Circle Formulas
With radius r and diameter d equal to 2r, the circumference is 2 pi r, and the area is pi r squared. The length of an arc with angle theta is theta over 360 times 2 pi r. The area of a sector is theta over 360 times pi r squared. If the radius doubles, the area becomes four times as large.
Circle Properties
A perpendicular from the centre to a chord bisects the chord. The angle at the centre is twice the angle at the circumference on the same arc. An angle in a semicircle is 90 degrees. Angles on the same arc are equal. Opposite angles of a cyclic quadrilateral add up to 180 degrees. A tangent is perpendicular to the radius at the point of contact. Two tangents from an external point are equal.
Plane Figures
A rectangle has area length times breadth, perimeter twice the sum, and diagonal the square root of length squared plus breadth squared. A square of side a has area a squared and diagonal a root 2. A parallelogram has area base times height. A rhombus has area half the product of its diagonals. A trapezium has area half the sum of the parallel sides times the height.
Solids
A cuboid has volume l b h, total surface 2 times lb plus bh plus hl, and diagonal the square root of l squared plus b squared plus h squared. A cube has volume a cubed, surface 6 a squared, and diagonal a root 3. A cylinder has volume pi r squared h, curved surface 2 pi r h and total surface 2 pi r times r plus h. A cone has slant height the square root of r squared plus h squared, volume one third pi r squared h, curved surface pi r l and total surface pi r times r plus l. A sphere has volume four thirds pi r cubed and surface 4 pi r squared. A hemisphere has volume two thirds pi r cubed, curved surface 2 pi r squared and total surface 3 pi r squared.
Shortcuts
Use pi as 22 over 7 when the radius is a multiple of 7. The area of a square is the diagonal squared over 2. When one solid is melted and recast into another, the volume stays equal.
Worked Examples
Example 1: Find the circumference and area of a circle of radius 7.
Solution: The circumference is 44 and the area is 154, using pi as 22 over 7.
Example 2: A chord of length 16 is 6 units from the centre. Find the radius.
Solution: Half the chord is 8. The radius squared is 64 plus 36, which is 100, so the radius is 10.
Example 3: Find the volume of a cylinder with radius 7 and height 10.
Solution: 22 over 7 times 49 times 10 equals 1540.
Quick Check
Question 1: What is the angle in a semicircle?
Answer: 90 degrees.
Question 2: If the radius doubles, how many times does the area grow?
Answer: Four times.
Question 3: What is the diagonal of a cube of side a?
Answer: a root 3.