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 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.

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Caption: : r = radius

Surface Area of 3D Shapes

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.

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Caption: : Surface Area of A Cylinder

Areas of 2D Shapes

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

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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 DAreaArea of a Circle = πr²Circumferencecircumference = πD   D = diameter

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Caption: : 2πr = πD

Areas of 2D Shapes

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

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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.

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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.

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Circle Theorems

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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)

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Caption: : x^2 = x²

A* Skills with Shapes

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

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