--- title: "Area of triangles" subtitle: "Area, volume, circles and Pythagoras · Area · Lesson 2" --- # Area of triangles **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Sails, roof gables and the slices of a pizza cut into wedges are triangles. How much fabric does a triangular sail need? :::teacher-note **In one sentence:** Students double a triangle with a turned copy to make a parallelogram, see why A = ½bh, find heights that fall outside obtuse triangles, and work backwards from an area (AC9M7M01, node 1). **Students should be able to say:** I can explain why a triangle's area is half of base × height. I can find the area of any triangle, including one whose height falls outside it. I can find a missing base or height from an area. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict How would two copies of the triangle fit together? :::keep-together **A triangle has base 6 cm and perpendicular height 4 cm. What is its area?** ☐ 20 cm² ☐ 24 cm² ☐ 10 cm² ☐ 12 cm² My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 12 cm². Two copies make a 6 by 4 parallelogram, 24 cm², so one triangle is half: 12 cm². 24 is the parallelogram. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Half of base times height :::keep-together ### Halve it Here is the reason, then the routine. 1. $6 \times 4 = 24$ — Two copies of the triangle make a parallelogram with the same base and height. 2. $A = \tfrac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$ — One triangle is half of it. 3. $A = \tfrac{1}{2} \times 10 \times 7 = 35 \text{ cm}^2$ — Multiply the base by the height, then halve (or halve first: 5 × 7). Now try one: base 10 cm, height 7 cm. **A triangle has base 10 cm and perpendicular height 7 cm. What is its area, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 35 ½ × 10 × 7 = 35 cm². ::: :::keep-together ### Find the mistake **Omar finds the area of a triangle. Click the line where his working goes wrong.** ``` 1 A triangle has base 8 cm and perpendicular height 5 cm. 2 Area = base × height 3 = 8 × 5 4 = 40 cm² ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: a triangle is half of base × height. ½ × 8 × 5 = 20 cm². ::: :::keep-together ### A height outside **Where is this triangle's perpendicular height?** ☐ It has no height, because the top is not above the base ☐ Along the longest side ☐ Outside the triangle: straight down from the top to the base, extended ::: :::teacher-note **Answer:** Outside the triangle: straight down from the top to the base, extended Extend the base (dashed) and drop the height from the top at a right angle. The area is still ½ × 4 × 4 = 8 cm². ::: :::typical ## One change at a time **A height outside.** Each task changes one thing. **Area 12.** **Another triangle.** **Backwards: the height.** **A right angle.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Triangles :::keep-together ### Halve it **Base 12 m, perpendicular height 5 m. Area, in m²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 30 ½ × 12 × 5 = 30 m². ::: :::keep-together ### A decimal answer **Base 7 cm, perpendicular height 3 cm. Area, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 10.5 ½ × 7 × 3 = 10.5 cm². ::: :::keep-together ### Half a rectangle **A 10 cm by 6 cm rectangle is cut along a diagonal. What is the area of one triangle, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 30 Half of 10 × 6 = 60: 30 cm². ::: :::keep-together ### Review: parallelograms **Review: a parallelogram has base 7 cm and perpendicular height 6 cm. Area, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 42 7 × 6 = 42 cm². ::: ## Make: area 15 Then find a second triangle with the same area and a different base. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A triangle has base 9 cm and height 6 cm. Find its area. A triangle with area 24 cm² has base 8 cm: find its height.** {{lines-3}} ::: :::teacher-note **Answer:** ½ × 9 × 6 = 27 cm². ½ × 8 × h = 24, so h = 6 cm. :::