# The Area is the width times the height: Area = w h

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1 Geometry Handout Rectangle and Square Area of a Rectangle and Square (square has all sides equal) The Area is the width times the height: Area = w h Example: A rectangle is 6 m wide and 3 m high; what is its Area? Area = 6 m 3 m = 18 m 2 Perimeter of a Rectangle The Perimeter is the distance around the edges. The Perimeter is 2 times the (width + height): Perimeter = 2(w+h) Example: A rectangle has a width of 12 cm and a height of 5 cm; what is its Perimeter? Perimeter = 2 (12 cm + 5 cm) = 2 17 cm = 34 cm Triangle Area of a Triangle The Area is half of the base times height. b is the distance along the base h is the height (measured at right angles to the base) Area = 1/2 b h The formula works for all triangles. Note: Another way of writing the formula is bh/2 1

2 Example: What is the Area of this triangle? (Note: 12 is the height, not the length of the left-hand side) Height = h = 12 Base = b = 20 Area = 1/2 b h = 1/ = 120 Perimeter of a Triangle The Perimeter is the distance around the edge of the triangle; just add up the three sides. Rhombus S B A altitude S S A B S p q Area of a Rhombus The Area can be calculated by: the altitude times the side length: Area = altitude s Perimeter of a Rhombus The Perimeter is the distance around the edges. The Perimeter is 4 times s (the side length) because all sides are equal in length: Perimeter = 4s 2

3 Parallelogram Area of a Parallelogram The Area is the base times the height: Area = b h (h is at right angles to b) Example: A parallelogram has a base of 6 m and is 3 m high; what is its Area? Area = 6 m 3 m = 18 m 2 Perimeter of a Parallelogram The Perimeter is the distance around the edges. The Perimeter is 2 times the (base + side length): Perimeter = 2(b+s) Example: A parallelogram has a base of 12 cm and a side length of 6 cm; what is its Perimeter? Perimeter = 2 (12 cm + 6 cm) = 2 18 cm = 36 cm Diagonals of a Parallelogram: The diagonals of a parallelogram bisect each other. In other words, the diagonals intersect each other at the half-way point. Trapezoid Area of a Trapezoid The Area is the average of the two base lengths times the altitude: Area = 1/2(a+b) x h Example: A trapezoid s two bases are 6 m and 4 m, and it is 3 m high. What is its Area? Area = 1/2(6 m + 4 m) 3 m = 5 m 3 m = 15 m 2 3

4 Perimeter of a Trapezoid The Perimeter is the distance around the edges. The Perimeter is the sum of all side lengths: Perimeter = a+b+c+d Example: A trapezoid has side lengths of 5 cm, 12 cm, 4 cm, and 15 cm. What is its Perimeter? Perimeter = 5 cm + 12 cm + 4 cm + 15 cm = 36 cm Circle The Radius is the distance from the center to the edge. The Diameter starts at one side of the circle, goes through the center, and ends on the other side. The Circumference is the distance around the edge of the circle. And here is the really cool thing: When you divide the circumference by the diameter you get , which is the number π (Pi). So when the diameter is 1, the circumference is We can say: Circumference = π Diameter Example: You walk around a circle which has a diameter of 100 m; how far have you walked? Distance walked = Circumference = π 100 m = 314 m (to the nearest m) Also note that the Diameter is twice the Radius: Diameter = 2 Radius And so this is also true: Circumference = 2 π Radius 4

5 Area of a Circle The Area of a circle is π times the Radius squared, which is written: A = π r 2 To help you remember, think Pie Are Squared (even though pies are usually round). Or, in relation to Diameter: A = (π/4) D 2 Example: What is the area of a circle with radius of 1.2 m? A = π r 2 A = π A = π ( ) A = = 4.52 (to 2 decimals) 5

6 Cuboid or Cube Volume of a Cuboid A cuboid is a 3-dimensional shape. So to work out the volume we need to know 3 measurements. Look at this shape. There are 3 different measurements: Length, Width, Height The volume is found using the formula: Volume = Length Width Height Which is usually shortened to: V = l w h Or more simply: V = lwh (in any order) It doesn t really matter which one is length, width, or height, so long as you multiply all three together. Example: Lengths in meters (m): The volume is: 10 m 5 m 4 m = 200 m 3 It also works out the same like this: 4 m 5 m 10 m = 200 m 3 Note: the result is in m 3 (cubic meters) because we have multiplied meters together three times. Perimeter The Perimeter of a cuboid equals: 2(length x width) + 2(length x height) + 2(width x height) Example problem from diagram shown above: Perimeter = 2(10 m x 5 m) + 2(10 m x 4 m) + 2(5 m x 4 m) Perimeter = 100 m m m 2 = 220 m 2 6

7 Sphere Sphere Facts Notice these interesting things: It is perfectly symmetrical It has no edges or vertices (corners) It is not a polyhedron All points on the surface are the same distance from the center Glass Sphere And for reference: Surface Area = 4 π r 2 Volume = (4/3) π r 3 Balls and marbles are shaped like spheres. Cylinder Just multiply the area of the circle by the height of the cylinder: Area of the circle: π r 2 Height: h Volume = Area Height = π r 2 h How to remember: Volume = pizza Imagine you just cooked a pizza. The radius is z, and the thickness a is the same everywhere... What is the volume? Answer: pi z z a (We would normally write pi as π, and z z as z 2, but you get the idea!) 7

8 Prism The Volume of a prism is the area of one end times the length of the prism. Volume = Area Length Example: What is the volume of a prism whose ends have an area of 25 m 2 and that is 12 m long? Answer: Volume = 25 m 2 12 m = 300 m 3 Cone Volume of a Cone vs. Cylinder The volume formulas for cones and cylinders are very similar: The volume of a cylinder is: π r 2 h The volume of a cone is: π r 2 (h/3) So, the only difference is that a cone s volume is one-third (1/3) of a cylinder. 8

9 Ellipses a a= 1/2 major axis b Area b = 1/2 minor axis Area = πab a = 1/2 the major axis b = 1/2 the minor axis Example: Find the Area of an elliptical swimming pool whose major axis is 9 meters and minor axis is 6 meters. a = 0.5(9 m) = 4.5 m b = 0.5(6 m) = 3 m Area = (4.5 m)(3 m) = 42.4 m 2 9

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