Properties of triangles, quadrilaterals and circles
Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths
What a learner can do afterwards
- List and verify properties of a parallelogram (opposite sides parallel/equal, opposite angles equal, diagonals bisect each other)
- Explain why a square is a special case of both a rectangle and a rhombus
- Derive the relationship between the radius, diameter, and circumference of a circle
1 · Read
A parallelogram is a four-sided shape where both pairs of opposite sides are parallel. In a parallelogram, opposite sides are always equal in length, and opposite angles are always equal to each other. The two diagonals also do something special: they bisect each other, meaning each diagonal cuts the other exactly in half at the point where they cross.
Picture a parallelogram ABCD, where AB is opposite CD, and BC is opposite AD. If AB measures 6 cm, then CD also measures 6 cm, since opposite sides are equal. If angle A measures 70 degrees, then angle C, its opposite angle, also measures 70 degrees. Now draw both diagonals, AC and BD: they cross at one point, and that point splits each diagonal into two equal halves.
A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square fits both descriptions at once: it has four right angles, like a rectangle, and four equal sides, like a rhombus. That is why a square counts as a special case of both a rectangle and a rhombus, rather than being a separate shape on its own.
For any circle, the diameter is always twice the radius: d = 2r. The circumference, the distance around the circle, is always about 3.14 times the diameter, written as C = pi times d. That number, pi, is the same for every circle, no matter how big or small.
Parallelograms have equal opposite sides and angles with diagonals that bisect each other, a square is a rectangle and a rhombus at once, and a circle's circumference is always pi times its diameter.
More to read
Types of triangles review
Triangles are sorted by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). The angles of any triangle add to 180 degrees, so knowing two angles reveals the third. That sum is the most-used triangle fact in problem solving. Special triangles earn their names by symmetry: equilateral has three equal sides and three 60-degree angles, isosceles has two of each. Spotting the equal marks tells you which angles match before any calculation starts.
Quadrilaterals review
Quadrilaterals are four-sided shapes with family rules. Parallelograms (both pairs of opposite sides parallel) guarantee equal opposite sides, equal opposite angles, and diagonals that bisect each other. Rectangles, rhombuses and squares all inherit these guarantees. Test a claimed property against the whole family: equal diagonals hold for rectangles but fail for general parallelograms, so they are not a parallelogram rule. Only properties surviving every member count as family traits.
Properties of shapes: FAQ
Shape families nest: every square is a rectangle (four right angles) and also a rhombus (four equal sides). A square qualifies for both clubs at once because it meets both entry rules. Hierarchy questions test definitions, not pictures. A rhombus need not be square (angles can slant), and a rectangle need not be a rhombus (sides can differ). The square sits in the overlap: the only shape passing both tests. Draw the overlap diagram to lock the logic in.
Hierarchy of quadrilaterals
The quadrilateral hierarchy runs from general to special: quadrilateral, then trapezoids and parallelograms, then rectangles and rhombuses, then the square at the peak. Each level adds one rule and keeps all earlier guarantees. Moving down never loses traits: a square still has bisecting diagonals from its parallelogram ancestors. Moving up drops rules: a general parallelogram need not have right angles. Inheritance flows one way, from broad families to special members.
3.1: Circumference of a Circle
Circles link three measures: the radius (centre to edge), the diameter (twice the radius, edge to edge through the centre) and the circumference (the distance round, pi times diameter). For radius 5 cm, the diameter is 10 cm and the round trip is about 31.4 cm. One turn of a wheel lays out exactly one circumference: diameter 40 cm rolls 3.14 x 40 = 125.6 cm. Radius, diameter and circumference move together, so finding one from another is one multiplication away.
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به انگلیسیTake it off screen
Where it sits
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.