R EVIEW OF A REA AND C IRCUMFERENCE Honors Geometry

Area of rectangle = l x b= 28 cm² 4 cm 7 cm = 7 x 4 Area of triangle = 14 cm² = ½ x 28 = ½ x Area of rectangle Can you see that the rectangle is split into two equal triangles?

T RIANGLE R ULE Area of rectangle = 10 x 4 = 40 Area of triangle = ½ x Area of rectangle = 40 ÷ 2 = 20 Area of rectangle = 8 x 5 = 40 Area of triangle = 40 ÷ 2 = 20 Area of rectangle = 6 x 6 = 36 Area of triangle = 36 ÷ 2 = 18

A REA OF A T RIANGLE Split a rectangle into two right angle triangles by drawing two lines from the top to bottom corners The top two triangles are the same size as the two bottom triangles This triangle is half the area of the rectangle Area of Triangle = ½ x Area of Rectangle = ½ x base x height base height

T HE A LTITUDE The “height” is more commonly called an “altitude” in a triangle.. The altitude may look different depending on the type of triangle.

O BTUSE T RIANGLE Area of a Triangle = ½ x Base x Height Base To get height move base up until it is level with the top corner Is the triangle half of this rectangle? Height

O BTUSE T RIANGLE Is the triangle half of this rectangle? Yes ….. from the six part shapes in the rectangle you could make two copies of the obtuse angle triangle The area of the obtuse angle triangle is half of the area of the rectangle

O BTUSE A NGLE T RIANGLE Do you see 2 right triangles? We can use these to prove the area formula for obtuse triangles. height base A B C D Area of ∆ADB = Area of ∆ABC – Area of ∆DBC Area of ∆ADB = ½ AC∙BC – ½ DC∙BC Area of ∆ADB = ½ BC∙(AC – DC) Area of ∆ADB = ½ BC∙AD = ½ base x height

O BTUSE A NGLE T RIANGLE Area of a Obtuse Angle Triangle = ½ x base x height base height

T HE CIRCUMFERENCE OF A CIRCLE For any circle, π = circumference diameter or, We can rearrange this to make a formula to find the circumference of a circle given its diameter. C = πd π = C d

F INDING THE CIRCUMFERENCE GIVEN THE RADIUS The diameter of a circle is two times its radius, or C = 2 πr d = 2 r We can substitute this into the formula C = πd to give us a formula to find the circumference of a circle given its radius.

T HE CIRCUMFERENCE OF A CIRCLE Use π = 3.14 to find the circumference of the following circles: C = πd 4 cm = 3.14 × 4 = cm C = 2 πr 9 m = 2 × 3.14 × 9 = m C = πd 23 mm = 3.14 × 23 = mm C = 2 πr 58 cm = 2 × 3.14 × 58 = cm

C IRCUMFERENCE PROBLEM The diameter of a bicycle wheel is 50 cm. How many complete rotations does it make over a distance of 1 km? 50 cm The circumference of the wheel = 3.14 × 50 Using C = πd and π = 3.14, = 157 cm The number of complete rotations = ÷ 157 = km = cm

F ORMULA FOR THE AREA OF A CIRCLE We can find the area of a circle using the formula radius Area of a circle = πr 2 Area of a circle = π × r × r or

T HE AREA OF A CIRCLE Use π = 3.14 to find the area of the following circles: A = πr 2 2 cm = 3.14 × 2 2 = cm 2 A = πr 2 10 m = 3.14 × 5 2 = 78.5 m 2 A = πr 2 23 mm = 3.14 × 23 2 = mm 2 A = πr 2 78 cm = 3.14 × 39 2 = cm 2

F IND THE AREA OF THIS SHAPE Use π = 3.14 to find area of this shape. The area of this shape is made up of the area of a circle of diameter 13 cm and the area of a rectangle of width 6 cm and length 13 cm. 6 cm 13 cm Area of circle = 3.14 × = cm 2 Area of rectangle = 6 × 13 = 78 cm 2 Total area = = cm 2