Converting Between Rectangular and Polar Forms of Complex Numbers

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| Attempts: 14 | Questions: 10 | Updated: Jan 20, 2026
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1) Convert 0 − 5i to polar form.

Explanation

r=5, point is on negative imaginary axis → 270°.

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About This Quiz
Converting Between Rectangular and Polar Forms Of Complex Numbers - Quiz

Complex numbers have two powerful forms—and knowing how to switch between them is key! In this quiz, you’ll practice converting rectangular form to polar form and back again. Take this quiz to master flexibility in representing complex numbers.

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2) Convert 1 + i to polar form.

Explanation

r = √(1²+1²)=√2 and θ = tan⁻¹(1/1)=45°.

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3) Convert 2 + 2√3 i to polar form.

Explanation

r = √(2²+(2√3)²)=√(4+12)=4, θ = tan⁻¹(2√3/2)=tan⁻¹(√3)=60°.

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4) Convert −3 + 3i to polar form.

Explanation

r = √((-3)²+3²)=√(18)=3√2; angle is in quadrant II with tan⁻¹(|3/−3|)=45°, so θ=135°.

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5) Convert polar form r = 5, θ = 0° to rectangular form.

Explanation

x=5cos0°=5, y=5sin0°=0.

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6) Convert polar form r = 10, θ = 180° to rectangular form.

Explanation

x=10cos180°=−10, y=10sin180°=0.

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7) Convert polar form r = 4, θ = 90° to rectangular form.

Explanation

x=4cos90°=0, y=4sin90°=4.

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8) Convert −4 − 4i to polar form.

Explanation

r=√(16+16)=4√2; both x and y negative → quadrant III; tan⁻¹(4/4)=45°, so θ=225°.

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9) Convert polar form r = 6, θ = 45° to rectangular form.

Explanation

x=6cos45°=6(√2/2)=3√2; y=6sin45°=3√2.

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10) Convert polar form r = 8, θ = 300° to rectangular form.

Explanation

x=8cos300°=8(½)=4√3, y=8sin300°=8(−½)=−4.

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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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Convert 0 − 5i to polar form.
Convert 1 + i to polar form.
Convert 2 + 2√3 i to polar form.
Convert −3 + 3i to polar form.
Convert polar form r = 5, θ = 0° to rectangular form.
Convert polar form r = 10, θ = 180° to rectangular form.
Convert polar form r = 4, θ = 90° to rectangular form.
Convert −4 − 4i to polar form.
Convert polar form r = 6, θ = 45° to rectangular form.
Convert polar form r = 8, θ = 300° to rectangular form.
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