A sphere is a three-dimensional object that is perfectly round, just like a ball. It is defined as the set of all points in space that are at a given distance (radius) from a given point (center). The distance from the center to any point on the sphere is constant.

The following are the key formulas for a sphere:

**Surface Area:**The surface area (A) of a sphere with radius (r) is given by the formula: A = 4πr^{2}**Volume:**The volume (V) of a sphere with radius (r) is given by the formula: V = (4/3)πr^{3}

The surface area of a sphere is the total area on the outside of the sphere. To find the surface area of a sphere, you can use the formula A = 4πr^{2}, where r is the radius of the sphere.

The volume of a sphere is the amount of space inside the sphere. To find the volume of a sphere, you can use the formula V = (4/3)πr^{3}, where r is the radius of the sphere.

- Find the surface area of a sphere with radius 5 units.
- Find the volume of a sphere with radius 3 units.
- A sphere has a surface area of 154π square units. Find its radius.
- A sphere has a volume of 36π cubic units. Find its radius.

Make sure to practice solving problems using the formulas for surface area and volume of a sphere to solidify your understanding of this topic.

Spheres are commonly found in real-world objects such as balls, planets, and bubbles. Understanding the properties of spheres and how to calculate their surface area and volume can be useful in fields such as architecture, engineering, and physics.

Remember to use the formulas and practice problems to become comfortable with working with spheres. Good luck!

.Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.