# COMBINING SHAPES

Date of the Problem
May 01, 2023

The radius of circle N is 3 inches, and the radius of circle O is 4 inches. What is the radius of circle P whose area is the sum of the areas of circles N and O?

The formula for the area of a circle is A = πr2. So, the area of circle N is 32π = 9π in2, and the area of circle O is 42π = 16π in2. The area of circle P is 9π + 16π = 25π in2. So, for circle P, we have πr2 = 25π → r2 = 25 → r = 5 inches.

A small square has side length 7 cm, and a medium square has side length 24 cm. What is the side length of a large square whose area is the sum of the areas of the small and medium squares?

The formula for the area of a square is A = s2.  So, the area of the small square is 72 = 49 cm2, and the area of the medium square is 242 = 576 cm2.  The area of the large square is 49 + 576 = 625 cm2. So, for the large square, we have s2 = 625 and s = 25 cm.

The side length of a small equilateral triangle is 10 feet, and the side length of a medium equilateral triangle is 24 feet. What is the side length of a large equilateral triangle whose area is the sum of the areas of the small and medium equilateral triangles?

The formula for the area of an equilateral triangle is A = s2√3/4. The area of the small equilateral triangle is 102√3/4 = 100√3/4 = 25√3 ft2, and the area of the medium equilateral triangle is 242√3/4 = 576√3/4 = 144√3 ft2. The area of the large equilateral triangle is 25√3 + 144√3 ft2 = 169√3 ft2. So, for the large equilateral triangle, we have s2√3/4 = 169√3 → s2 = 169 × 4 → s = √(169 × 4) = √169 × √4 = 13 × 2 = 26 cm.

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Math topic
CCSS (Common Core State Standard)
Difficulty