A triangle is exactly half of the rectangle that boxes it in
Start with a rectangle
The area of a rectangle is its length times its width. A rectangle 6 across and 4 up holds 4 rows of 6 unit squares, so its area is 6 × 4 = 24 squares. The area of a triangle comes from the rectangle drawn around it.
The 6 by 4 rectangle holds 24 unit squares.
Any triangle in the box
Draw a triangle whose base is the bottom edge of the rectangle and whose top corner touches the top edge. The top corner can be anywhere along the top edge, so this can be any triangle at all, not only a neat one.
The base of the triangle is 6. Its height is the distance straight up from the base to the top corner, which is 4, the height of the rectangle.
The triangle stands on the bottom edge, and its top corner is 2 squares along the top edge.
Two rectangles, and half of each
Drop a line straight down from the top corner to the base, at a right angle to it. The line splits the big rectangle into two smaller rectangles, one on each side: here a 2 by 4 rectangle and a 4 by 4 rectangle.
Each sloping side of the triangle runs from one corner of a small rectangle to the opposite corner, so it cuts that rectangle exactly in half. The triangle takes one half of each. Half of 2 × 4 = 8 is 4, and half of 4 × 4 = 16 is 8, so the triangle’s area is 4 + 8 = 12, which is half of 24.
The line from the top corner splits the box into a 2 by 4 rectangle and a 4 by 4 rectangle.
Slide the top corner at least 2 squares along the top edge, and add the two halves again
Drag the top corner along the top edge. The two small rectangles change size, but the triangle always takes half of each, so the two halves always add up to half of 24, which is 12.
Half the base times the height
Wherever the top corner sits, the triangle is half of the rectangle that boxes it in. The rectangle’s area is base × height, so the triangle’s area is half of that: area of a triangle = ½ × base × height.
A triangle with a base of 10 cm and a height of 7 cm has an area of ½ × 10 × 7 = 35 cm². The rectangle around it would be 10 × 7 = 70 cm², and 35 is half of 70.
The height is straight up
The height is always measured at a right angle to the base. It is not the length of a sloping side. Slide the top corner farther along and the sloping sides grow longer, but the height stays 4, so the area stays ½ × 6 × 4 = 12.
The top corner can even go past the end of the base. Then the line straight down from it lands outside the triangle, on the base line made longer, and the area is still 12. To see why, cut the triangle into thin strips that run along the base. Sliding the top corner slides each strip sideways, but no strip gets longer or shorter, so together they cover the same area.
Slide the apex along its rail until the perimeter passes 20
Drag the top corner along the dashed line. The base stays 6 and the height stays 4, so the area stays 12, while the two sloping sides and the perimeter grow.
Multiply, then halve
Two mistakes are common. 6 × 4 = 24 is the area of the whole rectangle, but the triangle fills only half of it, so its area is 12. And 6 + 4 = 10 adds the base to the height: area counts squares, so multiply the base by the height, then halve.
Composite areas · Two Triangles on One Base with Their Top Corners on a Parallel Line
QuestionTwo parallel lines are 5 cm apart. AB is a line 12 cm long on the lower line, and C and D are two different points on the upper line. (a) Find the area of triangle ABC. (b) Find the area of triangle ABD.
1.Both triangles have the same base, AB = 12 cm.
Both triangles stand on the same base, AB = 12 cm. 2.The height of a triangle is measured from its top corner straight down to the base line, at a right angle to it. C is on the upper line, and every point of the upper line is 5 cm above the lower line, so the height of triangle ABC is 5 cm.
Every point of the upper line is 5 cm above the lower line, so triangle ABC is 5 cm high. 3.(a) The area of triangle ABC is 12 × 12 × 5 = 30 cm2.
(a) Triangle ABC is 12 × 12 × 5 = 30 cm², wherever C is on the line. 4.D is on the upper line too, so triangle ABD has the same base, 12 cm, and the same height, 5 cm.
D is on the upper line too: the same base, 12 cm, and the same height, 5 cm. 5.(b) The area of triangle ABD is 12 × 12 × 5 = 30 cm2 as well. Sliding the top corner along the upper line changes the shape of the triangle, but not its area.
(b) Triangle ABD is 30 cm² as well. The shape changes; the area does not.
Answer: (a) 30 cm2; (b) 30 cm2
Watch out for
- Using a slanted side, such as BD, as the height. A slanted side leans over, so it is longer than 5 cm. The height is measured straight up from the base line to the top corner, and for both triangles that is the 5 cm between the lines.
- Deciding that the more stretched triangle must have the bigger area. Both triangles have the same base and the same height, and the area of a triangle depends only on those two, so both areas are 30 cm2.
Your turn
1.A triangle with base 22 and height 4. What is its area?