Name: Date: Period: Area of a Trapezoid: Compass Lab Area of Other Quadrilaterals Part 1: Constructing the Trapezoid Isosceles a. Using your straightedge, construct 2 intersecting lines in the space provided at the bottom of this page. b. Place the tip of your compass at the point of intersection formed by your 2 lines & extend the radius of your compass to any length so that it will cross 2 of your intersecting lines. c. Create 2 arcs, one on each line, on one side of the diagram (e.g. upper side, lower side, left side, or right side) d. Adjust the radius of your compass to be bigger or smaller than your 1 st radius length. e. Create 2 arcs, one on each line, on the opposite side of the diagram (e.g. if you used the upper side first, then make these two marks on the lower side). f. Connect all 4 intersection points of the arcs & intersecting lines to make your isosceles trapezoid. g. Erase the 2 intersecting lines used to create your trapezoid. Geometry Area of Figures -1- NJCTL.org
Part 2: Deriving the Area of a Trapezoid Formula a. Draw an altitude starting at a vertex corresponding to an obtuse angle. Label this segment as h. b. Label both of your parallel sides as your bases: b1 & b2. c. Draw a diagonal line connecting one pair of opposite vertices. You have created 2 triangles. d. True/False: The area of the trapezoid is equal to the sum of the areas of the 2 triangles? Explain. e. What is the area formula of any triangle? f. Write down the area formula for each of the triangles that you created using the variables that you have written down. Area of Top Triangle: Area of Bottom Triangle: g. Write down both area formulas (from the top & bottom triangles) in the blank spaces below: Area of a Trapezoid = + h. Look at your equation above & find any common factors between your two terms. Write down the final equation for the Area of a Trapezoid once you factor Area of a Trapezoid = Geometry Area of Figures -2- NJCTL.org
Name: Date: Period: Area of a Rhombus: Scissors Cutting out the Rhombus & Deriving its Area Formula: a. Cut along the dotted line below. b. Draw 2 diagonals in the rhombus, each one joining the vertices opposite of each other. Label the 2 sections of one diagonal as ½d1 & the 2 sections of the other diagonal as ½d2. c. Cut out the rhombus. d. Cut one of the diagonals completely, creating 2 congruent isosceles triangles. e. Cut one of the triangles in half by cutting along the other diagonal. f. Take the 2 smallest triangles and manipulate them so that they connect to the bigger triangle to form a rectangle. g. What is the area formula for any rectangle? h. What is the area formula of a rhombus? Use the rectangle formula above and the variables that are given in your triangles/rhombus to figure it out. Area of a Rhombus = --------------------------------------------------------------------------------------------------------------------- Geometry Area of Figures -3- NJCTL.org
Name: _ANSWERS Date: Period: Area of a Trapezoid: Compass Lab Area of Other Quadrilaterals Part 1: Constructing the Trapezoid Isosceles a. Using your straightedge, construct 2 intersecting lines in the space provided at the bottom of this page. b. Place the tip of your compass at the point of intersection formed by your 2 lines & extend the radius of your compass to any length so that it will cross 2 of your intersecting lines. c. Create 2 arcs, one on each line, on one side of the diagram (e.g. upper side, lower side, left side, or right side) d. Adjust the radius of your compass to be bigger or smaller than your 1 st radius length. e. Create 2 arcs, one on each line, on the opposite side of the diagram (e.g. if you used the upper side first, then make these two marks on the lower side). f. Connect all 4 intersection points of the arcs & intersecting lines to make your isosceles trapezoid. g. Erase the 2 intersecting lines used to create your trapezoid. Geometry Area of Figures -4- NJCTL.org
Part 2: Deriving the Area of a Trapezoid Formula a. Draw an altitude starting at a vertex corresponding to an obtuse angle. Label this segment as h. b. Label both of your parallel sides as your bases: b1 & b2. c. Draw a diagonal line connecting one pair of opposite vertices. You have created 2 triangles. d. True/False: The area of the trapezoid is equal to the sum of the areas of the 2 triangles? Explain. True. Since the trapezoid is formed by 2 triangles, the area of the 2 triangles will be equivalent to the area of the trapezoid. e. What is the area formula of any triangle? A = ½ bh f. Write down the area formula for each of the triangles that you created using the variables that you have written down. Note: answers could be switched, depending on how they labeled their figure Area of Top Triangle: A = ½ b1h Area of Bottom Triangle: A = ½ b2h g. Write down both area formulas (from the top & bottom triangles) in the blank spaces below: Area of a Trapezoid = ½ b1h + ½ b2h h. Look at your equation above & find any common factors between your two terms. Write down the final equation for the Area of a Trapezoid once you factor Area of a Trapezoid = _½ h(b1 + b2) b1 h b2 Geometry Area of Figures -5- NJCTL.org
Name: _ANSWERS Date: Period: Area of a Rhombus: Scissors Cutting out the Rhombus & Deriving its Area Formula: a. Cut along the dotted line below. b. Draw 2 diagonals in the rhombus, each one joining the vertices opposite of each other. Label the 2 sections of one diagonal as ½d1 & the 2 sections of the other diagonal as ½d2. c. Cut out the rhombus. d. Cut one of the diagonals completely, creating 2 congruent isosceles triangles. e. Cut one of the triangles in half by cutting along the other diagonal. Geometry Area of Figures -6- NJCTL.org
f. Take the 2 smallest triangles and manipulate them so that they connect to the bigger triangle to form a rectangle. g. What is the area formula for any rectangle? _ A = lw or A = bh h. What is the area formula of a rhombus? Use the rectangle formula above and the variables that are given in your triangles/rhombus to figure it out. Area of a Rhombus = _ A = ½ d1d2 Geometry Area of Figures -7- NJCTL.org