Materials & Prep
Prepare the board with one triangle labeled base 6 units, perpendicular height 4 units, and slanted side 5 units; one trapezoid labeled parallel bases 6 units and 10 units with height 4 units; grid paper, rulers, and scrap paper or mini-whiteboards. No special setup beyond the usual classroom materials is needed.
Opening prediction
Display the triangle with base 6, perpendicular height 4, and slanted side 5. Ask: “If you calculate its area, should the height be 4 or 5? Commit to one choice and explain why.” Students write 6 × 4 ÷ 2 or 6 × 5 ÷ 2 before discussing. Briefly recall that the area of a rectangle is length × width. Do not confirm the answer yet.
Turn and talkStudents compare their choice with a partner and point to the segment they believe is the height. Listen for students choosing the slanted side because it is the longest side.
Model the formulas
ConfrontTrace the perpendicular height from the base to the opposite vertex. Emphasize that height means the distance that meets the chosen base at a right angle. The slanted side is not the height just because it is labeled or longer.
Worked exampleImagine a rectangle with length 6 units and width 4 units. Its area is 6 × 4 = 24 square units. The triangle uses exactly half of that rectangle, so its area is 24 ÷ 2 = 12 square units. Using the slanted side would produce 6 × 5 ÷ 2 = 15 square units, which is wrong because that side is not perpendicular to the base.
ExplainThis gives the triangle formula, area = base × perpendicular height ÷ 2. The division by 2 is not a memorized extra step. It represents taking half of a matching rectangle or parallelogram.
Worked examplePair the trapezoid with an identical copy turned around. Together they form a parallelogram with base 6 + 10 = 16 units and height 4 units. The parallelogram area is 16 × 4 = 64 square units. One trapezoid is half of that pair, so its area is 64 ÷ 2 = 32 square units. Therefore, the trapezoid formula is area = (base 1 + base 2) × height ÷ 2.
Guided practice and check
ModelOn the board, draw a triangle with base 10 units, perpendicular height 6 units, and slanted side 8 units. Students identify the base and height by circling the base and drawing a right-angle mark at the height.
Check for understandingStudents show which expression is correct: 10 × 6 ÷ 2 or 10 × 8 ÷ 2. Then they calculate the area. The correct expression is 10 × 6 ÷ 2 = 30 square units. Ask, “What does the 6 represent, and why can’t we use 8?” Students must answer using the word perpendicular.
ScaffoldKeep a three-part frame visible: “The base is __ because __. The perpendicular height is __ because __. Area = __ × __ ÷ 2 = __ square units.” Students may use color coding: blue for the base, red for the perpendicular height, and gray for other sides.
Have students solve with a partner: A trapezoid has parallel bases of 5 units and 11 units and a perpendicular height of 6 units. Partners first identify the two bases and height, then explain why the two bases are added before multiplying by the height.
Collaborative explanation
Students work in pairs on grid paper. They draw or use the three figures below, mark the perpendicular height, calculate each area, and write one sentence explaining the formula’s connection to a rectangle or parallelogram.
- Triangle: base 12 units, perpendicular height 7 units.
- Triangle: base 9 units, perpendicular height 4 units.
- Trapezoid: parallel bases 5 units and 11 units, perpendicular height 6 units.
- Trapezoid: parallel bases 8 units and 14 units, perpendicular height 3 units.
CirculateAsk, “What does each number represent?” and “Where is the right angle?” Require students to name the units as square units, not just give a number.
ExtensionStudents explain why two trapezoids with the same height but different pairs of bases can have different areas. They may use the idea that the average of the two parallel bases determines the width of the matching parallelogram.
Independent practice
Students solve independently and show the perpendicular height in each problem. They must include one explanation, not only a formula substitution.
- A triangle has base 14 centimeters and perpendicular height 3 centimeters. Find its area.
- A trapezoid has parallel bases 4 meters and 12 meters and perpendicular height 5 meters. Find its area.
- A triangle has base 6 inches, perpendicular height 4 inches, and slanted side 5 inches. Find its area and explain why the slanted side is not used.
Look forStudents who write the slanted side as the height. Ask them to check whether that segment meets the base at a right angle, then revise the calculation rather than simply copying a formula.
Closing check
Exit ticketSolve both parts and explain the formula used.
A. Find the area of a triangle with base 14 units and perpendicular height 3 units. State why the height is the 3-unit segment. B. Find the area of a trapezoid with parallel bases 4 units and 12 units and perpendicular height 5 units. Explain how pairing it with a matching trapezoid helps justify the formula.
Collect the exit tickets. A complete response must identify the perpendicular height, calculate the area, and connect the formula to a rectangle or parallelogram.