This includes: - Volumes of 3D Shapes - Surface Area of 3D Shapes - Areas of 2D Shapes - Circle Theorems - Equations of Circles - To find the centre of a circle - To find the radius of a circle
- A* Skills with Shapes
Volumes of 3D Shapes
Cuboids Grade DVolume of Cuboid = length x width x heightPrisms Grade CVolume of Prism = cross-sectional area x length However cylinders can be written as V = πr²hSpheres Grade AVolume of Sphere = 4/3πr³ A hemisphere is half a sphere V = 2/3πr³Pyramids Grade AVolume of Pyramid = 1/3 x base area x vertical heightCone Grade AVolume of Cone = 1/3 x πr² x vertical heightDon't get confused with the slant height and vertical height.
Sphere Grade A S.A of a Cone = 4π²Cone Grade A S.A of a Cone = πrl + πr² πrl = curved area of a cone (l is the slant height) πr² = area of circular baseCylinder Grade A S.A of a Cylinder = 2πrh + 2πr²Remember the surface area of a solid shape = area of the net of the shape.
Triangle and Quadrilaterals Grade DArea of a Triangle = 1/2 x base x vertical height Alternatively you could use area = 1/2 ab sinC (Grade A)Area of a Parallelogram = base x vertical heightArea of a Trapezium = average of parallel sides x distance between them (vertical height) A = 1/2(a+b) x h
Areas of Sectors, Arcs and Segments Grade AArea of Sector = x/360 x area of full circle x = angleArea of Arc = x/360 x circumference of full circleArea of Segment = 1. Find the area of the sector 2. Then subtract the area of the triangle This can be done by using 1/2 ab sinC
Grade A*1. A tangent and a radius meet at 90º.2. Two radii form an isosceles triangle.3. The perpendicular bisector of a chord passes through the centre of the circle.4. The angle at the centre of a circle is twice the angle at the circumference.
8. Tangents from the same point are the same length.9. The alternative segment theorem.
Equations of Circles
To create the equation of a circle: Grade A1. Complete the square in the equation.2. Organise the numbers.3. Put the completed numbers on the left and the organised numbers on the right.4. The organised numbers should be a square number.Lets have and example:x² + y² + 2x - 8y + 8 = 0 display equation(x+1)² -1 + (y-4)² -16 + 8 = 0 complete the square(x+1)² + (y-4)² = 9 put the numbers on the leftradius = √9 = 3centre = (-1, +4)