This calculator solves a triangle using the law of cosines and the law of sines, the two rules that let you find unknown sides and angles in any triangle, not just right-angled ones. While basic trigonometry handles right triangles, real problems in surveying, navigation, engineering and physics involve triangles of every shape, and that is where these two laws come in. The law of cosines relates the three sides and one angle, and is the go-to when you know two sides and the angle between them, or all three sides. The law of sines relates each side to the sine of its opposite angle, and finishes off the remaining angles once you have a side-angle pair. This tool takes the common and reliable case where you know two sides and the angle between them, often called side-angle-side. You enter the two sides and the included angle in degrees, and the calculator uses the law of cosines to find the third side, then the law of sines to find the other two angles, and also returns the area of the triangle. The results update as you type, so you can explore how changing the angle or a side reshapes the triangle. Use it for geometry and trigonometry homework, for practical layout and surveying problems, or to check a hand calculation. The side-angle-side case always gives a single valid triangle, avoiding the ambiguous situations that can arise from other combinations. The calculations are exact for your inputs, with sides, angles and area rounded for display.
Uses side-angle-side: two sides and the angle between them. Angle A is opposite side a. Rounded for display.
The law of cosines finds the third side: a squared equals b squared plus c squared minus twice bc times the cosine of the included angle A. The law of sines then finds angle B from the ratio of side b to side a and the sine of A, and angle C is what remains to make 180 degrees. The area is half of bc times the sine of A.
With sides b = 5 and c = 7 and an included angle A of 60 degrees, the third side a is the square root of 25 plus 49 minus 2 times 35 times cosine 60, which is about 6.24. The law of sines then gives angle B about 43.9 degrees and angle C about 76.1 degrees, and the area is half of 5 times 7 times sine 60, about 15.16.
If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.
Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.
All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.
Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.
Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.
Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.
All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.
© 2019 to 2026 Calculate.co.nz. All rights reserved.