randRange(2, 8)
randRange(2, 8)
randRange(1, 6)
randRangeNonZero(-2, 2)
1/2 * (B1 + B2) * H
What is the area of this figure?
init({
range: [[-4, max(B1, B2) + 4], [-1, H + 1]],
scale: [30, 30]
});
style({ stroke: BLUE, fill: "#eee"});
path([[0, 0], [B1, 0], [B2 + SH, H], [SH, H], [0, 0]]);
label([B1/2, 0], B1 + "\\text{ " + UNIT + "}", "below");
label([B2/2 + SH, H], B2 + "\\text{ " + UNIT + "}", "above");
var x = min(B1, B2 + SH);
line([x, 0], [x, H], { strokeDasharray: "." });
label([x, H/2], H + "\\text{ " + UNIT + "}", "right");
rightAngleBox([[0, 0], [x, 0]], [[x, 0], [x, H]], { stroke: GRAY, opacity: 0.5 });
parallel([[0, 0], [B1, 0]], 1);
parallel([[SH, H], [B2 + SH, H]], 1);
K
square plural_form(UNIT_TEXT)
This figure is a quadrilateral with a pair of parallel sides (the top and bottom sides), so it's a trapezoid.
area of a trapezoid = \dfrac12 \cdot (b_1 + b_2) \cdot h
[Show me why]
Let's draw a line between the opposite ends of the two bases.
var showSubHint = function() {
graph.subhint.show();
$("a[data-subhint='area-trapezoid']")
.unbind("click", showSubHint)
.click(hideSubHint);
};
var hideSubHint = function() {
graph.subhint.hide();
$("a[data-subhint='area-trapezoid']")
.unbind("click", hideSubHint)
.click(showSubHint);
};
graph.subhint = raphael.set().push(
path([[0, 0], [B1, 0], [B2 + SH, H]], { stroke: BLUE, fill: ORANGE, opacity: 0.5 }),
path([[SH, H], [B2 + SH, H], [0, 0]], { stroke: BLUE, fill: RED, opacity: 0.5 })
);
hideSubHint();
Notice that the line divides the trapezoid into two triangles: one triangle with base b_1 = B1
, and another triangle with base b_2 = B2
. Both triangles have height h = H
.
The area of the trapezoid is equal to the sum of the areas of the two triangles.
A = \dfrac12 \cdot b_1 \cdot h + \dfrac12 \cdot b_2 \cdot h
Factor out \dfrac12 \cdot h
to get the formula for the area of a trapezoid:
A = \dfrac12 \cdot h \cdot (b_1 + b_2) = \dfrac12 \cdot (b_1 + b_2) \cdot h
Now use this formula to calculate the trapezoid's area.
b_1 = B1
b_2 = B2
h = H
A = \dfrac12 \cdot (B1 + B2) \cdot H = K
randRange(1, 7) * 2
randRange(1, 7) * 2
randFromArray(["v", "h"])
rand(3) !== 0 ? randRange(1, 5) : D1/2
1/2 * D1 * D2
SH === D1/2
What is the area of this figure?
