Drag a corner and watch the area. A triangle is half a rectangle. A parallelogram can be cut and slid into a rectangle. Play, solve missions, crack puzzles and beat the clock.
Start here. The area of a shape is how many unit squares fit inside it. A rectangle that is 4 squares wide and 3 squares tall holds 4 × 3 = 12 squares. That is the rule for rectangles: area = base × height.
A triangle is half a rectangle. Draw a rectangle around a triangle so that the triangle uses the whole bottom edge and touches the top edge. Two copies of the triangle fill the rectangle exactly, so one triangle is half. Area of a triangle = ½ × base × height. For a base of 4 and a height of 3 the rectangle is 12, so the triangle is 6. Try it in Play: you can drag the top corner anywhere along the top edge and the area never changes, because the base and the height do not change.
The height is not the slanted side. The height is the straight-up distance from the base to the top corner, at a right angle to the base. A slanted side is always longer than the height, so using it gives an area that is too big. In the game the height is the dashed orange line.
Obtuse triangles. When the top corner leans out past the end of the base, the height line falls outside the triangle. You extend the base with a dotted line and measure straight up to that line. The formula ½ × base × height still works.
Parallelograms: cut and slide. Cut a right-angled triangle off one slanted end and slide it to the other end. You get a rectangle with the same base and the same height, so the area of a parallelogram = base × height. Leaning the parallelogram more or less does not change its area. A diagonal cuts any parallelogram into two equal triangles, which is another way to see the ½.
Shapes made of pieces. A house shape is a rectangle with a triangle on top. Find the area of each piece and add them. If a triangle has an odd base and an odd height the area ends in .5, such as 7.5 square units.
Area = ½ × base × height. Any side can be the base, but the height must be measured straight up from that base at a right angle. For a triangle with base 6 and height 5 the area is ½ × 6 × 5 = 15 square units.
Area = base × height. The height is the straight distance between the two parallel sides, not the length of the slanted side. A parallelogram with base 7 and height 4 has an area of 28 square units, whatever its lean.
Put two identical triangles together and they make a rectangle or a parallelogram with the same base and height. So each triangle is half of base × height. That is where the ½ comes from.
You can pick any side. Once you pick it, the height is the perpendicular distance from that side to the opposite corner. The area comes out the same whichever side you choose.
Yes. A triangle with a base of 5 and a height of 3 has an area of ½ × 5 × 3 = 7.5 square units. In Challenge me some answers end in .5.
Just starting: right triangles on a grid, half of a rectangle. I know the basics: ½ × base × height and parallelograms. Challenge me: obtuse triangles, slanted sides that are not the height, and shapes made of pieces.
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