randRange(2, 9) randRange(2, 9) randRange(L2 + 2, 12) randRange(W2 + 2, 12) L1 * W1 L2 * W2

A \green{L2 \times W2} rectangle sits inside a \blue{L1 \times W1} rectangle.

What is the area of the shaded region?

init({ range: [[-1, L1 + 1], [-1, W1 + 1]], scale: 30 }); rect(0, 0, L1, W1, { stroke: BLUE, fill: BLUE, "fill-opacity": 0.5 }); label([L1 / 2, W1], L1, "above"); label([L1, W1 / 2], W1, "right"); var x = (L1 - L2) / 2; var y = (W1 - W2) / 2; rect(x, y, L2, W2, { stroke: GREEN, fill: '#fdfdfd' }); label([x + L2 / 2, y + W2], L2, "above"); label([x + L2, y + W2 / 2], W2, "right");
A1 - A2

First, calculate the area of the whole figure, including the unshaded area.

rect(0, 0, L1, W1, { stroke: BLUE, fill: BLUE, opacity: 0.5 });

The area of a rectangle is the length times the width.
\qquad \blue{L1 \times W1 = A1}

Next, calculate the area of the inner figure.

rect((L1 - L2) / 2, (W1 - W2) / 2, L2, W2, { stroke: GREEN, fill: '#fdfdfd' });

\qquad \green{L2 \times W2 = A2}

Finally, subtract the area of the inner rectangle from the area of the outer rectangle.

\qquad \blue{A1} - \green{A2} = A1 - A2

randRange(2, 9) randRange(R2 + 2, 12) R1 * R1 R2 * R2

A circle with radius of \green{R2} sits inside a circle with radius of \blue{R1}.

What is the area of the shaded region?

var d = R1 + 1; init({ range: [[-d, d], [-d, d]], scale: 15 }); ellipse([0, 0], [R1, R1], { stroke: BLUE, fill: BLUE, "fill-opacity": 0.5 }); ellipse([0, 0], [R2, R2], { stroke: BLUE, fill: '#fdfdfd', }); path([[0, 0], [R1, 0]], { stroke: BLUE }); path([[0, 0], [0, R2]], { stroke: GREEN }); label([(R1 + R2) / 2, 0], R1, "above"); label([0, R2 / 2], R2, "right");
(A1 - A2) * Math.PI

First, calculate the area of the whole figure, including the unshaded area.

ellipse([0, 0], [R1, R1], { stroke: BLUE, fill: BLUE, opacity: 0.5 });

The area of a circle is \pi r^2.
\qquad \blue{\pi \times R1 \times R1 = A1\pi}

Next, calculate the area of the inner figure.

ellipse([0, 0], [R2, R2], { stroke: GREEN, fill: '#fdfdfd' }); path([[0, 0], [0, R2]], { stroke: GREEN }); label([0, R2 / 2], R2, "right");

\qquad \green{\pi \times R2 \times R2 = A2\pi}

Finally, subtract the area of the inner circle from the area of the outer circle.

\qquad \blue{A1\pi} - \green{A2\pi} = A1 - A2\pi

randRange(1, 5) randRange(2 * R + 1, 12) randRange(2 * R + 1, 12) L * W R * R roundTo(2, A1 - A2 * Math.PI)

A circle with radius of \green{R} sits inside a \blue{L \times W} rectangle.

What is the area of the shaded region?

Round to the nearest hundredth.

init({ range: [[-1, L + 1], [-1, W + 1]], scale: 30 }); rect(0, 0, L, W, { stroke: BLUE, fill: BLUE, "fill-opacity": 0.5 }); label([L / 2, W], L, "above"); label([L, W / 2], W, "right"); ellipse([L / 2, W / 2], [R, R], { stroke: GREEN, fill: '#fdfdfd', }); path([[L / 2, W / 2], [L / 2 + R, W / 2]], { stroke: GREEN }); label([(L + R) / 2, W / 2], R, "above");
A1 - A2 * Math.PI

First, calculate the area of the whole figure, including the unshaded area.

rect(0, 0, L, W, { stroke: BLUE, fill: BLUE, opacity: 0.5 });

The area of a rectangle is the length times the width.
\qquad \blue{L \times W = A1}

Next, calculate the area of the inner figure.

ellipse([L / 2, W / 2], [R, R], { stroke: GREEN, fill: '#fdfdfd', }); path([[L / 2, W / 2], [L / 2 + R, W / 2]], { stroke: GREEN });

The area of a circle is \pi r^2.
\qquad \green{\pi \times R \times R = A2\pi}

Finally, subtract the area of the inner circle from the area of the outer rectangle.

\qquad \blue{A1} - \green{A2\pi} \approx A

randRange(2, 12) randRange(2, 12) ceil(sqrt(L * L + W * W) / 2) + 1 randRange(D, 12) R * R L * W roundTo(2, A1 * Math.PI - A2)

A \green{L \times W} rectangle sits inside a circle with radius of \blue{R}.

What is the area of the shaded region?

Round to the nearest hundredth.

var d = R + 1; init({ range: [[-d, d], [-d, d]], scale: 15 }); ellipse([0, 0], [R, R], { stroke: BLUE, fill: BLUE, "fill-opacity": 0.5 }); rect(-L / 2, - W / 2, L, W, { stroke: GREEN, fill: '#fdfdfd' }); label([0, W / 2], L, "above"); label([-L / 2, 0], W, "left"); path([[0, 0], [R, 0]], { stroke: BLUE }); label([L / 4 + R / 2, 0], R, "above");
A1 * Math.PI - A2

First, calculate the area of the whole figure, including the unshaded area.

ellipse([0, 0], [R, R], { stroke: BLUE, fill: BLUE, opacity: 0.5 });

The area of a circle is \pi r^2.
\qquad \blue{\pi \times R \times R = A1\pi}

Next, calculate the area of the inner figure.

rect(-L / 2, - W / 2, L, W, { stroke: GREEN, fill: '#fdfdfd' });

The area of a rectangle is the length times the width.
\qquad \green{L \times W = A2}

Finally, subtract the area of the inner rectangle from the area of the outer circle.

\qquad \blue{A1\pi} - \green{A2} \approx A