var range, v;
var drawCongruencies, drawD1, drawD2;
if (ORIENT === "h") {
range = [[-1, D1 + 2], [-D2/2 - 1, D2/2 + 1]];
v = [[0, 0], [SH, D2/2], [D1, 0], [SH, -D2/2], [0, 0]];
drawCongruencies = function(style) {
congruent([[0, 0], [SH, D2/2]], 1, style);
congruent([[0, 0], [SH, -D2/2]], 1, style);
congruent([[SH, D2/2], [D1, 0]], RHOMBUS ? 1 : 2, style);
congruent([[SH, -D2/2], [D1, 0]], RHOMBUS ? 1 : 2, style);
};
drawD1 = function(style) {
return {
label: label([D1/2, 0], D1 + "\\text{ " + UNIT + "}", style),
path: path([[0, 0], [D1, 0]], style)
};
};
drawD2 = function(style) {
return {
label: label([D1, 0], D2 + "\\text{ " + UNIT + "}", "right", style),
path: path([[D1, -D2/2], [D1, D2/2]], style)
};
};
} else {
range = [[-D2/2 - 1, D2/2 + 1], [-1, D1 + 2]];
v = [[0, 0], [D2/2, SH], [0, D1], [-D2/2, SH], [0, 0]];
drawCongruencies = function(style) {
congruent([[0, 0], [D2/2, SH]], 1);
congruent([[0, 0], [-D2/2, SH]], 1);
congruent([[D2/2, SH], [0, D1]], RHOMBUS ? 1 : 2);
congruent([[0, D1], [-D2/2, SH]], RHOMBUS ? 1 : 2);
};
drawD1 = function(style) {
return {
label: label([0, D1/2], D1 + "\\text{ " + UNIT + "}", style),
path: path([[0, 0], [0, D1]], style)
};
};
drawD2 = function(style) {
return {
label: label([0, D1], D2 + "\\text{ " + UNIT + "}", "above", style),
path: path([[-D2/2, D1], [D2/2, D1]], style)
};
};
}
init({ range: range, scale: 20 });
path(v, { stroke: BLUE, fill: "#eee" });
drawCongruencies({ stroke: BLUE });
style({ stroke: BLUE, strokeDasharray: "." }, function() {
graph.d1 = drawD1();
graph.d2 = drawD2();
});
rightAngleBox(graph.d1.path.graphiePath, graph.d2.path.graphiePath, { stroke: GRAY, opacity: 0.5 });
K
square plural_form(UNIT_TEXT)
This figure is a quadrilateral with perpendicular diagonals and two pairs of congruent, adjacent sides, so it is a kite.
In fact, because this shape's sides are all congruent, it is also a rhombus.
area of a kite = \dfrac12 \cdot d_1 \cdot d_2
[Show me why]
The horizontal diagonal in the center splits the kite into two congruent triangles.
The vertical diagonal in the center splits the kite into two congruent triangles.
var showSubHint = function() {
graph.subhint.show();
$("a[data-subhint='area-kite']")
.unbind("click", showSubHint)
.click(hideSubHint);
};
var hideSubHint = function() {
graph.subhint.hide();
$("a[data-subhint='area-kite']")
.unbind("click", hideSubHint)
.click(showSubHint);
};
if (ORIENT === "h") {
graph.subhint = raphael.set().push(
path([[0, 0], [SH, D2/2], [D1, 0], [0, 0]], { fill: ORANGE, opacity: 0.5 }),
path([[0, 0], [SH, -D2/2], [D1, 0], [0, 0]], { fill: GREEN, opacity: 0.5 })
);
} else {
graph.subhint = raphael.set().push(
path([[0, 0], [D2/2, SH], [0, D1], [0, 0]], { fill: ORANGE, opacity: 0.5 }),
path([[0, 0], [-D2/2, SH], [0, D1], [0, 0]], { fill: GREEN, opacity: 0.5 })
);
}
hideSubHint();
Let d_1 = D1
, the diagonal that bisects the kite. Then let d_2 = D2
.
Notice that d_1
is the base of both triangles, and d_2
is the combined height of the two triangles, so \dfrac{d_2}{2}
is the height of each triangle.
So the area of each triangle is:
A_T = \dfrac12 \cdot b \cdot h = \dfrac12 \cdot d_1 \cdot \dfrac{d_2}{2} = \dfrac14 \cdot d_1 \cdot d_2
The area of both triangles combined, 2A_T
, is the total area of the kite:
2A_T = 2(\dfrac14 \cdot d_1 \cdot d_2) = \dfrac12 \cdot d_1 \cdot d_2 = A
Now use this formula to calculate the kite's area.
d_1 = D1
d_2 = D2
A = \dfrac12 \cdot D1 \cdot D2 = K