Mathematics: Shapes

Mathematics: Shapes

• 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 D
• Volume of Cuboid = length x width x height
• Prisms Grade C
• Volume of Prism = cross-sectional area x length
• However cylinders can be written as V = πr²h
• Spheres Grade A
• Volume of Sphere = 4/3πr³
• A hemisphere is half a sphere
• V = 2/3πr³
• Pyramids Grade A
• Volume of Pyramid = 1/3 x base area x vertical height
• Cone Grade A
• Volume of Cone = 1/3 x πr² x vertical height
• Don't get confused with the slant height and vertical height.
• 
• Caption: r = radius

Surface Area of 3D Shapes

• Sphere Grade A
• S.A of a Cone = 4π²
• Cone Grade A
• S.A of a Cone = πrl + πr²
• πrl = curved area of a cone (l is the slant height)
• πr² = area of circular base
• Cylinder Grade A
• S.A of a Cylinder = 2πrh + 2πr²
• Remember the surface area of a solid shape = area of the net of the shape.
• 
• Caption: Surface Area of A Cylinder

Areas of 2D Shapes

• Triangle and Quadrilaterals Grade D
• Area 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 height
• Area of a Trapezium = average of parallel sides x distance between them (vertical height)
• A = 1/2(a+b) x h
• 
• Caption: The area of a trapezium is on the formula sheet, but you should learn it anyway.

Areas of 2D Shapes

• Area and Circumference of Circles Grade D
• Area
• Area of a Circle = πr²
• Circumference
• circumference = πD
• D = diameter
• 
• Caption: 2πr = πD

Areas of 2D Shapes

• Areas of Sectors, Arcs and Segments Grade A
• Area of Sector = x/360 x area of full circle
• x = angle
• Area of Arc = x/360 x circumference of full circle
• Area 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
• 

Circle Theorems

• 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.
• 

Circle Theorems

• 5. The angle in a semi circle is 90º.
• 6. Angles in the same segment are equal.
• 7. Opposite angles in a cyclic quadrilateral add up to 180º.
• 

Circle Theorems

• 
• 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 A
• 1. 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 left
• radius = √9 = 3
• centre = (-1, +4)
• 
• Caption: x^2 = x²

A* Skills with Shapes

• Volumes of Frustums Grade A*
• Volume = Volume of Original Cone - Volume of Removed Cone
• Volume = 1/3 x π x r² x h - 1/3 x π x r² x h
•