and asking children to sort them into overlapping hoops — has four equal sides, has four right angles — puts the square in the overlap physically. That is the one-way relationship the exam asks about, discovered rather than recited.
Where this comes from
Scope follows the shape headings of four syllabuses: WBBPE's Primary TET (Mathematics, Content: Shapes), CBSE's CTET Paper I (Shapes and spatial understanding; Solids around us) and Paper II (Understanding elementary shapes, 2-D and 3-D), and the Upper Primary headings for shapes and spatial relationships. Properties and the faces-edges-vertices counts follow NCERT's elementary mathematics usage. The worked question and the error list are written for this lesson from the pattern of past papers rather than quoted from one, and which errors recur is our own reading rather than a Board statement.
Also on this subtopic
Other material filed under Shapes.